High School

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Besides optimism, there are other benefits associated with exercise. A doctor claims that 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 of 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 proportions are different?

The random and 10% conditions for this problem are met, but what about the large counts condition? Calculate [tex]$b_c=\frac{x_1+x_2}{n_1+n_2}$[/tex].

Enter 3 decimal places: [tex]$b_c=\square$[/tex].

Answer :

To solve this problem, we are trying to find the combined proportion, denoted as [tex]\( b_c \)[/tex], which will help us in evaluating the large counts condition in hypothesis testing for comparing two population proportions.

Here's a step-by-step solution:

1. Understand the Scenario:
- We have two groups of adults:
- 50 adults who exercise regularly.
- 75 adults who do not exercise regularly.

2. Number of People who got Sick:
- From the group of people who exercise regularly, 18 got sick.
- From the group of people who do not exercise regularly, 56 got sick.

3. Combine the Data:
- To find the combined proportion [tex]\( b_c \)[/tex], we need to calculate the total number of people who got sick from both groups and the total number of people in both groups.

4. Calculate the Combined Proportion [tex]\( b_c \)[/tex]:
- Add the number of people who got sick in both groups: [tex]\( x_1 + x_2 = 18 + 56 = 74 \)[/tex].
- Add the total number of people in both groups: [tex]\( n_1 + n_2 = 50 + 75 = 125 \)[/tex].
- The combined proportion [tex]\( b_c \)[/tex] is given by:
[tex]\[
b_c = \frac{x_1 + x_2}{n_1 + n_2} = \frac{74}{125}
\][/tex]

5. Final Answer:
- When you calculate the above expression, you get [tex]\( b_c = 0.592 \)[/tex].

This combined proportion of 0.592 means that, overall, about 59.2% of the people in the samples got sick, which can be used further in statistical tests to examine if there's a significant difference between the two groups.

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Rewritten by : Barada