The Practice of Statistics Third Edition Chapter 12: Significance Tests in Practice Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

Tests about a Population Proportion This test is called a one-proportion z test or a large- sample test (because it is based on a Normal approximation to the sampling distribution of phat that becomes more accurate as the sample size increases).

Note that p o is used in the SE formula NOT phat!

Ex 12.7 Work Stress A national survey of restaurant employees found that 75% said that work stress had a negative impact on their personal lives. A random sample of 100 employees from a large restaurant chain finds that 68 answer “yes” when asked if work stress negatively impacted their personal lives. Is there good reason to think that the proportion of all employees in this chain who would say “yes” differs from the national proportion of 0.75?

Significance Test Steps 1 - Hypotheses Identify the population and parameter. State H o and H a. 2 - Conditions Determine appropriate inference procedure and check conditions. 3 - Calculations Calculate test statistic. Find p-value. 4 – Interpretation Answer in context, interpreting the p-value or making a decision about Ho using statistical significance.

There is over a 10% chance of obtaining a sample result as unusual or even more unusual than we did (phat =0.68) when the null hypothesis is true. We have insufficient evidence to suggest that the proportion of this chain restaurant’s employees who suffer from work stress is different from the national survey result, 0.75.

Ex 12.8 Work Stress, Again A national survey of restaurant employees found that 25% said that work stress did not have a negative impact on their personal lives. A random sample of 100 employees from a large restaurant chain finds that 32 answer “No” when asked if work stress negatively impacted their personal lives. Is there good reason to think that the proportion of all employees in this chain who would say “No” differs from the national proportion of 0.25?

Significance Test Steps 1 - Hypotheses Identify the population and parameter. State H o and H a. 2 - Conditions Determine appropriate inference procedure and check conditions. 3 - Calculations Calculate test statistic. Find p-value. 4 – Interpretation Answer in context, interpreting the p-value or making a decision about Ho using statistical significance.

z = 1.62 and p = Note that only the sign of the statistic z changed. Our conclusion does not depend on an arbitrary choice of success.

CI’s Provide Additional Info If the sample of restaurant workers is sufficiently large, we will have sufficient power to detect a very small difference between the proportion for our chain of restaurant workers and the national worker’s proportion. However, if n is very small, we may be unable to detect differences that could be very important. This is why we often include a CI in our analysis.

CI’s Provide Additional Info Calculate a 95% CI for proportion of the restaurant chain’s employees that feel work stress is damaging to their personal lives. Remember to check conditions using phat not p o. Remember to use phat in your CI SE not p o.

CI’s Provide Additional Info (0.59, 0.77) This CI gives an approximate range of p o ’s that would NOT be rejected by a test at the α = 0.05 significance level. That is, the CI gives a range of plausible values for the true population parameter p.