Question: (AF). A national survey conducted by the National Sleep Foundation in 2002 reports that 58% of the US population has problems with insomnia. Two researchers (A and B) would like to determine if the proportion of people having problems with insomnia in the state of Massachusetts is the same as the national proportion. They each decide to test p = 0.58 versus p \(\ne \) 0.58, using the z-statistics for large sample analyses and the rejection region of |z| \(\ge \) 2.576. Researcher A plans to collect a simple random sample of 100 Massachusetts residences, and Researchers B plans to collect a simple random sample of 400 from the same population.

1. Find the probability of making a type I error in the test.
2. If the null hypothesis is true, how many individuals are expected to have insomnia problems in Researcher A’s sample, and how many individual are expected to have insomnia problems in Researcher B’s sample?
3. If the true Massachusetts proportion is 0.65,
   - find the power of the test described above in Researcher AÕs study. - find the power of the test described above in Researcher BÕs study.
4. Whose test is more powerful? Why?

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