Question

The 20 to 34 age groups are approximately normally distributed with µ = 110

Answer 1

Answer:

21.94% of people aged 20 to 34 have IQs between 125 and 150.

Step-by-step explanation:

The complete question is: Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed with  μ = 110  and  σ = 25.

What percent of people aged 20 to 34 have IQs between 125 and 150?

Let X = Scores on the standard IQ test for the 20 to 34 age group

SO, X ~ Normal($\mu=110,\sigma^{2} =25^{2}$)

The z-score probability distribution for the normal distribution is given by;

                            Z  =  $\frac{X-\mu}{\sigma}$  ~ N(0,1)

where, $\mu$ = population mean = 110

            $\sigma$ = standard deviation = 25

Now, the percent of people aged 20 to 34 have IQs between 125 and 150 is given by = P(125 < X < 150) = P(X < 150) - P(X $\leq$ 125)

       P(X < 150) = P( $\frac{X-\mu}{\sigma}$ < $\frac{150-110}{25}$ ) = P(Z < 1.60) = 0.9452

       P(X $\leq$ 125) = P( $\frac{X-\mu}{\sigma}$ $\leq$ $\frac{125-110}{25}$ ) = P(Z $\leq$ 0.60) = 0.7258

The above probability is calculated by looking at the value of x = 1.60 and x = 0.60 which has an area of 0.9452 and 0.7258.

Therefore, P(125 < X < 150) = 0.9452 - 0.7258 = 0.2194 or 21.94%