STA 291Summer 2010

Lecture 18Dustin Lueker

The p-value for testing H1: µ≠100 is p=.001. This indicates that…1. There is strong evidence that μ=1002. There is strong evidence that μ≠1003. There is strong evidence that μ>1004. There is strong evidence that μ<100

Example

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The p-value for testing H1: µ≠100 is p=.001. In addition you know that the test statistic was z=3.29. This indicates that…1. There is strong evidence that μ=1002. There is strong evidence that μ>1003. There is strong evidence that μ<100

Example

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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

Rejection Region

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Testing µ with n large

◦ Just like finding a confidence interval for µ with n large Reasons for choosing test statistics are the same as

choosing the correct confidence interval formula

Test Statistic

5

ns

xz 0

STA 291 Summer 2010 Lecture 18

Testing µ with n small

◦ Just like finding a confidence interval for µ with 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

Test Statistic

6

ns

xt 0

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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

Normality Assumption

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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 5.7 and a standard deviation of 1.58. 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?

Example

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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

Correlation Between Tests and Confidence Intervals

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A survey of 35 cars that just left their metered parking spaces produced a mean of 18 minutes remaining on the meter 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.

Example

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