Lecture Three Notes

Hypothesis Testing on Two
Population Parameters



1. Goals and Objectives


a. You should be able to calculate confidence intervals and perform hypothesis tests for the difference in two population means when population variances are known.
b. You should be able to calculate confidence intervals and perform hypothesis tests for the difference in two population means when population variances are unknown and sample come from large, independent samples.
c. You should be able to calculate confidence intervals and perform hypothesis tests for the difference in two population means when population variances are unknown and samples come from small independent samples where we assume normal population distributions.
d. You should be able to calculate confidence intervals and perform hypothesis tests for the difference in two population means when population variances are unknown and samples come from small independent samples where we assume normal population distributions and constant population variances.
e. You should be able to calculate confidence intervals and perform hypothesis tests for the difference in two population proportions.
f.  You should be able to calculate confidence intervals and perform hypothesis tests for the difference in two population means for two dependent (or matched pair) samples



2. The sampling distribution of the difference between two population means

 

The expected value of the difference between two sample means  = Ux1 - Ux2, but the standard deviation of the sampling distribution takes on different shapes depending on how the sample is collected.

 

When we analyze the difference between two sample means we must construct the sampling distribution of the difference in those means. We know the mean of the sampling distributions (Ux1 - Ux2), but the standard deviation of the distribution will depend on whether the samples are independent, are large or small, and come from a normally distributed population with equal variances.  In each case we must have an understanding of the standard deviation of the sampling distribution.

 

 

You can have a hypothesized distribution that is not zero when testing the difference between two means.

 


3. A Flow Chart of Possible Sampling Distributions










 

 

 






4. Sampling distribution of the difference between two means. Population variance known with  independent samples




Standard deviation of sampling distribution

 

 


Z =

 

 

 

5.  Sampling distribution of the difference between two means. Population variance unknown.  Independent samples from normal distributions



Standard deviation of sampling distribution:







t = 
 

 

 



df =

 

 

 




6.  Sampling distribution of the difference between two means. Population variance unknown.  Independent samples from populations with normal distributions and equal variances



Standard deviation of sampling distribution:







t =




 

 


df = 






 





7.  Sampling distribution of the difference between two means. Population variance unknown.  Dependent samples from normal distributions. Matched pairs



 

 

t =






df = 










8.  Difference between two sample proportions.  You must have large samples!  Valid to test only with a hypothesized difference of zero.


 

 

Z =

 




9. General formulae for confidence intervals.














10. Examples






a. Students from the economics department are surveying residents of Weber County on how much of their taxes they would be willing to go towards the construction of a gondola from WSU to downtown Ogden. They surveyed 10 citizens from Ogden and 15 citizens from Weber County, but outside of the Ogden City limits. They gathered the following data. Assume the population variances of willingness to pay are equal.



Test if their is a difference in the amount city and county residents are willing to pay. Use 95% confidence.



Construct a 90% confidence interval for the difference in the amount city and county residents are willing to pay.



                      City Residents                 County Residents 

mean                       19.45                             7.80

standard 
deviation                 12.00                             8.00





b. The Provost's office is trying to determine if the general education requirement at WSU is helping students become better writers. The university tests 10 students when they enroll at WSU and again when the students have completed their general education program. Each student is given a 1-25 score judging the quality of their writing. Test the Provost's belief that the general education program at WSU helps students learn how to write. Report the p-value.

Student     Pre-test Score     Post Gen Ed Score

1                 9                             13
2                 6                             15
3                 13                            10
4                 8                              16
5                 4                              7
6                 20                             20
7                 17                             23
8                 14                             19 
9                 9                               14
10             10                               11





c. The Dean of the GSBE believes that WSU business graduates earn at least $5,000 more in salary after graduating from WSU than do graduates of USU. A sample of WSU and USU graduates yields the following data:

School         Mean         Standard Deviation     sample size

USU             $34,000             $ 8,000                 100

WSU             $40000             $12,000                 80


Test the Dean's hypothesis and report the p-value.






d. A national poll indicated that of 300 registered republicans, 135 approved of the president's job performance, and of 200 registered democrats, 60 approved of his performance. Find a 96% confidence interval for the difference in the proportion of democrats and republicans that approve of the president's job performance.