Question

The distribution of grade point average for students at a large high school is skewed to the left with a mean of 3.53 and a standard deviation of 1.02.

a. Describe the shape of the sampling distribution of x¯ for SRSs of size n = 4 from the population of students at this high school. Justify your answer.

b. Describe the shape of the sampling distribution of x¯ for SRSs of size n = 50 from the Page Number: 480 population of students at this high school. Justify your answer.

Short answer

a. The sample mean's sampling distribution is skewed to the left.

b. The sampling distribution of the sample mean is approximately normal.

Step-by-step solution

Part (a) : Step 1 : Given information 

Given :

Mean :$3.53$

Standard deviation :$1.02.$

Size $n=4$

Part (a) : Step 2 : Simplification 

The distribution of the population is skewed to the left.

$n=4$

Theorem of the center limit: The sampling distribution of the sample mean x is roughly normal if the sample size is $30$or greater. The central limit theorem cannot be applied since the sample size of four is less than thirty. The sample mean's sampling distribution has the same shape as the population distribution in this example, hence the sample mean's sampling distribution is skewed to the left.

Part (b) : Step 1 : Given information 

Given :

Mean :$3.53$

Standard deviation : $1.02.$

Size$n=50$

Part (b) : Step 2 : Simplification 

The distribution of the population is skewed to the left.

$n=50$people

Theorem of the center limit: The sampling distribution of the sample mean is roughly normal if the sample size is $30$or greater. Because the sample size of $50$is greater than $30$, the central limit theorem can be used, and the sampling distribution of the sample mean is approximately normal.