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The Power of a Statistical Test

Date: 02/11/99 at 08:10:53
From: John Kohut III
Subject: Power of a statistical test

Dear Dr. Math,

Is there a formula to determine the power of a statistical test which 
relates alpha and beta error?

For example, does the sample group of numbers 12, 14, 15, 16, 20 
belong to the population of numbers with a mean of 17 with standard 
deviation of 2? What would be the power of the z-test performed on 
this sample using alpha = 95% and beta = 80%? Thank you.

John Kohut III

Date: 02/11/99 at 10:50:02
From: Doctor Anthony
Subject: Re: Power of a statistical test

Below is an example of how the power of a test is calculated.

A population is known to have a variance of 9. Investigate whether the 
population mean is equal to 10. The sample size is 49 and the
probability of a type I error is taken to be alpha = .1. If the true 
value of the population mean is 8, calculate the power of the test.

If we reject the null hypothesis when it is true we make a type 1 
error, and its probability is denoted by alpha.

The power of the test is the probability of rejecting the hypothesis 
when it is false, and it is denoted by 1 - beta.  

With a .1 significance level, the probability of rejecting the null 
hypothesis will be 0.05 in each tail, so we use z = +/- 1.645.

If the null hypothesis is true, the mean is 10. We want to find an 
acceptance region where:

   P(reject Ho | Ho is true) = P(reject Ho | mean = 10) = .1

Using the null hypothesis of mean = 10, the acceptance region is given 

                      xm - mean              xm - 10       
   +/-1.645 = --------------------------  =  -------
              std. dev/sqrt(sample size)       3/7

So the acceptance region would be

   xm = 10 +/- (3/7)(1.645)

which is

   9.295 < xm < 10.705        

The probability of rejecting the null hypothesis when the null is 
false (i.e. mean = 8) is given by the z-value

        9.295 - 8   1.295
   z =  --------- = ----- = 3.02   
           3/7       3/7

which has a probability of

    A(z) = 0.9987

So, in this situation, the power of the test is  0.9987.

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Statistics

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