# Type II Error in Two-Tailed Test of Population Mean with Unknown Variance

In a two-tailed test of the population mean, the null hypothesis claims that the true population mean μ is equal to a given hypothetical value μ0.

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 different from μ0.

Let s2 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 mean weight of King Penguins found in an Antarctic colony last year was 15.4 kg. Assume in a random sample 35 penguins, the standard deviation of the weight is 2.5 kg. If actual mean penguin weight is 15.1 kg, 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.

> n = 35                # sample size
> s = 2.5               # sample standard deviation
> SE = s/sqrt(n); SE    # standard error estimate
[1] 0.42258

We next compute the lower and upper bounds of sample means for which the null hypothesis μ = 15.4 would not be rejected.

> alpha = .05           # significance level
> mu0 = 15.4            # hypothetical mean
> I = c(alpha/2, 1-alpha/2)
> q = mu0 + qt(I, df=n-1) * SE; q
[1] 14.541 16.259

Therefore, so long as the sample mean is between 14.541 and 16.259 in a hypothesis test, the null hypothesis will not be rejected. Since we assume that the actual population mean is 15.1, we can compute the lower tail probabilities of both end points.

> mu = 15.1             # assumed actual mean
> p = pt((q - mu)/SE, df=n-1); p
[1] 0.097445 0.995168

Finally, the probability of type II error is the probability between the two end points.

> diff(p)               # p[2]-p[1]
[1] 0.89772