Some IQ tests are standardized based on the assumption that the population mean is 100 and the standard deviation is 15. Test graders decide to reject this hypothesis if a random sample of 25 people has a mean IQ greater than 110. Suppose the true mean of the population is actually 105. Assuming that IQ scores are normally distributed, what's the probability of a Type II error?
A. 952
B. 9995
C. 63
D. .0005
Ε.048
The critical value for rejecting the null hypothesis is 110. The standard error of the mean is 15/√25 = 3.
To calculate the probability of a Type II error, we need to find the probability that the sample mean falls between 100 and 110 given that the true mean is 105. This is equivalent to finding the Z-score for 110 and 100:
Z(110) = (110 - 105) / 3 = 5/3 = 1.67
Z(100) = (100 - 105) / 3 = -5/3 = -1.67
Using a standard normal distribution table, we can find the probability that Z is between 1.67 and -1.67. This is the same as finding the probability that Z < 1.67 - the probability that Z < -1.67.
This probability is approximately 0.952 or 95.2%. Therefore, the answer is:
A. 95.2%