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:

nᵣ =

To determine if the proportion of all runners who are optimistic is greater than the proportion of all walkers who are optimistic, we can conduct a hypothesis test. We will use a significance level of 0.05.

Explanation:

To determine if the proportion of all runners who are optimistic is greater than the proportion of all walkers who are optimistic, we can conduct a hypothesis test. Let's define the null hypothesis as the proportion of optimistic runners being equal to the proportion of optimistic walkers, and the alternative hypothesis as the proportion of optimistic runners being greater than the proportion of optimistic walkers. We will use a significance level of 0.05.

nr = 80 (number of runners)