# Type II Error in Lower Tail Test of Population Mean with Unknown Variance

In a lower tail test of the population mean, the null hypothesis claims that the true
population mean μ is greater than a given hypothetical value μ_{0}.

A type II error occurs if the hypothesis test based on a random sample fails to
reject the null hypothesis even when the true population mean μ is in fact less than
μ_{0}.

Let s^{2} be the sample variance. For sufficiently large n, the population of the
following statistics of all possible samples of size n is approximately a Student t
distribution with n - 1 degrees of freedom.

This allows us to compute the range of sample means for which the null hypothesis will not be rejected, and to obtain the probability of type II error. We demonstrate the procedure with the following:

#### Problem

Suppose the manufacturer claims that the mean lifetime of a light bulb is more than 10,000 hours. Assume in a random sample of 30 light bulbs, the standard deviation of the lifetime is 125 hours. If actual mean light bulb lifetime is 9,950 hours, what is the probability of type II error for a hypothesis test at .05 significance level?

#### Solution

We begin with computing the standard error estimate, SE.

> s = 125 # sample standard deviation

> SE = s/sqrt(n); SE # standard error estimate

[1] 22.822

We next compute the lower bound of sample means for which the null hypothesis μ ≥ 10000 would not be rejected.

> mu0 = 10000 # hypothetical lower bound

> q = mu0 + qt(alpha, df=n-1) * SE; q

[1] 9961.2

Therefore, so long as the sample mean is greater than 9961.2 in a hypothesis test, the null hypothesis will not be rejected. Since we assume that the actual population mean is 9950, we can compute the probability of the sample mean above 9961.2, and thus found the probability of type II error.

#### Answer

If the light bulbs sample size is 30, the sample standard variance is 125 hours and the actual mean light bulb lifetime is 9,950 hours, then the probability of type II error for testing the null hypothesis μ ≥ 10000 at .05 significance level is 31.3%, and the power of the hypothesis test is 68.7%.

#### Exercise

Under same assumptions as above, if the actual mean light bulb lifetime is 9,965 hours, what is the probability of type II errors? What is the power of the hypothesis test?