Question

Question:Consider a sample statistic A. As with all sample statistics, A is computed by utilizing a specified function (formula) of the sample measurements. (For example, if A were the sample mean, the specified formula would sum the measurements and divide by the number of measurements.

a. Describe what we mean by the phrase "the sampling distribution of the sample statistic A."

b. Suppose A is to be used to estimate a population parameter$\theta$. What is meant by the assertion that A is an unbiased estimator of $\theta$?

c. Consider another sample statistic, B. Assume that B is also an unbiased estimator of the population parameter$\alpha$. How can we use the sampling distributions of A and B to decide which is the better estimator of $\theta$?

d. If the sample sizes on which A and B are based are large, can we apply the Central Limit Theorem and assert that the sampling distributions of A and B are approximately normal? Why or why not?

Short answer

1. The probability distribution of the sample statistic A.
2. The expected value of the sample statistic is$\theta$.
3. The sample statistic will have a minimum standard deviation.
4. We can't apply the central limit theorem in this case.

Step-by-step solution

(a)Sampling distribution

Let A be a sample statistic.

The sampling distribution of sample statistic A means the probability distribution of the sample statistic A.

(b) Unbiased estimator

Given that, A be a sample statistic

A is an unbiased estimator representing the expected value of sample statistic$\theta$.

(c) Sampling distribution of a and b are better estimators

Let B be another sample statistic.

B is an unbiased estimator of the population parameters$\alpha$ .

We select the sample statistic with a minimum standard deviation for the sampling distribution of A and B to decide the better estimate$\theta$.

(d) Sampling distribution of a and b are approximately normal.

According to the central limit theorem, it is applicable for the sample mean.

Here, A and B are not the same. We can't apply the central limit theorem if the sample sizes on which A and B are based are large.