Question

You’re a brilliant inventor who recently won a noble prize, but you are obsessed with intelligence. You start a company and as part of hiring practices you institute an IQ test. You know the tests are designed to have an average of 100 and standard deviation upon retaking of the tests of 15. You want to be 99 percent sure that no one with an average IQ or lower is employed in each batch of 20 employees you test.

A) What is the critical IQ value that you use for this assuming you are using the Bonferroni method for achieving a level of .01 for every 10 tests.

B) What is the power you have for detecting an IQ of 125 given the standard deviation of tests is 25 for the sub population with a mean of 125 (This was the measured IQ of the person who this describes at age 14).

Answer 1

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Answer:

Given data:

a)

Mean() =100

Standard deviation( )= 15

n=10

Ho : = 100

Ha : <100

= 0.01

Then Z=

Critical value of Z = -2.33.

If calculated z value <-2.33,

Then we reject n.

b)

true mean ,    µ =    125              
hypothesis mean,   µo =    100              
significance level,   α =    0.01   
sample size,   n =   10              
std dev,   σ =    20              
δ=   µ - µo =    25              
std error of mean,   σx = σ/√n =    20.0000   / √    10   =   6.32456

Zα =       2.3263   (right tailed test)          

We will fail to reject the null (commit a Type II error) if we get a Z statistic < 2.326      
this Z-critical value corresponds to X critical value( X critical), such that                      
(x̄ - µo)/σx ≤ Zα                      
x̄ ≤ Zα*σx + µo                      
x̄ ≤    2.326* 6.3246+100  

x̄ ≤    114.7131   (acceptance region)      
now, type II error is ,ß =    P( x̄ ≤    114.713   given that µ =   125   )  
= P ( Z < (x̄-true mean)/σx )                  
=P( Z < (   114.713   -   125   ) /   6.3246   )
= P ( Z <    -1.626   ) =   0.0519   [excel fucntion: =normsdist(z)  
power =    1 - ß =   0.9481