Question

1. It is desired to test the null hypothesis µ = 40 against the alternative hypothesis µ < 40 on the basis of a random sample from a population with standard deviation 4. If the probability of a Type I error is to be 0.04 and the probability of Type II error is to be 0.09 for µ = 38, find the required size of the sample.

Answer 1

For 0.04 level of significance, that is the type I error probability of 0.04, as this is a lower tailed test, we have from standard normal tables:
P(Z < -1.751 ) = 0.04

Therefore the critical mean value here is computed as:

Note that 40 is the hypothesized mean value above.

Now, for type II error, we do not reject a false null hypothesis, therefore for a true mean of 38, the probability here is computed as:

Note that the hypothesized mean value now has changed to 38 as we are given that 38 is the true mean.

From standard normal tables, we have:
P(Z < 1.341) = 0.91, therefore P(Z > 1.341) = 0.09

Therefore the z score for the above mean is 1.341 here.

therefore 39 is the required sample size here.