Calculating the Sample Size Needed to Achieve Specified Values of α and β
Assume that the sampled population is normally distributed, or that a large sample will be taken. Consider testing H0: μ = μ0 versus one of Ha: μ > μ0, Ha: μ < μ0, or Ha: μ ≠ μ0. Then, in order to make the probability of a Type I error equal to α and the probability of a Type II error corresponding to the alternative value μa of μ equal to β, we should take a sample of size
Here z* equals zα if the alternative hypothesis is one-sided (μ > μ0 or μ < μ0), and z* equals zα/2 if the alternative hypothesis is two-sided (μ ≠ μ0). Also, zβ is the point on the scale of the standard normal curve that gives a right-hand tail area equal to β.
EXAMPLE 9.10
Again consider the coffee fill example and suppose we wish to test H0: μ ≥ 3 (or μ = 3) versus Ha: μ < 3. If we wish α to be .05 and β for the alternative value μa = 2.995 of μ to be .05, we should take a sample of size
Here, z* = zα = z.05 = 1.645 because the alternative hypothesis (μ < 3) is one-sided, and zβ = z.05 = 1.645.
Although we have set both α and β equal to the same value in the coffee fill situation, it is not necessary for α and β to be equal. As an example, again consider the Valentine’s Day chocolate case, in which we are testing H0: μ = 330 versus Ha: μ ≠ 330. Suppose that the candy company decides that failing to reject H0: μ = 330 when μ differs from 330 by as many as 15 valentine boxes (that is, when μ is 315 or 345) is a serious Type II error. Furthermore, suppose that it is also decided that this Type II error is more serious than a Type I error. Therefore, α will be set equal to .05 and β for the alternative value μa = 315 (or μa = 345) of μ will be set equal to .01. It follows that the candy company should take a sample of size
Here, z* = zα/2 = z.05/2 = z.025 = 1.96 because the alternative hypothesis (μ ≠ 330) is two-sided, and zβ = z.01 = 2.326 (see the bottom row of the t table on page 865).
To conclude this section, we point out that the methods we have presented for calculating the probability of a Type II error and determining sample size can be extended to other hypothesis tests that utilize the normal distribution. We will not, however, present the extensions in this book.
Exercises for Section 9.5
CONCEPTS
9.76 We usually take action on the basis of having rejected the null hypothesis. When we do this, we know the chances that the action has been taken erroneously because we have prespecified α, the probability of rejecting a true null hypothesis. Here, it is obviously important to know (prespecify) α, the probability of a Type I error. When is it important to know the probability of a Type II error? Explain why.