Question

1. Suppose you want to test H₀: u <=100 against H₁: u> 100 using a significance level of 0.05. The population is normally distributed with a standard deviation of 75. A random sample size of n = 40 will be used. If u = 130, what is the probability of correctly rejecting a false null hypothesis? What is the probability that the test will incorrectly fail to reject a false null hypothesis?

Answer 1

H₀: u <=100 against H₁: u> 100

z value for significance level of 0.05 is 1.96

Standard error of mean = = 75 / = 11.85854

Critical value to reject H0 = 100 + 1.96 * 11.85854 = 123.24

We reject the null hypothesis if X > 123.24

Probability of correctly rejecting a false null hypothesis = P(X > 123.24 | u = 130)

= P[Z > (123.24 - 130) / 11.85854]

= P[Z > -0.57]

= 0.7157

Probability that the test will incorrectly fail to reject a false null hypothesis = 1 - probability of correctly rejecting a false null hypothesis = 1 - 0.7157 = 0.2843