Preparing to Check the Large Counts Condition for a Proportion z-Test

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:

\[ n_R = \square \]

\[ n_W = \square \]

To address the question, we need to prepare to check the large counts condition for a proportion z-test. This involves identifying the values for successes and failures in the samples of runners and walkers.

In the problem, we have:

1. Number of Runners Surveyed:
- Total runners surveyed: [tex]\( n_R = 80 \)[/tex]

2. Number of Walkers Surveyed:
- Total walkers surveyed: [tex]\( n_W = 100 \)[/tex]

3. Number of Optimistic Runners:
- Runners who scored as "optimistic": [tex]\( 68 \)[/tex]

4. Number of Optimistic Walkers:
- Walkers who scored as "optimistic": [tex]\( 72 \)[/tex]

To check the large counts condition, you need the expected number of successes and failures for both groups.

- For Runners:
- Expected number of successes: The number of optimistic runners, which is [tex]\( 68 \)[/tex].
- Expected number of failures: Total runners minus optimistic runners: [tex]\( n_R - \)[/tex] optimistic runners [tex]\( = 80 - 68 = 12 \)[/tex].

- For Walkers:
- Expected number of successes: The number of optimistic walkers, which is [tex]\( 72 \)[/tex].
- Expected number of failures: Total walkers minus optimistic walkers: [tex]\( n_W - \)[/tex] optimistic walkers [tex]\( = 100 - 72 = 28 \)[/tex].

Therefore, for your calculation, you have:

- Runners: [tex]\( n_R = 80 \)[/tex], 68 successes, and 12 failures.
- Walkers: [tex]\( n_W = 100 \)[/tex], 72 successes, and 28 failures.

These values will help determine if the sample sizes are sufficiently large to use a z-test for comparing proportions.