Question

In a certain large population of adults, the distribution of IQ scores is strongly left skewed with a mean of 122 and a standard deviation of 5. Suppose 200 adults are randomly selected from this population for a market research study. For SRSs of size 200, the distribution of sample mean IQ score is

a. left-skewed with mean 122 and standard deviation 0.35.

b. exactly Normal with mean 122 and standard deviation 5.

c. exactly Normal with mean 122 and standard deviation 0.35.

d. approximately Normal with mean 122 and standard deviation 5.

e. approximately Normal with mean 122 and standard deviation 0.35.

Short answer

(e) Approximately normal with mean is 122 and standard deviation is 0.35

Step-by-step solution

Given Information 

The mean is $\mathrm{\mu }=122$

Standard deviation is $\mathrm{\sigma }=5$

The number of persons $\mathrm{n}=200$

The following concept was used:

${\mathrm{\sigma }}_{\overline{\mathrm{x}}}=\frac{\mathrm{\sigma }}{\sqrt{\mathrm{n}}}$

Explanation for correct option 

The sample size is larger than 33.

By Know that the sampling distribution of the central limit theorem sample mean$\overline{\mathrm{x}}$is concerned with the ordinary.

$$\begin{gathered} \mathrm{\mu }\overline{\mathrm{x}}=\mathrm{\mu }=122 \\ {\mathrm{\sigma }}_{\overline{\mathrm{x}}}=\frac{\mathrm{\sigma }}{\sqrt{\mathrm{n}}}=\frac{5}{\sqrt{200}}=0.35 \end{gathered}$$

As a result, the sample mean's sampling distribution is close to normal, with a mean of 122 and a standard deviation of 0.35.

Explanation for incorrect option 

a. left-skewed with not provided mean 122 and standard deviation 0.35

b. exactly Normal with not provided mean 122 and standard deviation 5

c. exactly Normal with not provided mean 122 and standard deviation 0.35

d. approximately Normal with not provide mean 122 and standard deviation 5