Q. 3 - The Practice of Statistics for AP

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  adults are randomly selected from this population for a market research study. The distribution of the sample mean of IQ scores is

(a) left-skewed with a mean of $122$ and a standard deviation of $0.35$.

(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 a mean$122$ and standard deviation $0.35$.

Step-by-Step Solution

Answer

The distribution of the sample mean of IQ scores is option (e) Approximately normal with mean$122$ and standard deviation $0.35$.

Step 1: Given information

The distribution of IQ scores is strongly left-skewed with a mean of $122$ and a standard deviation of $5$.

$200$ adults are randomly selected and the IQ score is found.

Step 2: Explanation

The population is strongly left-skewed 

$\mu =Mean=122$$\sigma =S\tan dard deviation =5$

$n=Sample size=200$

If the sample is large enough, the central limit theorem states that the sampling distribution of the sample mean is approximately normal. When a sample size of$30$ or more is used, it is regarded as adequately large.

The sample size  $200$ in this situation is an adequate sample size.

We know that the sampling distribution of the sample mean $\overline{x}$is approximately normal because of the central limit theorem.

${\mu }_{\overline{x}}=\mu =122$

The population standard deviation divided by the square root of the sample size yields the standard deviation of the sampling distribution of the sample mean $\overline{x}$

${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}=\frac{5}{\sqrt{200}}\approx 0.35$

Thus it is approximately normal with a mean $122$ and standard deviation $0.35.$