Question: Suppose that a researcher conducts a political survey. She makes 1000 random phone calls and asks whether the recipients are in favor of the healthcare bill that is being discussed at the Senate currently. Let \(X_{i}\) be 1 if the \(i\) -th recipient of the call is in favor of the bill and 0 otherwise. Then, she sets up the null hypothesis and the alternative hypothesis as

Hence the null hypothesis is that about the half of the population is in favor of the bill. Suppose that the researcher finds that \((n=1000)\)

and the standard deviation \(\left(=\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{n}\right)^{2}}\right)\) is equal to 0.55.

1. Construct a test statistic using the sample mean of \(X_{i}\).
2. Find a critical value that gives Type I error of the test approximately 0.05 when the sample size is large.
3. See if the researcher would reject the null hypothesis at \(5 \%\) significance level based on the test defined through (a) and (b).
4. Compute the \(p\) -value using the test statistic and its asymptotic distribution. Would the researcher reject the null hypothesis at \(1 \%\) ?

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