How many colleges did you apply to?
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How many colleges did you apply to?
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10-2Estimating a Population Mean(σ Unknown)
Confidence Intervals in the Calculator
• High School students who take the SAT Mathematics exam a second time generally score higher than on their first try. The change in the score has a Normal distribution with standard deviation σ=50. A random sample of 250 students gain on average x-bar=22 points on their second try.
• Construct a 95% Confidence interval for μ
Confidence Intervals Involving ZUsing the Calculator
What if we don’t know In common practice, we would never know the
population standard deviation. • Instead, we would use an estimate of : the
sample standard deviation, s. • We then estimate the standard deviation of
using • This is called the standard error of the sample
mean
“Standard error”: You are estimating the standard deviation…but there will likely be
some error involved because we are estimating it from sample data.
In other words… the standard error is (most likely) an inaccurate estimate of a (population)
standard deviation.
The t distributions
When we substitute the standard error of ()for its standard deviation () we get the distribution of
the resulting statistic, t.
We call it the t distribution.
The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland ("Student" was his pen
name). Gosset devised the t-test as a way to cheaply monitor the quality of stout.
The t distributionsThere is a different t-distribution for each sample
size n.
We specify a t distribution by giving its degrees of freedom, which is equal to n-1
We will write the t distribution with k degrees of freedom as t(k) for short.
We also will refer to the standard Normal distribution as the z-distribution.
Comparing t and z distributions
Compare the shape, center, and spread of the t-distribution with
the z-distribution.
As the degrees of freedom k increase, (the sample size increases), the t-distribution is
increasingly Normal.
Our formula is the same as it was for z-intervals EXCEPT we replace sigma with s!!!
Finding t with Table CSuppose you
want to construct a 95%
confidence interval for the mean μ of a
population based on a SRS of size
n=12. What critical value t
should you use?
Finding t with Table CSuppose you want to construct a 95% confidence interval for the mean μ of a
population based on a SRS of size n=12. What critical value t should you use?
Finding t with Table C
Suppose you want to construct a 90% confidence interval for the mean μ of a population based on a
SRS of size n=15. What critical value t should you use?
Finding t with Table CSuppose you want to construct a 99% confidence interval for the mean μ of a population based on a SRS of size n=34. What critical value t should you
use?
Suppose you want to construct a 80% confidence interval for the mean μ of a population based on a
SRS of size n=95. What critical value t should you use?
a) 1.290b) .846c) 1.292c) .845
One sample t interval for 1)SRS2) Normality
- n < 15 : Use t procedures if data are close to Normal with no outliers
- n ≥ 15 : Use t procedures except in cases of outliers of strong skew
- n ≥ 30 : Use t-procedures even for clearly skewed distributions (cannot have extreme
outliers)3) Independence
One sample t interval for
Let’s use our class data to construct a 95% confidence interval for the true mean number of colleges that high school seniors applied to in
2013.
One sample t interval for mu
Step 1: STATEStep 2: PLAN
Step 3: CALCULATIONSStep 4: INTERPERATION
State: We are estimating ________, the true mean
______________________________________________________________.
Plan:Procedure:Conditions: 1)
2)
3)
Calculations:
Interpretation: We are 95% confident that the true mean
“Last year, 750,000 applicants submitted 3 million applications, an average of four per student”
College Decision Day: More Applications, More Problems|TIME.com
http://nation.time.com/2013/05/01/as-college-applications-rise-so-does-indecision/#ixzz2sr0ANbp4
Which of the following changes will make a t-distribution more Normal?
a) Decrease b) Increase the Confidence Levelc) Decrease the margin of error. d) Increase
Paired t-proceduresTo compare the responses of the two treatments in a
matched pairs design or before and after measurements on the same subjects, apply the one sample t procedures to the differences observed between the pairs.
• µ = the mean difference between each pair
Ex) Mrs. Skaff gave a new study tool to her students to see if it would improve their test scores. She matched students based on current grade and randomly gave one student in each pair the study tool.
Paired t-procedures• µ = the mean difference between each pair Ex) Mrs. Skaff gave a new study tool to her students to see if it would improve their test
scores. She matched students based on current grade and randomly gave one student in each pair the study tool. She wants to know if the study tool improved test scores.
92 73 81 89 95 90 96 72 85 8890 73 84 84 88 91 93 70 80 882 0 -3 5 7 -1 3 2 5 0
Given Study ToolNo Study ToolStudy– No Study
Confidence Intervals in the Calculator
You still need all other steps!!!!
Ronald McDonald’s sister Diana Rhea is the purchasing manager for domestic hamburger
outlets. The company has decided to provide a free package of Tums to any complaining
customer. In order to estimate monthly demand, she took a sample of 5 outlets and found the number of Tums distributed to customers in a
month was250, 280, 220, 280, 320
(a)Find the sample mean and sample standard deviation
(b)Construct a 90% confidence interval on the average monthly demand per outlet.
Homework!