A doctor claims that runners tend to be optimistic, but are they more likely to be optimistic than those who walk? A study selected independent random samples of 80 runners and 100 walkers and found that 68 of the runners and 72 of the walkers scored as "optimistic" on a personality test. Do these data provide convincing evidence that the proportion of all runners who are optimistic is greater than the proportion of all walkers who are optimistic?

To prepare for calculating the expected number of successes and failures for the large counts condition, identify these values:

[tex]\[
\begin{array}{l}
n_R=\square \\
n_W=\square
\end{array}
\][/tex]

We are given that the study sampled 80 runners and 100 walkers. Therefore, the sample sizes we need for the large counts condition are

[tex]$$
n_R = 80 \quad \text{and} \quad n_W = 100.
$$[/tex]

To elaborate:

1. For runners, the study selected 80 independent subjects, so the sample size for runners is
[tex]$$n_R = 80.$$[/tex]

2. For walkers, the study selected 100 independent subjects, so the sample size for walkers is
[tex]$$n_W = 100.$$[/tex]

These values are used later in the analysis to check that the expected counts for successes (optimism) and failures (non-optimism) are sufficiently large, which is a necessary condition for using large-sample approximations in hypothesis testing.