# Type II Error in Lower Tail Test of Population Mean with Known 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}.

Assume that the population has a known variance σ^{2}. By the Central Limit
Theorem, the population of all possible means of samples of sufficiently large size n
approximately follows the normal distribution. Hence we can compute the range of
sample means for which the null hypothesis will not be rejected, and then obtain an
estimate of 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 actual mean light bulb lifetime is 9,950 hours and the population standard deviation is 120 hours. At .05 significance level, what is the probability of having type II error for a sample size of 30 light bulb?

#### Solution

We begin with computing the standard deviation of the mean, sem.

> sigma = 120 # population standard deviation

> sem = sigma/sqrt(n); sem # standard error

[1] 21.909

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 = qnorm(alpha, mean=mu0, sd=sem); q

[1] 9964

Therefore, so long as the sample mean is greater than 9964 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 9964, and thus found the probability of type II error.

#### Answer

If the light bulbs sample size is 30, the actual mean light bulb lifetime is 9,950 hours and the population standard deviation is 120 hours, then the probability of type II error for testing the null hypothesis μ ≥ 10000 at .05 significance level is 26.2%, and the power of the hypothesis test is 73.8%.

#### Exercise

Under same assumptions as above, if the actual mean light bulb lifetime is 9,965 hours, what is the probability of type II error at .05 significance level? What is the power of the hypothesis test?