Question

The Wechsler IQ test is designed so that the mean is 100 and the standard deviation is 15 for the population of normal adults. Listed below are IQ scores of randomly selected professional pilots. It is claimed that because professional pilots are a more homogeneous group than the general population, they have IQ scores with a standard deviation less than 15. Test that claim using a 0.05 significance level. 121 116 115 121 116 107 127 98 116 101 130 114 Identify the test statistic, -value or critical value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.

Answer 1

Solution:

Given: The Wechsler IQ test is designed so that the mean is 100 and the standard deviation is 15 for the population of normal adults. That is: X follows Normal distribution with Mean = and Standard deviation =

We are given that the IQ scores of randomly selected professional pilots.

Level of significance =

Claim: because professional pilots are a more homogeneous group than the general population, they have IQ scores with a standard deviation less than 15.

That is we have to test if population standard deviation is less than 15

Step 1) State H0 and H1:

Vs

Step 2) Find the test statistic

where

Thus we need to make following table:

x: IQ scores	x^2
121	14641
116	13456
115	13225
121	14641
116	13456
107	11449
127	16129
98	9604
116	13456
101	10201
130	16900
114	12996

Thus we get:

Step 3) Find Chi-square critical value:

df = n - 1 = 12 -1 = 11

Level of significance =

This is left tailed test

0.05 area in chi-square table is right tail area but we want 0.05 area in left tail

so if left tail area is 0.05 then its right side area is 1 - 0.05 = 0.95

so we need to look in Chi-square table for Area = 0.95

Chi-square critical value = 4.575

Step 4) Decision rule:

Reject H0, if Chi-square test statistic value < Chi-square critical value, otherwise we fail to reject H0.

Since Chi-square test statistic value = < Chi-square critical value = 4.575 , we reject H0.

Step 5) Conclusion:

Since we have rejected null hypothesis H0, there is sufficient evidence to support the claim that: because professional pilots are a more homogeneous group than the general population, they have IQ scores with a standard deviation less than 15.