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]\[ n_R = \square \][/tex]

[tex]\[ n_W = \square \][/tex]

To determine if runners are more likely to be optimistic than walkers based on the collected data, we need to identify the expected number of successes and failures for both groups (runners and walkers). Here is the step-by-step solution:

1. Determine sample sizes and number of successes:

- Number of runners sampled ([tex]\(n_R\)[/tex]): 80
- Number of walkers sampled ([tex]\(n_W\)[/tex]): 100

- Number of optimistic runners: 68
- Number of optimistic walkers: 72

2. Calculate the number of failures for each group:

For runners:
- Number of failures = Total number of runners - Number of optimistic runners
- Number of failures = 80 - 68 = 12

For walkers:
- Number of failures = Total number of walkers - Number of optimistic walkers
- Number of failures = 100 - 72 = 28

3. Summarize the counts:

- For runners:
- Number of optimistic runners (successes) = [tex]\(n_{R\_successes}\)[/tex] = 68
- Number of non-optimistic runners (failures) = [tex]\(n_{R\_failures}\)[/tex] = 12

- For walkers:
- Number of optimistic walkers (successes) = [tex]\(n_{W\_successes}\)[/tex] = 72
- Number of non-optimistic walkers (failures) = [tex]\(n_{W\_failures}\)[/tex] = 28

Using these values, we can observe and later run the appropriate statistical tests to determine whether the observed difference in optimism between runners and walkers provides convincing evidence that runners are more optimistic.

To summarize the expected numbers:
- [tex]\(n_R = 80\)[/tex]
- [tex]\(n_W = 100\)[/tex]

With these values calculated, we can proceed to statistical analysis to further investigate the doctor's claim.