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Preparing to Check the Large Counts Condition for a Two-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:

\[

\begin{array}{l}

n_R=\square \\

n_W=\square

\end{array}

\]

Answer :

- Identify the sample size of runners: $n_R = 80$.
- Identify the sample size of walkers: $n_W = 100$.
- State the final answer: $n_R = 80$ and $n_W = 100$.

### Explanation
1. Identify the given information
We are given the sample sizes for runners and walkers and need to identify them.

2. Determine the sample size for runners
The problem states that a random sample of 80 runners was selected. Therefore, $n_R = 80$.

3. Determine the sample size for walkers
The problem also states that a random sample of 100 walkers was selected. Therefore, $n_W = 100$.

4. Final Answer
Thus, we have identified the sample sizes for runners and walkers as $n_R = 80$ and $n_W = 100$, respectively.

### Examples
Understanding sample sizes is crucial in various real-world scenarios. For instance, in market research, companies need to determine the right number of customers to survey to get reliable insights about their products. Similarly, in clinical trials, researchers need to select appropriate sample sizes to test the effectiveness of new drugs. Knowing the sample sizes helps in making informed decisions and drawing accurate conclusions.

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