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Besides optimism, there are other benefits associated with exercise. A doctor claims the proportion of those who exercise and got sick in the past year is smaller than the proportion of those who do not exercise.

To investigate, an analyst selects independent random samples: 50 adults who exercise regularly and 75 adults who do not exercise regularly. Of those who exercise regularly, 18 got sick in the past year, and of those who do not exercise regularly, 56 got sick in the past year. Do these data provide convincing evidence that these two population proportions differ?

The random and [tex]10\%[/tex] conditions for this problem are met, but what about the large counts condition? Calculate [tex]\hat{p}_c=\frac{x_1+X_2}{n_1+n_2}[/tex].

Enter 3 decimal places.

[tex]\hat{p}_c = \square[/tex]

Answer :

To determine if the large counts condition for this problem is met, we first need to calculate the combined sample proportion. This is an important step as it helps in conducting hypothesis tests for comparing two proportions.

Here’s how you can calculate the combined sample proportion, [tex]\(\hat{p}_c\)[/tex], using the given data:

1. Identify the Sample Sizes and Counts:
- The total number of adults who exercise regularly, [tex]\( n_1 = 50 \)[/tex].
- The number of adults who exercise and got sick, [tex]\( x_1 = 18 \)[/tex].
- The total number of adults who do not exercise regularly, [tex]\( n_2 = 75 \)[/tex].
- The number of adults who do not exercise and got sick, [tex]\( x_2 = 56 \)[/tex].

2. Calculate the Combined Proportion:
- First, sum the number of "successes" (individuals who got sick) from both groups: [tex]\( x_1 + x_2 = 18 + 56 = 74 \)[/tex].
- Then, sum the total number of individuals from both groups: [tex]\( n_1 + n_2 = 50 + 75 = 125 \)[/tex].
- Finally, use the formula for the combined sample proportion:

[tex]\[
\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2} = \frac{74}{125}
\][/tex]

3. Compute [tex]\(\hat{p}_c\)[/tex]:
Divide 74 by 125 to get [tex]\(\hat{p}_c\)[/tex]. This calculation results in:

[tex]\[
\hat{p}_c \approx 0.592
\][/tex]

Thus, the combined sample proportion, [tex]\(\hat{p}_c\)[/tex], is 0.592. This value will help determine if the large counts condition for performing a hypothesis test is satisfied, ensuring the sample proportion estimates are approximately normally distributed.

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