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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 X¯is approximately equal to the mean of X.

b. When the sample size is large, X¯is approximately normally distributed.

c. When the sample size is large, the standard deviation of X¯is approximately the same as the standard deviation of X.

Short Answer

Expert verified

All the statements are true.

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

Step by step solution

01

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 X¯ is approximately equal to the mean of X."

02

Part (a) Step 2: Explanation 

When the sample size is big, the central limit theorem holds true. The sample mean is almost equal to X¯.

03

Part (b) Step 1: Given Information 

We need to find this statement is true or false. " When the sample size is large, X¯is approximately normally distributed."

04

Part (b) Step 2: Explanation 

When the sample size is big, the central limit theorem holds true. Then X¯has a roughly normal distribution.

05

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 X¯is approximately the same as the standard deviation of X."

06

Part (c) Step 2: Explanation 

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

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