EXAMPLE 5 Time spent eating

The Bureau of Labor Statistics conducts the American Time Use Survey (ATUS). This survey is intended to provide nationally representative estimates of how, where, and with whom Americans spend their time, and more than 159,000 individuals have been surveyed between 2003 and 2014. In 2014, it was reported that Americans spend an average of 1.23 hours per day eating and drinking. An administrator at a college campus in Ohio conducts a survey of a random sample of 150 college students and finds that the average amount of time these students report eating and drinking per day is hours, with a standard deviation of s = 0.25 hour. Is this evidence that the college students eat and drink, on average, a different amount of time each day when compared to the general population?

The hypotheses. The null hypothesis is “no difference’’ from the national mean. The alternative is two-sided because the college administrator did not have a particular direction in mind before examining the data. So, the hypotheses about the unknown mean μ of the population are

H₀: μ = 1.23

H_a: μ ≠ 1.23

The sampling distribution. If the null hypothesis is true, the sample mean has approximately the Normal distribution with mean μ = 1.23 and standard error

The data. The sample mean is . The standard score, or test statistic, for this outcome is

534

We know that an outcome 1 standard error away from the mean of a Normal distribution is not very surprising. The last step is to make this formal.

The P-value. Figure 22.5 locates the sample outcome 1 (in the standard scale) on the Normal curve that represents the sampling distribution if H₀ is true. The two-sided P-value is the probability of an outcome at least this far out in either direction. This is the shaded area under the curve. Table B says that a standard score of 1.0 is the 84.13 percentile of a Normal distribution. This means the area to the left of 1.0 in the Normal curve is 0.8413. The area to the right is, therefore, 0.1587. The area to the right of 1.0 and to the left of −1.0 is double this, or about 0.3174. This is our approximate P-value.

The conclusion. The large P-value gives us no reason to think that the mean time spent eating and drinking per day among the Ohio college students differs from the national average.