Question: In 2003, the General Social Survey found that 24% of Americans were willing to pay much higher taxes in order to protect the environment. Suppose a similar study will be conducted in 2008 and researchers would like to test whether the proportion in 2008 is the same as in 2003. A difference of 10% would be regarded as practically important.

Let p denote the proportion of Americans willing to pay much higher taxes in order to protect the environment.

1. Find the sample size needed to design a test of p = 0.24 versus p \(\ne \) 0.24 at the 1% significance level so that there is an 85% chance of rejecting the null hypothesis if the true population proportion is 0.34 (10% higher than 24%).
2. Find the sample size needed to design a test of p = 0.24 versus p \(\ne \) 0.24 at the 1% significance level so that there is an 85% chance of rejecting the null hypothesis if the true population proportion is 0.14 (10% lower than 24%).
3. What sample size should the researchers use so that there is at least an 85% chance of rejecting the null hypothesis if the true proportion in 2008 is 34% and if the true proportion is 14% in 2008?

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