Textbook Problem

In statistics we often use observed data to test a hypothesis about a population or populations. The basic method uses the observed data to calculate a test statistic (a single number). If the magnitude of this test statistic is sufficiently large, the null hypothesis is rejected in favor of the research hypothesis. As an example, consider a researcher who believes teenage girls sleep longer than teenage boys on average. She collects observations on n = 40 randomly selected girls and n = 40 randomly selected boys. (Each observation is the average sleep time over several nights for a given person. The averages are ${\bar{X}}_1$ = 7.9 hours for the girls and ${\bar{X}}_2$ = 7.6 hours for the boys. The standard deviation of the 40 observations for girls is s₁ = 0.5 hour; for the boys it is s₂ = 0.7 hour. The researcher, consulting a statistics textbook, then calculates the test statistic

$\frac{{\bar{X}}_1-{\bar{X}}_2}{\sqrt{s_1^2/40+s_2^2}/40}=\frac{7.9-7.6}{\sqrt{0.25/40+0.49/40}}=202.6$

Based on the fact that 2.206 is “large,” she claims that her research hypothesis is confirmed—girls do sleep longer than boys.

You are skeptical of this claim, so you check it out by running a simulation. In your simulation you assume that girls and boys have the same mean and standard deviation of sleep times in the entire population, say, 7.7 and 0.6. You also assume that the distribution of sleep times is normal. Then you repeatedly simulate observations of 40 girls and 40 boys from this distribution and calculate the test statistic. The question is whether the observed test statistic, 2.206, is “extreme.” If it is larger than most or all of the test statistics you simulate, then the researcher is justified in her claim; otherwise, this large a statistic could have happened easily by chance, even if the girls and boys have identical population means. Use @RISK to see which of these possibilities occurs.