Question: Suppose that a researcher is interested in estimating the probability of being unemployed for a white male. She collected observations from a survey of the employment status of 10,000 men living in Pennsylvania. The survey was designed and performed well so that it is reasonable to assume that the observations \(Y_{i}, i=1, \cdots, n\), are i.i.d. The random variables \(Y_{i}\) are such that \(Y_{i}=1\) if the \(i\) -th male is unemployed and \(Y_{i}=0\) if otherwise. Hence the parameter of interest is

Suppose that the researcher uses the following estimator:

as the estimator of the probability of being unemployed.

1. Write constants \(a\) and \(b\) (in terms of the mean or variance of \(\left.Y_{i}\right)\) such that for each \(t \in \mathbf{R}\),
   \[P\left\{\frac{1}{a \sqrt{n}} \sum_{i=1}^{n}\left(Y_{i}-b\right) \leq t\right\} \rightarrow \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{t} e^{-\frac{x^{2}}{2}} d x\]
   (HINT: Look at the Central Limit Theorem.)
2. From the result of (a), we can approximate the distribution of \(\hat{\mu}\) by a normal distribution. Find the mean and the variance of this normal distribution. The approximation is reliable when the sample size is large. (This approximate distribution is called the asymptotic distribution of \(\hat{\mu}\).)
3. Find an asymptotic \(95 \%\) confidence interval for \(\mu\) using the estimator \(\hat{\mu}\). (Recall that an asymptotic confidence interval is a confidence interval that uses the asymptotic distribution of \(\hat{\mu}\) instead of the distribution of \(\hat{\mu}\).) Recall that when \(Z\) has a standard normal distribution,

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