A certain statistic will be used as an unbiased estimator of a parameter. Let J represent the sampling distribution of the estimator for samples of size 40, and let K represent the sampling distribution of the estimator for samples of size 100. Which of the following must be true about J and K ?

a. The expected values of J and K will be equal, and the variability of J will be less than the variability of K.
b. The expected values of J and K will be equal, and the variability of J will be equal to the variability of K.
c. The expected values of J and K will be equal, and the variability of J will be greater than the variability of K.
d. The expected value of J will be greater than the expected value of K, and the variability of R will be greater than the variability of K.
e. The expected value of J will be greater than the expected value of K, and the variability of R will be less than the variability of K.

The correct statement can be identified by using certain limit theorem.

The correct option is (c).

Given:

The sample size of J is 40.

The sample size of K is 100.

As per the central limit theorem, if the population with mean, standard deviation and take large random sample from population  with replacement, then distribution of sample mean will approximately distributed.

Now from the above definition of central limit theorem, both J and K will have same means.

Write the expression for standard deviation.

From the above expression standard deviation is inversely proportional to the sample size . Thus, as the sample size increases, the standard deviation, related to the variability, decreases.

Thus, the correct statement is  The expected values of J and K will be equal, and the variability of J will be greater than the variability of K.

Learn more about certain limit theorem here:

link

c. The expected values of J and K will be equal, and the variability of J will be greater than the variability of K.

Step-by-step explanation:

This is a theoretical question, that is solved using concepts of the central limit theorem.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question:

J has samples of size 40, and K has samples of size 100.

By the Central Limit Theorem, both will have the same means(expected value), while the standard deviation is inversely proportional to the sample size n, that is, as the sample size increases, the standard deviation, related to the variability, decreases.

So J and K have the same expected values, while K, due to the higher sample size, will have less variability. So the correct answer is given by option c.