Suppose that Y has a binomial distribution based on n trials...

Suppose that $Y$ has a binomial distribution based on $n$ trials and success probability $p$. Then $\hat{p}_{n}=Y / n$ is an unbiased estimator of $p .$ Use Theorem 9.3 to prove that the distribution of $\left(\hat{p}_{n}-p\right) / \sqrt{\hat{p}_{n} \hat{q}_{n} / n}$ converges to a standard normal distribution. [Hint: Write $Y$ as we did in Section 1.5. ( ) $\begin{array}{ll}\text { (?) } & \text { (?) } 1 \\ \text { (?) }\end{array}$

Suppose that $Y$ has a binomial distribution based on $n$ trials and success probability $p$. Then
$\hat{p}_{n}=Y / n$ is an unbiased estimator of $p .$ Use Theorem 9.3 to prove that the distribution of
$\left(\hat{p}_{n}-p\right) / \sqrt{\hat{p}_{n} \hat{q}_{n} / n}$ converges to a standard normal distribution. [Hint: Write $Y$ as we did in Section
1.5. ( ) $\begin{array}{ll}\text { (?) } & \text { (?) } 1 \\ \text { (?) }\end{array}$

Key Concepts

Binomial Distribution

A binomial distribution describes the total number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is a discrete probability distribution that is used to model the number of successes in a series of binary experiments and is characterized by the number of trials and the success probability.

Unbiased Estimator

An unbiased estimator is a statistical estimator whose expected value is equal to the true parameter it is estimating. In this context, using Y/n as an estimator for the success probability p is unbiased because its expectation accurately recovers p, ensuring that, on average, the estimator does not systematically overestimate or underestimate the parameter.

Central Limit Theorem

The central limit theorem is a fundamental statistical theorem stating that, as the sample size becomes large, the standardized sum of independent random variables converges in distribution to a normal distribution, regardless of the original distribution of the variables. This concept is crucial for assessing the asymptotic behavior of estimators based on sums or averages of independent variables, such as the estimator for p in the binomial model.

Delta Method

The delta method is a technique used to derive the asymptotic distribution of a function of an estimator, especially when the estimator itself is asymptotically normal. In this problem context, it allows the transformation of the normally distributed approximation of the sample proportion into the standardized form involving the plug?in variance estimator, thereby facilitating the proof that the standardized estimator converges to a standard normal distribution.

Convergence in Distribution

Convergence in distribution is a type of convergence where the distribution of a sequence of random variables approaches the distribution of another random variable as the sample size increases. This concept is used to justify that the standardized estimator, when appropriately normalized, will have a limiting distribution that is normal, making it applicable for asymptotic inference.

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