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]$\hat{\rho}_{c}=\frac{x_1+x_2}{n_1+n_2}$[/tex].

Enter 3 decimal places.

[tex]$\hat{\rho}_{c}=$[/tex] [tex]$\square$[/tex]

Sure, let's go through the problem step-by-step to solve it and find [tex]$\hat{\rho}_{c}$[/tex].

Step 1: Identify the given information.
- Number of adults who exercise regularly ([tex]$n_1$[/tex]): 50
- Number of adults who do not exercise regularly ([tex]$n_2$[/tex]): 75
- Number of adults who exercise regularly and got sick ([tex]$x_1$[/tex]): 18
- Number of adults who do not exercise regularly and got sick ([tex]$x_2$[/tex]): 56

Step 2: Calculate the combined sample size.
The combined sample size is the sum of the two groups:
[tex]\[ n_{total} = n_1 + n_2 = 50 + 75 = 125 \][/tex]

Step 3: Calculate the combined number of people who got sick.
The combined number of people who got sick is:
[tex]\[ x_{total} = x_1 + x_2 = 18 + 56 = 74 \][/tex]

Step 4: Calculate the combined proportion [tex]$\hat{\rho}_{c}$[/tex].
The combined proportion is calculated using the formula:
[tex]\[ \hat{\rho}_{c} = \frac{x_{total}}{n_{total}} = \frac{74}{125} \][/tex]

Step 5: Simplify the fraction and convert to decimal.
[tex]\[ \hat{\rho}_{c} = \frac{74}{125} = 0.592 \][/tex]

So the combined proportion, [tex]$\hat{\rho}_{c}$[/tex], is 0.592 when rounded to three decimal places.