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Answer :
To evaluate the claim that the proportion of runners who are optimistic is greater than the proportion of walkers who are optimistic, we need to identify the necessary values for the large counts condition, which is part of the process for testing hypotheses about proportions.
Here's a step-by-step explanation to prepare for calculating the expected number of successes and failures:
1. Identify Sample Sizes:
- For the group of runners, the sample size is denoted as [tex]\( n_R \)[/tex].
- For the group of walkers, the sample size is denoted as [tex]\( n_W \)[/tex].
2. Assign the Sample Sizes:
- [tex]\( n_R = 80 \)[/tex]: This is the number of runners in the sample.
- [tex]\( n_W = 100 \)[/tex]: This is the number of walkers in the sample.
With these values established, the large counts condition can be further assessed by calculating the expected number of successes (those who are optimistic) and failures (those who are not optimistic) in each group. This is critical when determining if it's appropriate to use normal approximation methods for hypothesis testing.
Remember that for the large counts condition (or np ≥ 10 and n(1-p) ≥ 10 for both groups), the number of expected successes and failures should be at least 10 to proceed with certain types of hypothesis tests. You would calculate these expected numbers based on sample proportions and total counts, if you were to continue the hypothesis testing process.
Here's a step-by-step explanation to prepare for calculating the expected number of successes and failures:
1. Identify Sample Sizes:
- For the group of runners, the sample size is denoted as [tex]\( n_R \)[/tex].
- For the group of walkers, the sample size is denoted as [tex]\( n_W \)[/tex].
2. Assign the Sample Sizes:
- [tex]\( n_R = 80 \)[/tex]: This is the number of runners in the sample.
- [tex]\( n_W = 100 \)[/tex]: This is the number of walkers in the sample.
With these values established, the large counts condition can be further assessed by calculating the expected number of successes (those who are optimistic) and failures (those who are not optimistic) in each group. This is critical when determining if it's appropriate to use normal approximation methods for hypothesis testing.
Remember that for the large counts condition (or np ≥ 10 and n(1-p) ≥ 10 for both groups), the number of expected successes and failures should be at least 10 to proceed with certain types of hypothesis tests. You would calculate these expected numbers based on sample proportions and total counts, if you were to continue the hypothesis testing process.
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