Question

IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

a) Find the proportion of the population that has an IQ higher than 94

b) Find the proportion of the population that has an IQ between 82 and 88

c) Find the IQ score that seperates the highest scoring 67% from the rest of the population

Critical Values

Z0.05= 1.645

Z0.025=1.96

Z0.01=2.325

Z0.005=2.575

Answer 1

Given Mean = 100

Standard Deviation = 15

z-score = (X - ) /

Question (a)

proportion of the population that has an IQ higher than 94

We need to find P(X > 94)

z-score = (94 - 100) / 15

= -6/15

= -0.4

P(X < 94) will be the area to the left of z-score -0.4 which can be found out from the below attached negative z-table

P(X < 94) = 0.34458

We know that P(X < 94) + P(X > 94) = 1

So P(X > 94)= 1 - 0.34458

= 0.65542

proportion of the population that has an IQ higher than 94 is 0.65542

Question (b)

proportion of the population that has an IQ between 82 and 88

We need to find P(82 < X < 88)

So we find P(X<82) and P(X<88) and subtract P(X<82) from P(X<88) to get P(82 < X < 88)

P(X<82) is calculated as follows

z-score = (82 - 100) / 15

= -18/15

= -1.2

P(X < 82) will be the area to the left of z-score -1.2 which can be found out from the below attached negative z-table

P(X < 82) = 0.11507

P(X<88) is calculated as follows

z-score = (88 - 100) / 15

= -12/15

= -0.8

P(X < 88) will be the area to the left of z-score -0.8 which can be found out from the below attached negative z-table

P(X < 88) = 0.21186

P(82 < X < 88) = P(X<88) - P(X<82)

= 0.21186 - 0.11507

= 0.09679

proportion of the population that has an IQ between 82 and 88 is 0.09679

Question (c)

The IQ score that seperates the highest scoring 67% from the rest of the population

So we need to find a z-score first that has an area of 0.67 to its left

The z-score is 0.43991 from the online calculator sicne the z-table gives only approximate value rounded to 2 decimal places

z-score = (X - ) /

0.43991 = (X - 100) / 15

X - 100 = 15 * 0.43991

X - 100 = 6.59865

X = 106.59865

So The IQ score of 106.59865 seperates the highest scoring 67% from the rest of the population