1) Economists often track employment trends by measuring the proportion of people who are "underemployed," meaning they are either unemployed or would like to work full time but are only working part-time. In the summer of 2013, 17.6% of Americans were "underemployed." The mayor of Thicksburg wants to show the voters that the situation is not as bad in his town as it is in the rest of the country. His staff takes a simple random sample of 300 Thicksburg residents and finds that 45 of them are underemployed.

(a) Do the data give convincing evidence that the proportion of underemployed in Thicksburg is lower than elsewhere in the country? Support your answer with a significance test.

(b) Interpret theP-value from your test in the context of the problem.

2)When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution with =7hours. A quality control supervisor selects a

simple random sample ofnbatteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded.

(a)Define the parameter of interest and state appropriate hypotheses for the quality control supervisor to test.

(b)Since testing the lifetime of a battery requires draining the battery completely, the supervisor wants to sample as few batteries as possible from each hour's production. She is considering a sample size ofn= 4. Explain why this sample size may lead to problems in carrying out the significance test from (a).

(c)Describe a Type I and a Type II error in this situation and the consequences of each.

(d) The quality control officer is considering changing the significance level of the test to 1%. Discuss the impact this might have on error probabilities and the power of the test, and describe the practical consequences of this change.