Answer :
- Calculate the sum of sick adults: $18 + 56 = 74$.
- Calculate the sum of sample sizes: $50 + 75 = 125$.
- Calculate the combined sample proportion: $\frac{74}{125} = 0.592$.
- The combined sample proportion is $\boxed{0.592}$.
### Explanation
1. Understand the problem and provided data
We are given two independent random samples. The first sample has $n_1 = 50$ adults who exercise regularly, and $X_1 = 18$ of them got sick in the past year. The second sample has $n_2 = 75$ adults who do not exercise regularly, and $X_2 = 56$ of them got sick in the past year. We need to calculate the combined sample proportion $\hat{p}_{c} = \frac{X_1 + X_2}{n_1 + n_2}$ to check the large counts condition for a two-proportion z-test.
2. Calculate the sum of sick adults
First, we calculate the sum of the number of adults who got sick in both samples: $X_1 + X_2 = 18 + 56 = 74$.
3. Calculate the sum of sample sizes
Next, we calculate the sum of the sample sizes: $n_1 + n_2 = 50 + 75 = 125$.
4. Calculate the combined sample proportion
Now, we calculate the combined sample proportion: $\hat{p}_{c} = \frac{X_1 + X_2}{n_1 + n_2} = \frac{74}{125} = 0.592$.
5. State the final answer
Therefore, the combined sample proportion is $\hat{p}_{c} = 0.592$.
### Examples
Understanding combined proportions is useful in many real-world scenarios. For instance, if you're a marketing analyst trying to determine the overall success rate of two different advertising campaigns, calculating the combined proportion of successful conversions helps you evaluate the effectiveness of your marketing strategies. This approach provides a clear, single metric to compare and optimize your campaigns.
- Calculate the sum of sample sizes: $50 + 75 = 125$.
- Calculate the combined sample proportion: $\frac{74}{125} = 0.592$.
- The combined sample proportion is $\boxed{0.592}$.
### Explanation
1. Understand the problem and provided data
We are given two independent random samples. The first sample has $n_1 = 50$ adults who exercise regularly, and $X_1 = 18$ of them got sick in the past year. The second sample has $n_2 = 75$ adults who do not exercise regularly, and $X_2 = 56$ of them got sick in the past year. We need to calculate the combined sample proportion $\hat{p}_{c} = \frac{X_1 + X_2}{n_1 + n_2}$ to check the large counts condition for a two-proportion z-test.
2. Calculate the sum of sick adults
First, we calculate the sum of the number of adults who got sick in both samples: $X_1 + X_2 = 18 + 56 = 74$.
3. Calculate the sum of sample sizes
Next, we calculate the sum of the sample sizes: $n_1 + n_2 = 50 + 75 = 125$.
4. Calculate the combined sample proportion
Now, we calculate the combined sample proportion: $\hat{p}_{c} = \frac{X_1 + X_2}{n_1 + n_2} = \frac{74}{125} = 0.592$.
5. State the final answer
Therefore, the combined sample proportion is $\hat{p}_{c} = 0.592$.
### Examples
Understanding combined proportions is useful in many real-world scenarios. For instance, if you're a marketing analyst trying to determine the overall success rate of two different advertising campaigns, calculating the combined proportion of successful conversions helps you evaluate the effectiveness of your marketing strategies. This approach provides a clear, single metric to compare and optimize your campaigns.
