Question

IQ test scores Scores on the Wechsler Adult Intelligence Scale (an IQ test) for the 20- to 34-year-old age group are approximately Normally distributed with μ=110 and σ=25 .

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

b. MENSA is an elite organization that admits as members people who score in the top 2% on IQ tests. What score on the Wechsler Adult Intelligence Scale would an individual aged 20 to 34 have to earn to qualify for MENSA membership?

Short answer

Part (a) $21.95\%$of the people aged $20$to $34$have IQs between $125$and $150$

Part (b) An individual requires score of at least $161.25$ to qualify for MENSA membership.

Step-by-step solution

Part (a) Step 1: Given information

IQ score,$x=125or150$

Mean,$\mu =110$

Standard deviation,$\sigma =25$

Part (a) Step 2: Concept

The formula used:$z=\frac{x-\mu }{\sigma }$

Part (a) Step 3: Calculation

Calculate the $z-$ score,

$z=\frac{x-\mu }{\sigma }=\frac{125-110}{25}=0.60$

Or

$z=\frac{x-\mu }{\sigma }=\frac{150-110}{25}=1.60$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with 1.6 and the column that starts with .00 of the standard normal probability table for P (z < 1.60)

Or

See the row that starts with 0.6 and the column that starts with .00 of the standard normalprobability table for P (z < 0.60)

$$\begin{gathered} P(125<x<150)=P(0.60<z<1.60)=P(z<1.60)-P(z<0.60) \\ =0.9452-0.7257 \\ =0.2195 \\ =21.95\% \end{gathered}$$

Therefore,

$21.95\%$ of the people aged $20$ to $34$ have IQs between $125$ and $150$

Part (b) Step 1: Calculation

Find the z − score corresponding to the probability of 98% (or 0.98) in the normal probability table of the appendix.

Note that

The closest probability would be 0.9798 which lies in row 2.0 and in column .05 of the normal probability table.

Then

The corresponding z − score,

$z=2.05$

Calculate the z − score,

$z=\frac{x-\mu }{\sigma }=\frac{x-110}{25}$

Equate the above two expressions:

$\frac{x-110}{25}=2.05$

Multiply both sides by 25:

$x-110=51.25$

Add 110 to both sides:

$x-110+110=51.25+110$

That becomes

$x=161.25$

Therefore,

An individual have to score at least 161.25 to become eligible for MENSA membership.