Q.7.6 - The Practice of Statistics for AP

Question

IQ tests The Wechsler Adult Intelligence Scale (WAIS) is a common "IQ test" for adults. The distribution of WAIS scores for persons over $16$ years of age is approximately Normal with mean 100 and standard deviation $15$.

(a) What is the probability that a randomly chosen individual has a WAIS score of $105$ or higher? Show your work.

(b) Find the mean and standard deviation of the sampling distribution of the average WAIS score $\overline{x}$ for an SRS of $60$ people.

(c) What is the probability that the average WAIS score of an SRS of $60$ people is $105$ or higher? Show your work.

(d) Would your answers to any of parts (a), (b), or (c) be affected if the distribution of WAIS scores in the adult population were distinctly non-Normal? Explain.

Step-by-Step Solution

Answer

a). The probability is $0.3707$.

b). The required mean and standard deviation are $100$ and$1.937$.

c). The probability is $0.0049$.

d). The distribution is said to be non-normal .

Part (a) Step 1: Given Information

Population mean $\left(\mu \right)=100$.

Population standard deviation $\left(\sigma \right)=15$.

Sample size $\left(n\right)=60$.

Part (a) Step 1: Explanation

The probability that selected individuals' WAIS scores are $105$ or higher can be computed as follows:

$P(x>29)=P\left(Z>\frac{105-100}{15}\right)$

                $=P(Z>0.33)$

                $=0.3707$

Part (b) Step 1: Given Information

Population mean $\left(\mu \right)=100$.

Population standard deviation $\left(\sigma \right)=15$.

Sample size $\left(n\right)=60$.

Part (b) Step 2: Explanation

Calculate the mean and standard deviation as follows:

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

$\mu =100$

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

    $=\frac{15}{\sqrt{60}}$

   $=1.937$

Part (c) Step 1: Given Information

Population mean $\left(\mu \right)=100$.

Population standard deviation $\left(\sigma \right)=15$.

Sample size $\left(n\right)=60$.

Part (c) Step 2: Explanation

The probability that the average WAIS score is $105$ or higher can be estimated as follows:

$P(\overline{x}>105)=P\left(Z>\frac{105-100}{\frac{15}{\sqrt{60}}}\right)$

                  $=P(Z>2.58)$

                 $=0.0049$

Part (d) Step 1: Given Information

Population mean $\left(\mu \right)=100$.

Population standard deviation $\left(\sigma \right)=15$.

Sample size $\left(n\right)=60$.

Part (d) Step 2: Explanation

If the distribution is said to be non-normal, (a) will be influenced.

However, because the sample size is more than $30$, the answers of (b) and (c) would be unaffected.