The Power of a Statistical TestDate: 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 by 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 http://mathforum.org/dr.math/ |
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