HS 1678: Comparing Two Means1 Two Independent Means Unit 8.
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Transcript of HS 1678: Comparing Two Means1 Two Independent Means Unit 8.
HS 167 8: Comparing Two Means 2
Sampling ConsiderationsOne sample or two?If two samples, paired or independent?Is the response variable quantitative or categorical?Am I interested in the mean difference?
This chapter → two independent samples → quantitative response → interest in mean difference
HS 167 8: Comparing Two Means 3
One sample
Comparisons made to an external reference population
SRS from one population
HS 167 8: Comparing Two Means 4
Paired Sample
Just like a one-sample problem except inferences directed toward within-pair differences DELTA
Two samples with each observation in sample 1 matched to a unique observation in sample 2
HS 167 8: Comparing Two Means 5
Independent sample inference
No matching or pairing
Independent samples from two populations
HS 167 8: Comparing Two Means 6
What type of sampling method?
1. Measure vitamin content in loaves of bread and see if the average meets national standards.
2. Compare vitamin content of bread immediately after baking versus 3 days later (same loaves are used on day one and 3 days later)
3. Compare vitamin content of bread immediately after baking versus loaves that have been on shelf for 3 days
1 = single sample2 = paired samples3 = independent samples
HS 167 8: Comparing Two Means 7
Illustrative example: independent samples
Fasting cholesterol (mg/dl) Group 1 (type A personality): 233, 291, 312, 250, 246, 197, 268, 224, 239, 239, 254, 276, 234, 181, 248, 252, 202, 218, 212, 325
Group 2 (type B personality) 344, 185, 263, 246, 224, 212, 188, 250, 148, 169, 226, 175, 242, 252, 153, 183, 137, 202, 194, 213
Goal: compare response variable in two groups
HS 167 8: Comparing Two Means 8
Data setup for independent samples
Two columns Response variable in one columnExplanatory variable in other column
HS 167 8: Comparing Two Means 9
Side-by-side boxplots
2020N =
GROUP
21
Ch
ole
ste
rol (
mg
/dl)
400
300
200
100
21
20
Interpretation:
(1) Different locations (group 1 > group 2)
(2) Different spreads (group 1 < group 2)
(3) Shape: fairly symmetrical (but both with outside values)
Compare locations, spreads, and shapes
HS 167 8: Comparing Two Means 10
Summary statistics by group
Group
n mean std dev
1 20 245.05 36.64
2 20 210.30 48.34Take time to look at your results.
If no major departures from Normality, report means and standard deviations (and sample sizes)
HS 167 8: Comparing Two Means 11
Notation for independent samples
Parameters (population)
Group 1 N1 µ1 σ1
Group 2 N2 µ2 σ2
Statistics (sample)
Group 1 n1 s1
Group 2 n2 s2
2121
21
estimates
differencemean sample theis
xx
xx
1x
2x
HS 167 8: Comparing Two Means 12
Sampling distribution of mean difference
),(~212121 xxSENxx
The sampling distribution of the mean difference is key to inference
{FIGURE DRAWN ON BOARD}
The SDM difference tends to be Normal with expectation μ1 − μ2 and standard deviation SE; (SE discussed next
slide)
HS 167 8: Comparing Two Means 13
Pooled Standard Error
381919
191201
191201
21
22
11
dfdfdf
ndf
ndf
Illustrative data (summary statistics)
Group ni si xbari
1 20 36.64 245.05
2 20 48.34 210.30
623.183938
)34.48)(19()64.36)(19(
))(())((
22
222
2112
df
sdfsdfspooled
56.1320
1
20
11839.623
11
21
2
21
nn
sSE pooledxx
HS 167 8: Comparing Two Means 14
Confidence interval for µ1 – µ2
)( 2121,21 xxdf SEtxx
62.14) (7.36,
39.2775.34
)13.56)(02.2()30.21005.245(
))(()(21975,.121
xxn SEtxx
Illustrative example (Cholesterol in type A and B men)
(1−αlpha)100% confidence interval for µ1 – µ2
HS 167 8: Comparing Two Means 15
Comparison of CI formulas
)*)((estimate)(point SEt
Type of sample
point estimat
e
df for t* SE
single
paired
independent
21 xx
dx
xn
1n
1dn
)1()1( 21 nn
ndelta
11
21
2
nn
spooled
HS 167 8: Comparing Two Means 16
Independent t testA. H0: µ1 = µ2
vs. H1: µ1 > µ2 or H1: µ1 < µ2 or H1: µ1 µ2
B. Independent t statistic
C. P-value – use t table or software utility to convert tstat to P- value
D. Significance level
)2()1(
with
)(
21
21
21stat
21
nn
dfdfdf
SE
xxt
xx
Pooled t statistic
Illustrative example
381919
75.3430.21005.245
13.56
21
21
df
xx
SE xx
0.02 and 0.01between sided-Two
0.005 and 0.01between sided-One
2.56 56.13
75.34
21
21stat
P
P
SE
xxt
xx
HS 167 8: Comparing Two Means 17
SPSS output
These are the pooled (equal variance) statistics calculated in HS 167
HS 167 8: Comparing Two Means 18
Conditions necessary for t procedures
Validity assumptions good information (no information bias) good sample (“no selection bias”) good comparison (“no confounding” – no
lurking variables)
Distributional assumptions Sampling independence Normality Equal variance
HS 167 8: Comparing Two Means 19
Sample size requirements for confidence intervals
296.1
d
n
This will restrict the margin of error to no bigger than plus or minus d
HS 167 8: Comparing Two Means 20
Sample size requirement for CI
Suppose, you have a variable with = 15
365
154 use ,5For
2
2
nd
1445.2
154 use ,5.2For
2
2
nd
9001
154 use ,1For
2
2
ndSample size requirements increases when you need precision
HS 167 8: Comparing Two Means 21
Sample size for significance test
Goal: to conduct a significance test with adequate power to detect “a difference worth detecting”The difference worth detecting is a difference difference worth finding.
In a study of an anti-hypertensivesfor instance, a drop of 10 mm Hg might be worth detecting, while a drop of 1 mm Hg might not be worth detecting.
In a study on weight loss, a drop of 5 pounds might be meaningful in a population of runway models, but may be meaningless in a morbidly obese population.
HS 167 8: Comparing Two Means 22
Determinants of sample size requirements
“Difference worth detecting” () Standard deviation of data ()Type I error rate ( We consider only two-sided
Power of test (we consider on 80% power)