Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute...
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![Page 1: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/1.jpg)
Last Time
• Interpretation of Confidence Intervals
• Handling unknown μ and σ
• T Distribution
• Compute with TDIST & TINV
(Recall different organization)
(relative to NORMDIST & NORMINV)
![Page 2: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/2.jpg)
Reading In Textbook
Approximate Reading for Today’s Material:
Pages 420-427, 86-94
Approximate Reading for Next Class:
Pages 101-105 , 447-465, 511-516
![Page 3: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/3.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
Usually σ is unknown, so replace with an estimate, s.
For n large, should be “OK”, but what about:
• n small?
• How large is n “large”?
nNX /,0~
![Page 4: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/4.jpg)
Unknown SD
Then
So can write:
Replace by , then
has a distribution named:
“t-distribution with n-1 degrees of freedom”
nNX /,~
1,0~ N
n
X
sn
sX
![Page 5: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/5.jpg)
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST & TINV
Caution: these are set up differently from NORMDIST & NORMINV
![Page 6: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/6.jpg)
EXCEL Functions
Summary:
Normal:
plug in: get out:
NORMDIST: cutoff area
NORMINV: area cutoff
(but TDIST is set up really differently)
![Page 7: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/7.jpg)
EXCEL Functions
t distribution:
Area
2 tail:
plug in: get out:
TDIST: cutoff area
TINV: area cutoff
(EXCEL note: this one has the inverse)
![Page 8: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/8.jpg)
t - Distribution
Application 1: Confidence Intervals
![Page 9: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/9.jpg)
t - Distribution
Application 1: Confidence Intervals
Recall: mX
![Page 10: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/10.jpg)
t - Distribution
Application 1: Confidence Intervals
Recall:
margin of error
mX
![Page 11: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/11.jpg)
t - Distribution
Application 1: Confidence Intervals
Recall:
margin of error
from NORMINV
mX
![Page 12: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/12.jpg)
t - Distribution
Application 1: Confidence Intervals
Recall:
margin of error
from NORMINV
or CONFIDENCE
mX
![Page 13: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/13.jpg)
t - Distribution
Application 1: Confidence Intervals
Recall:
margin of error
from NORMINV
or CONFIDENCE
Using TINV?
mX
![Page 14: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/14.jpg)
t - Distribution
Application 1: Confidence Intervals
Recall:
margin of error
from NORMINV
or CONFIDENCE
Using TINV? Careful need to standardize
mX
![Page 15: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/15.jpg)
t - DistributionUsing TINV? Careful need to standardize
![Page 16: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/16.jpg)
t - DistributionUsing TINV? Careful need to standardize
mXmXbyveredcoP ,95.0
![Page 17: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/17.jpg)
t - DistributionUsing TINV? Careful need to standardize
mXmXbyveredcoP ,95.0
mXmXP
![Page 18: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/18.jpg)
t - DistributionUsing TINV? Careful need to standardize
# spaces on number line
mXmXbyveredcoP ,95.0
mXmXP
mXP
![Page 19: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/19.jpg)
t - DistributionUsing TINV? Careful need to standardize
# spaces on number line
Need to work into use TINV
mXmXbyveredcoP ,95.0
mXmXP
mXP ns
![Page 20: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/20.jpg)
t - DistributionUsing TINV? Careful need to standardize
# spaces on number line
Need to work into use TINV
mXmXbyveredcoP ,95.0
mXmXP
mXP
ns
mns
XP
ns
![Page 21: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/21.jpg)
t - Distribution
ns
mns
XP
95.0
![Page 22: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/22.jpg)
t - Distribution
distribution
ns
mns
XP
95.0
nsX
![Page 23: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/23.jpg)
t - Distribution
distribution
ns
mns
XP
95.0
nsm
nsX
![Page 24: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/24.jpg)
t - Distribution
distribution
So want:
ns
mns
XP
95.0
nsm
nTINV )1,05.0( nsm
nsX
![Page 25: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/25.jpg)
t - Distribution
distribution
So want:
i.e. want:
ns
mns
XP
95.0
nsm
nTINV )1,05.0(
ns
nTINVm )1,05.0(
nsm
nsX
![Page 26: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/26.jpg)
t - Distribution
Class Example 15, Part Ihttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Old text book problem 7.24
![Page 27: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/27.jpg)
t - Distribution
Class Example 15, Part Ihttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Old text book problem 7.24:
In a study of DDT poisoning, researchers fed several rats a measured amount. They measured the “absolutely refractory period” required for a nerve to recover after a stimulus. Measurements on 4 rats gave:
![Page 28: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/28.jpg)
t - Distribution
Class Example 15, Part Ihttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Old text book problem 7.24:
Measurements on 4 rats gave:
1.6 1.7 1.8 1.9
a) Find the mean refractory period, and the standard error of the mean
b) Give a 95% CI for the mean “absolutely refractory period” for all rats of this strain
![Page 29: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/29.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
![Page 30: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/30.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Note: small sample size (n = 4)
![Page 31: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/31.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Note: small sample size (n = 4),
population sd, σ, unknown
![Page 32: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/32.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Note: small sample size (n = 4),
population sd, σ, unknown,
so use sample sd, s
![Page 33: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/33.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Note: small sample size (n = 4),
population sd, σ, unknown,
so use sample sd, s,
and t distribution
![Page 34: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/34.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Center CI at Sample Mean
![Page 35: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/35.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Center CI at Sample Mean
Measure Sample Spread by S. D.
![Page 36: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/36.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Center CI at Sample Mean
Measure Sample Spread by S. D.
Divide by to get Standard Errorn
![Page 37: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/37.jpg)
t - Distribution
Class Example 15, Part I
Data in cells B9:E9
Center CI at Sample Mean
Measure Sample Spread by S. D.
Divide by to get Standard Error
Which
answers (a)
n
![Page 38: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/38.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
![Page 39: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/39.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
![Page 40: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/40.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compute using TINV
![Page 41: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/41.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Recall:
d.f. = n – 1
= 4 – 1
![Page 42: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/42.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compare to old Normal CIs
![Page 43: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/43.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compare to old Normal CIs
Compute using
CONFIDENCE
![Page 44: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/44.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compare to old Normal CIs
T CIs are wider
![Page 45: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/45.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compare to old Normal CIs
T CIs are wider
(as expected)
![Page 46: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/46.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compare to old Normal CIs
Left End
![Page 47: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/47.jpg)
t - Distribution
Class Example 15, Part I (b) 95% CI for μ
Data in cells B9:E9
CI Radius = Margin of Error
Compare to old Normal CIs
Left End
Right End
![Page 48: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/48.jpg)
t - Distribution
Confidence Interval HW:
7.24 (a. Q-Q roughly linear, so OK, b. 43.17, 4.41, 0.987 c. [41.1, 45.2])
7.25
![Page 49: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/49.jpg)
And now for something completely different
An extreme “sport” video:
![Page 50: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/50.jpg)
t - Distribution
Application 2: Hypothesis Tests
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t - Distribution
Application 2: Hypothesis Tests
Idea: Calculate P-values using TDIST
![Page 52: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/52.jpg)
t - Distribution
Application 2: Hypothesis Tests
Idea: Calculate P-values using TDIST
![Page 53: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/53.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example
![Page 54: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/54.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example
Recall Data in cells B9:E9
![Page 55: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/55.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example
Recall Data in cells B9:E9
As above: t – distribution appropriate
![Page 56: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/56.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example
Recall Data in cells B9:E9
As above: t – distribution appropriate
(small sample, and using s ≈ σ)
![Page 57: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/57.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3
![Page 58: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/58.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3
(recall observed in data)
![Page 59: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/59.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3. DDT poisoning should slow nerve recovery, and so increase this period.
![Page 60: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/60.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3. DDT poisoning should slow nerve recovery, and so increase this period. Do the data give good evidence for this supposition?
![Page 61: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/61.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
Let = population mean absolutely
refractory period for poisoned rats.
![Page 62: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/62.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
Let = population mean absolutely
refractory period for poisoned rats.
(checking strong evidence for
this)
3.1:0 H
3.1: AH
![Page 63: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/63.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26
Let = population mean absolutely
refractory period for poisoned rats.
(from before)
3.1:0 H
3.1: AH
75.1X
![Page 64: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/64.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
![Page 65: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/65.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
3.1|75.1 XP
![Page 66: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/66.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
3.1|75.1 XP
3.1|
3.175.1 nsns
XP
![Page 67: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/67.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
3.1|75.1 XP
3.1|
3.175.1 nsns
XP
ns
tP3.175.1
3
![Page 68: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/68.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
ns
tP3.175.1
3
![Page 69: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/69.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
Now use TDIST
ns
tP3.175.1
3
![Page 70: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/70.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
Now use TDIST
ns
tP3.175.1
3
![Page 71: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/71.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
Now use TDIST
ns
tP3.175.1
3
![Page 72: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/72.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
Now use TDIST
ns
tP3.175.1
3
![Page 73: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/73.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
Now use TDIST
Degrees of Freedom = n – 1 = 4 - 1
ns
tP3.175.1
3
![Page 74: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/74.jpg)
t – Distribution Hypo Testing
E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}
Now use TDIST
Tails
ns
tP3.175.1
3
![Page 75: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/75.jpg)
t – Distribution Hypo TestingE.g. Old Textbook Example 7.26
From Class Example 27, part 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
P - value = 0.003
![Page 76: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/76.jpg)
t – Distribution Hypo TestingE.g. Old Textbook Example 7.26
From Class Example 27, part 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
P - value = 0.003
Interpretation: very strong evidence, for either yes-no or gray-level
![Page 77: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/77.jpg)
t – Distribution Hypo TestingVariations:
• For “opposite direction” hypotheses:
:AH
![Page 78: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/78.jpg)
t – Distribution Hypo TestingVariations:
• For “opposite direction” hypotheses:
P-value =
:AH
tP
![Page 79: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/79.jpg)
t – Distribution Hypo TestingVariations:
• For “opposite direction” hypotheses:
P-value =
:AH
tP
![Page 80: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/80.jpg)
t – Distribution Hypo TestingVariations:
• For “opposite direction” hypotheses:
P-value =
[wrong way for TDIST(…,1)]
:AH
tP
![Page 81: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/81.jpg)
t – Distribution Hypo TestingVariations:
• For “opposite direction” hypotheses:
P-value =
Then use symmetry
:AH
tP
![Page 82: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/82.jpg)
t – Distribution Hypo TestingVariations:
• For “opposite direction” hypotheses:
P-value =
Then use symmetry, i.e. put - into TDIST.
:AH
tP
![Page 83: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/83.jpg)
t – Distribution Hypo TestingVariations:
• For 2-sided hypotheses
![Page 84: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/84.jpg)
t – Distribution Hypo TestingVariations:
• For 2-sided hypotheses:
H0: μ =
H1: μ ≠
![Page 85: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/85.jpg)
t – Distribution Hypo TestingVariations:
• For 2-sided hypotheses:
H0: μ =
H1: μ ≠
![Page 86: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/86.jpg)
t – Distribution Hypo TestingVariations:
• For 2-sided hypotheses:
H0: μ =
H1: μ ≠
Use 2-tailed version of TDIST
![Page 87: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/87.jpg)
t – Distribution Hypo TestingVariations:
• For 2-sided hypotheses:
H0: μ =
H1: μ ≠
Use 2-tailed version of TDIST,
i.e. TDIST(…,2)
![Page 88: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/88.jpg)
t – Distribution Hypo Testing
HW: Interpret P-values:
(i) yes-no
(ii) gray-level
7.21e ((i)significant, (ii) significant, but not
very strongly so)
7.22e (0.0619, (i) not significant (ii) not sig.,
but nearly significant)
![Page 89: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/89.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
![Page 90: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/90.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
From Matthew Campbell
UNC Master’s Student
In Geological Sciences
![Page 91: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/91.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
![Page 92: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/92.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
(fossil beds up and down Eastern Seaboard)
![Page 93: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/93.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
• Dated (by fossil bed)
![Page 94: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/94.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
• Dated (by fossil bed)
• Biologically categorized
![Page 95: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/95.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
• Dated (by fossil bed)
• Biologically categorized
(family – genus – species, etc.)
![Page 96: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/96.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
• Dated (by fossil bed)
• Biologically categorized
Goal: study extinctions over long periods
![Page 97: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/97.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Data points:
• Fossilized shells
• Dated (by fossil bed)
• Biologically categorized
Goal: study extinctions over long periods
(via last time saw each)
![Page 98: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/98.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Oversmoothed:
nothing interesting
![Page 99: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/99.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Undersmoothed:
many bumps appear
![Page 100: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/100.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Undersmoothed:
many bumps appear
but not statistically significant
![Page 101: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/101.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Intermediate Smoothing
two bumps appear
![Page 102: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/102.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Intermediate Smoothing
two bumps appear
SiZer result: not statistically significant
![Page 103: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/103.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Matthew’s Comment:
Whoah, those are times
of mass extinctions
![Page 104: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/104.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Matthew’s Comment:
Whoah, those are times
of mass extinctions
Any way to show these are “really there”?
![Page 105: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/105.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Any way to show these
are “really there”?
![Page 106: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/106.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Any way to show these
are “really there”?
Standard Answer:
Get more data
![Page 107: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/107.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Any way to show these
are “really there”?
Challenge:
Took 100s of year to get these!
![Page 108: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/108.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Any way to show these
are “really there”?
Alternate Approach:
Refined from Genus level to Species
![Page 109: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/109.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Species level result
![Page 110: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/110.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Species level result:
Now both bumps are
significant
![Page 111: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/111.jpg)
Research Corner
Another SiZer analysis:
Mollusk Extinction Data
Species level result:
Now both bumps are
significant
Consistent with Global Climactic Events
![Page 112: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/112.jpg)
Variable Relationships
Chapter 2 in Text
![Page 113: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/113.jpg)
Variable Relationships
Chapter 2 in Text
Idea: Look beyond single quantities
![Page 114: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/114.jpg)
Variable Relationships
Chapter 2 in Text
Idea: Look beyond single quantities, to how quantities relate to each other.
![Page 115: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/115.jpg)
Variable Relationships
Chapter 2 in Text
Idea: Look beyond single quantities, to how quantities relate to each other.
E.g. How do HW scores “relate”
to Exam scores?
![Page 116: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/116.jpg)
Variable Relationships
Chapter 2 in Text
Idea: Look beyond single quantities, to how quantities relate to each other.
E.g. How do HW scores “relate”
to Exam scores?
Section 2.1: Useful graphical device:
Scatterplot
![Page 117: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/117.jpg)
Plotting Bivariate Data
Toy Example: Ordered pairs
(1,2)
(3,1)
(-1,0)
(2,-1)
![Page 118: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/118.jpg)
Plotting Bivariate Data
Toy Example: Ordered pairs
Captures relationship between X & Y
(1,2) as (X,Y)
(3,1)
(-1,0)
(2,-1)
![Page 119: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/119.jpg)
Plotting Bivariate Data
Toy Example: Ordered pairs
Captures relationship between X & Y
(1,2) as (X,Y)
(3,1) e.g. (height, weight)
(-1,0)
(2,-1)
![Page 120: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/120.jpg)
Plotting Bivariate Data
Toy Example: Ordered pairs
Captures relationship between X & Y
(1,2) as (X,Y)
(3,1) e.g. (height, weight)
(-1,0) e.g. (MT Score, Final Exam Score)
(2,-1)
![Page 121: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/121.jpg)
Plotting Bivariate Data
Toy Example:
(1,2) Think in terms of:
(3,1)
(-1,0) X coordinates
(2,-1)
![Page 122: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/122.jpg)
Plotting Bivariate Data
Toy Example:
(1,2) Think in terms of:
(3,1)
(-1,0) X coordinates
(2,-1) Y coordinates
![Page 123: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/123.jpg)
Plotting Bivariate Data
Toy Example:
(1,2) Think in terms of:
(3,1)
(-1,0) X coordinates
(2,-1) Y coordinates
And plot in x,y plane, to see relationship
![Page 124: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/124.jpg)
Plotting Bivariate Data
Toy Example:
(1,2)
(3,1)
(-1,0)
(2,-1)
Toy Scatterplot, Separate Points
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3 4
x
y
![Page 125: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/125.jpg)
Plotting Bivariate Data
Toy Example:
(1,2)
(3,1)
(-1,0)
(2,-1)
Toy Scatterplot, Separate Points
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3 4
x
y
![Page 126: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/126.jpg)
Plotting Bivariate Data
Toy Example:
(1,2)
(3,1)
(-1,0)
(2,-1)
Toy Scatterplot, Separate Points
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3 4
x
y
![Page 127: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/127.jpg)
Plotting Bivariate Data
Toy Example:
(1,2)
(3,1)
(-1,0)
(2,-1)
Toy Scatterplot, Separate Points
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3 4
x
y
![Page 128: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/128.jpg)
Plotting Bivariate Data
Sometimes:
Can see more
insightful patterns
by connecting
points
Toy Scatterplot, Connected points
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3 4
x
y
![Page 129: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/129.jpg)
Plotting Bivariate Data
Sometimes:
Useful to switch off
points, and only
look at lines/curves
Toy Scatterplot, Lines Only
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3 4
x
y
![Page 130: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/130.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
![Page 131: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/131.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel
![Page 132: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/132.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
![Page 133: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/133.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
Charts
![Page 134: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/134.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
Charts
Scatter Button
![Page 135: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/135.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
Charts
Scatter Button
Choose Dots
![Page 136: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/136.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
Charts
Scatter Button
Choose Dots
(but note other options)
![Page 137: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/137.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
Charts
Scatter Button
Choose Dots
Manipulate plot as done before for bar plots
![Page 138: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/138.jpg)
Plotting Bivariate Data
Common Name: “Scatterplot”
A look under the hood in Excel:
Insert Tab
Charts
Scatter Button
Choose Dots
Manipulate plot as done before for bar plots
(e.g. titles, labels, colors, styles, …)
![Page 139: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/139.jpg)
Scatterplot E.g.Class Example 16:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Data from related Intro. Statistics Class
![Page 140: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/140.jpg)
Scatterplot E.g.Class Example 16:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Data from related Intro. Statistics Class
(actual scores)
![Page 141: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/141.jpg)
Scatterplot E.g.Class Example 16:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Data from related Intro. Statistics Class
(actual scores)
A. How does HW score predict Final Exam?
![Page 142: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/142.jpg)
Scatterplot E.g.Class Example 16:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Data from related Intro. Statistics Class
(actual scores)
A. How does HW score predict Final Exam?
xi = HW, yi = Final Exam
![Page 143: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/143.jpg)
Scatterplot E.g.Class Example 16:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Data from related Intro. Statistics Class
(actual scores)
A. How does HW score predict Final Exam?
xi = HW, yi = Final Exam
(Study Relationship
using scatterplot)
![Page 144: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/144.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
![Page 145: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/145.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
(Scatterplot View)
![Page 146: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/146.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
i. In top half of HW scores
![Page 147: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/147.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
i. In top half of HW scores:Better HW Better Final
![Page 148: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/148.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
i. In top half of HW scores:Better HW Better Final
ii. For lower HW
![Page 149: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/149.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
i. In top half of HW scores:Better HW Better Final
ii. For lower HW:Final is more “random”
![Page 150: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/150.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
![Page 151: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/151.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
Call X the “Explanatory Var.”
![Page 152: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/152.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
Call X the “Explanatory Var.” or “Indep. Var.”
![Page 153: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/153.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
Call X the “Explanatory Var.” or “Indep. Var.”
Call Y the “Response Var.”
![Page 154: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/154.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
Call X the “Explanatory Var.” or “Indep. Var.”
Call Y the “Response Var.” or “Dep. Var.”
![Page 155: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/155.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
Call X the “Explanatory Var.” or “Indep. Var.”
Call Y the “Response Var.” or “Dep. Var.”
(think of “Y as function of X”)
![Page 156: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/156.jpg)
Scatterplots
Common Terminology:
When thinking about “X causes Y”,
Call X the “Explanatory Var.” or “Indep. Var.”
Call Y the “Response Var.” or “Dep. Var.”
(think of “Y as function of X”)
(although not always sensible)
![Page 157: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/157.jpg)
Scatterplots
Note: Sometimes think about causation
![Page 158: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/158.jpg)
Scatterplots
Note: Sometimes think about causation,
Other times: “Explore Relationship”
![Page 159: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/159.jpg)
Scatterplots
Note: Sometimes think about causation,
Other times: “Explore Relationship”
HW: 2.9
![Page 160: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/160.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
![Page 161: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/161.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
(Replace Final above
with 1st Midterm)
![Page 162: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/162.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
![Page 163: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/163.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
i. Better HW better Exam
(general upwards
tendency still
the same)
![Page 164: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/164.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
i. Better HW better Exam
ii. Wider range MT1 scores
(for each range
of HW scores)
(relative to final scores)
![Page 165: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/165.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
i. Better HW better Exam
ii. Wider range MT1 scores
iii. HW doesn’t predict MT1
(as well as HW
predicted the final)
![Page 166: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/166.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
i. Better HW better Exam
ii. Wider range MT1 scores
iii. HW doesn’t predict MT1
iv. “Outliers” in scatterplot
![Page 167: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/167.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
i. Better HW better Exam
ii. Wider range MT1 scores
iii. HW doesn’t predict MT1
iv. “Outliers” in scatterplot
e.g. HW = 72, MT1 = 94
![Page 168: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/168.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
B. How does HW predict Midterm 1?
xi = HW, yi = MT1
i. Better HW better Exam
ii. Wider range MT1 scores
iii. HW doesn’t predict MT1
iv. “Outliers” in scatterplot may not be outliers in either individual variable
e.g. HW = 72, MT1 = 94
(bad HW, but good MT1?, fluke???)
![Page 169: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/169.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
![Page 170: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/170.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
(Different choice of x and y, since
studying different relationship)
![Page 171: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/171.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
(Study Relationship
using tool of
scatterplot)
![Page 172: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/172.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
(don’t expect better MT1
to lead to better MT2)
![Page 173: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/173.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
(again clear overall
upwards trend)
![Page 174: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/174.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
iii. Wider range of MT2
(for each range of MT1)
![Page 175: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/175.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
iii. Wider range of MT2
i.e. “not good predictor”
(MT1) (of MT2)
![Page 176: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/176.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
iii. Wider range of MT2
i.e. “not good predictor”
iv. Interesting Outliers
![Page 177: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/177.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
iii. Wider range of MT2
i.e. “not good predictor”
iv. Interesting Outliers:MT1 = 100, MT2 = 56 (oops!)
![Page 178: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/178.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
iii. Wider range of MT2
i.e. “not good predictor”
iv. Interesting Outliers:MT1 = 100, MT2 = 56
MT1 = 23, MT2 = 74 (woke up!)
![Page 179: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/179.jpg)
Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
C. How does MT1 predict MT2?
xi = MT1, yi = MT2
i. Idea: less “causation”, more “exploration”
ii. High MT1 High MT2
iii. Wider range of MT2
i.e. “not good predictor”
iv. Interesting Outliers:MT1 = 100, MT2 = 56
MT1 = 23, MT2 = 74
MT1 70s, MT2 90s (moved up!)
![Page 180: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/180.jpg)
And now for something completely different
A thought provoking movie clip:
http://www.aclu.org/pizza/
![Page 181: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/181.jpg)
Important Aspects of Relations
I. Form of Relationship
(Linear or not?)
![Page 182: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/182.jpg)
Important Aspects of Relations
I. Form of Relationship
II. Direction of Relationship
(trending up or down?)
![Page 183: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/183.jpg)
Important Aspects of Relations
I. Form of Relationship
II. Direction of Relationship
III. Strength of Relationship
(how much of data “explained”?)
![Page 184: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/184.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
![Page 185: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/185.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
![Page 186: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/186.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
![Page 187: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/187.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final vs. High values of HW is “best”
![Page 188: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/188.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final vs. High values of HW is “best”
But saw others with
“rough linear trend”
![Page 189: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/189.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final vs. High values of HW is “best”
But saw others with
“rough linear trend”
![Page 190: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/190.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final vs. High values of HW is “best”
But saw others with
“rough linear trend”
Interesting question:
Measure strength of
linear trend
![Page 191: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/191.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final vs. High values of HW is “best”
• Nonlinear: Data follows different pattern
(non-linear)
![Page 192: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/192.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final vs. High values of HW is “best”
• Nonlinear: Data follows different pattern
Nice Example: Bralower’s Fossil Data
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
![Page 193: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/193.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
![Page 194: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/194.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
![Page 195: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/195.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate
![Page 196: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/196.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago
![Page 197: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/197.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
• Small shells from ocean floor cores
![Page 198: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/198.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
• Small shells from ocean floor cores
• Ratios of Isotopes of Strontium
![Page 199: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/199.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
• Small shells from ocean floor cores
• Ratios of Isotopes of Strontium
• Reflects Ice Ages, via Sea Level
![Page 200: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/200.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
• Small shells from ocean floor cores
• Ratios of Isotopes of Strontium
• Reflects Ice Ages, via Sea Level
(50 meter difference!)
![Page 201: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/201.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
• Small shells from ocean floor cores
• Ratios of Isotopes of Strontium
• Reflects Ice Ages, via Sea Level
(50 meter difference!)
• As function of time
![Page 202: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/202.jpg)
Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
• Small shells from ocean floor cores
• Ratios of Isotopes of Strontium
• Reflects Ice Ages, via Sea Level
(50 meter difference!)
• As function of time
• Clearly nonlinear relationship
![Page 203: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/203.jpg)
II. Direction of Relationship
• Positive Association
![Page 204: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/204.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
![Page 205: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/205.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
E.g. Class Scores Data above
![Page 206: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/206.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
• Negative Association
![Page 207: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/207.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
• Negative Association
X bigger Y smaller
![Page 208: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/208.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
• Negative Association
X bigger Y smaller
E.g. X = alcohol consumption, Y = Driving Ability
Clear negative association
![Page 209: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/209.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
• Negative Association
X bigger Y smaller
Note: Concept doesn’t always apply:
![Page 210: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/210.jpg)
II. Direction of Relationship
• Positive Association
X bigger Y bigger
• Negative Association
X bigger Y smaller
Note: Concept doesn’t always apply:
Bralower’s Fossil Data
![Page 211: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/211.jpg)
III. Strength of Relationship
Idea: How close are points to lying on a line?
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
![Page 212: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/212.jpg)
III. Strength of Relationship
Idea: How close are points to lying on a line?
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• Final Exam is “closely related to HW”
![Page 213: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/213.jpg)
III. Strength of Relationship
Idea: How close are points to lying on a line?
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• Final Exam is “closely related to HW”
• Midterm 1 less closely related to HW
![Page 214: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/214.jpg)
III. Strength of Relationship
Idea: How close are points to lying on a line?
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• Final Exam is “closely related to HW”
• Midterm 1 less closely related to HW
• Midterm 2 even less related to Midterm 1
![Page 215: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/215.jpg)
III. Strength of Relationship
Idea: How close are points to lying on a line?
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• Final Exam is “closely related to HW”
• Midterm 1 less closely related to HW
• Midterm 2 even less related to Midterm 1
Interesting Issue:
Measure this strength
![Page 216: Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative.](https://reader035.fdocuments.net/reader035/viewer/2022062322/5697bfc41a28abf838ca5e3c/html5/thumbnails/216.jpg)
Linear Relationship HW
HW:
2.11, 2.13, 2.15, 2.17