1. STA 291 Spring 2008
Lecture 19
Dustin Lueker

2. Example
The p-value for testing H1: µ≠100 is p= This indicates that…
There is strong evidence that μ=100
There is strong evidence that μ≠100
There is strong evidence that μ>100
There is strong evidence that μ<100
STA 291 Spring 2008 Lecture 19

3. Example
The p-value for testing H1: µ≠100 is p=.001. In addition you know that the test statistic was z= This indicates that…
There is strong evidence that μ=100
There is strong evidence that μ>100
There is strong evidence that μ<100
STA 291 Spring 2008 Lecture 19

4. Rejection Region
Range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothesis
Type of test determines which tail(s) the rejection region is in
Left-tailed
Right-tailed
Two-tailed
STA 291 Spring 2008 Lecture 19

5. Test Statistic Testing µ with σ unknown and n large
Just like finding a confidence interval for µ with σ unknown and n large
Reasons for choosing test statistics are the same as choosing the correct confidence interval formula
STA 291 Spring 2008 Lecture 19

6. Test Statistic Testing µ with σ unknown and n small
Just like finding a confidence interval for µ with σ unknown and n small
Reasons for choosing test statistics are the same as choosing the correct confidence interval formula
Note: It is difficult for us to find p-values for this test statistic because of the way our table is set up
STA 291 Spring 2008 Lecture 19

7. Normality Assumption
An assumption for the t-test is that the population distribution is normal
In practice, it is impossible to be 100% sure if the population distribution is normal
It may be useful to look at histogram or stem-and-leaf plot (or normal probability plot) to check whether the normality assumption is reasonable
Good news
t-test is relatively robust against violations of this assumption
Unless the population distribution is highly skewed, the hypotheses tests and confidence intervals are valid
However, the random sampling assumption must never be violated, otherwise the test results are completely invalid
STA 291 Spring 2008 Lecture 19

8. Example
A courier service advertises that its average delivery time is less than 6 hours for local deliveries. A random sample of times for 12 deliveries found a mean of and a standard deviation of Is this sufficient evidence to support the courier’s advertisement at α=.05?
State and test the hypotheses using the rejection region method.
Why wouldn’t the p-value method be good to use?
STA 291 Spring 2008 Lecture 19

9. Correlation Between Tests and Confidence Intervals
Results of confidence intervals and of two- sided significance tests are consistent
Whenever the hypothesized mean is not in the confidence interval around the sample mean, then the p-value for testing H0: μ=μ0 is smaller than 5% (significance at the 5% level)
Why does this make sense?
In general, a 100(1-α)% confidence interval corresponds to a test at significance level α
STA 291 Spring 2008 Lecture 19

10. Example
A survey of 35 cars that just left their metered parking spaces produced a mean of 18 and a standard deviation of 22. Test the parking control officer’s claim that the average time left on meters is equal to 15 minutes.
State and test the hypotheses with a level of significance of 5% using the confidence interval method.
STA 291 Spring 2008 Lecture 19