Question

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

a. When the sample size is large, the mean of $\overline{X}$is approximately equal to the mean of $X$.

b. When the sample size is large, $\overline{X}$is approximately normally distributed.

c. When the sample size is large, the standard deviation of $\overline{X}$is approximately the same as the standard deviation of $X$.

Short answer

All the statements are true.

The statements a,b,c of the question are true.

Step-by-step solution

Part (a) Step 1: Given Information 

We need to find this statement is true or false. " When the sample size is large, the mean of $\overline{X}$ is approximately equal to the mean of $X$."

Part (a) Step 2: Explanation 

When the sample size is big, the central limit theorem holds true. The sample mean is almost equal to $\overline{X}$.

Part (b) Step 1: Given Information 

We need to find this statement is true or false. " When the sample size is large, $\overline{X}$is approximately normally distributed."

Part (b) Step 2: Explanation 

When the sample size is big, the central limit theorem holds true. Then $\overline{X}$has a roughly normal distribution.

Part (c) Step 1: Given Information 

We need to find the statement is true or false. " When the sample size is large, the standard deviation of $\overline{X}$is approximately the same as the standard deviation of $X$."

Part (c) Step 2: Explanation 

When the sample size is big, the central limit theorem holds true. The standard deviation of $\overline{X}$is hence roughly equal to the sample mean's standard deviation.