Question

Exercise 4. You are given the following null and alternative hypotheses:

?0: ? ≤ 500

? A   : ? > 500

? = 0.01

Calculate the probability of committing a Type II error (Beta ?) when the population mean is 505, the sample size is 64, and the population standard deviation is known to be 36.

Answer 1

From standard normal tables, we have here:

P( Z > 2.325) = 0.01,

Therefore the critical value here for 1% level of significance is 2.325

The critical mean value thus is computed here as:

The probability of committing a type II error is retaining a false null hypothesis. Therefore here it would mean given the true mean is 505, the probability that we dont reject the null hypothesis here is computed as:

P(X < 510.4625)

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.8876 is the required probability of type II error here.