1. STA 291 Spring 2010
Lecture 21
Dustin Lueker

2. Example
If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 5% level of significance?
Yes No
Maybe
STA 291 Spring 2010 Lecture 21

3. Example
If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 1% level of significance?
Yes No
Maybe
STA 291 Spring 2010 Lecture 21

4. Significance Test for a Proportion
Same process as with population mean
Value we are testing against is called p0
Test statistic
P-value
Calculation is exactly the same as for the test for a mean
Sample size restrictions:
STA 291 Spring 2010 Lecture 21

5. Testing Difference Between Two Population Proportions
Similar to testing one proportion
Hypotheses are set up like two sample mean test
H0:p1-p2=0
Same as H0: p1=p2
Test Statistic
STA 291 Spring 2010 Lecture 21

6. Testing the Difference Between Means from Different Populations
Hypothesis involves 2 parameters from 2 populations
Test statistic is different
Involves 2 large samples (both samples at least 30)
One from each population
H0: μ1-μ2=0
Same as H0: μ1=μ2
Test statistic
STA 291 Spring 2010 Lecture 21

7. Example
In the 1982 General Social Survey, 350 subjects reported the time spent every day watching television. The sample yielded a mean of 4.1 and a standard deviation of 3.3.
In the 1994 survey, 1965 subjects yielded a sample mean of 2.8 hours with a standard deviation of 2.
Set up hypotheses of a significance test to analyze whether the population means differ in 1982 and and test at α=.05 using the p-value method.
STA 291 Spring 2010 Lecture 21

8. Correspondence Between Confidence Intervals and Tests
Constructing a confidence interval to do a hypothesis test with 2 samples works the same as it did when we were dealing with 1 sample
The confidence interval shows plausible values for the difference between the two means
STA 291 Spring 2010 Lecture 21

9. Small Sample Tests for Two Means
Used when comparing means of two samples where at least one of them is less than 30
Normal population distribution is assumed for both samples
Equal Variances
Both groups have the same variability
Unequal Variances
Both groups may not have the same variability
STA 291 Spring 2010 Lecture 21

10. Small Sample Test for Two Means, Equal Variances
Test Statistic
Degrees of freedom
n1+n2-2
STA 291 Spring 2010 Lecture 21

11. Small Sample Confidence Interval for Two Means, Equal Variances
Degrees of freedom
n1+n2-2
STA 291 Spring 2010 Lecture 21

12. Small Sample Test for Two Means, Unequal Variances
Test statistic
Degrees of freedom
STA 291 Spring 2010 Lecture 21

13. Small Sample Confidence Interval for Two Means, Unequal Variances
STA 291 Spring 2010 Lecture 21

14. Method 1 (Equal Variances) vs. Method 2 (Unequal Variances)
How to choose between Method 1 and Method 2?
Method 2 is always safer to use
Definitely use Method 2
If one standard deviation is at least twice the other
If the standard deviation is larger for the sample with the smaller sample size
Usually, both methods yield similar conclusions
STA 291 Spring 2010 Lecture 21