EXAMPLE 7.23

Is the sample size large enough? Recall Example 7.2 (page 413) on the average time that U.S. college students spend watching traditional television. The sample mean of n = 8 students was four hours lower than the U.S. average of 18- to 24-year-olds but not found significantly different. Suppose a new study is being planned using a sample size of n = 50 students. Does this study have adequate power when the population mean is four hours less than the U.S. average?

We wish to compute the power of the t test for

H₀: μ = 18.5

H_a: μ < 18.5

against the alternative that μ = 18.5 − 4 = 14.5 when n = 50. This gives us most of the information we need to compute the power. The other important piece is a rough guess of the size of σ. In planning a large study, a pilot study is often run for this and other purposes. In this case, we can use the standard deviation from the earlier survey. Similar to Example 7.21, we will round up and use σ = 17.5 and s = 17.5 in the approximate calculation.

Step 1. The t test with 50 observations rejects H₀ at the 5% significance level if the t statistic

is less than the lower 5% point of t(49), which is −1.677. Taking s = 17.5, the event that the test rejects H₀ is, therefore,

Step 2. The power is the probability that when μ = 14.5. Taking σ = 17.5, we find this probability by standardizing :

= P(Z ≤ −0.061)

= 0.4761

A mean value of 14.5 hours per week will produce significance at the 5% level in only 47.6% of all possible samples. Figure 7.18 shows Minitab output for the exact power calculation. It is about 48% and is represented by a dot on the power curve at a difference of −4. This curve is very informative. For many studies, 80% is considered the standard value for desirable power. We see that with a sample size of 50, the power is greater than 80% only for reductions larger than 6.25 hours per week. If we want to detect a reduction of only four hours, we definitely need to increase the sample size.