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Sagot :
To compare the population proportions of males over the age of 30 that have been married at least once between two countries, we will conduct a hypothesis test. Our goal is to compute the test statistic for this hypothesis test.
### Step-by-Step Solution:
1. Identify the Sample Proportions and Sample Sizes:
- For Country 1: [tex]\( n1 = 200 \)[/tex]
- Sample proportion [tex]\( p1 = 0.87 \)[/tex] (87% of 200 males have been married at least once)
- For Country 2: [tex]\( n2 = 100 \)[/tex]
- Sample proportion [tex]\( p2 = 0.81 \)[/tex] (81% of 100 males have been married at least once)
2. Calculate the Pooled Proportion:
- The pooled proportion [tex]\( \hat{p}_{\text{pool}} \)[/tex] is calculated by combining the two sample proportions weighted by their respective sample sizes.
- [tex]\( \hat{p}_{\text{pool}} = \frac{p1 \times n1 + p2 \times n2}{n1 + n2} \)[/tex]
- Substituting the values: [tex]\( \hat{p}_{\text{pool}} = \frac{0.87 \times 200 + 0.81 \times 100}{200 + 100} \)[/tex]
- After computation, we find that [tex]\( \hat{p}_{\text{pool}} = 0.85 \)[/tex]
3. Calculate the Standard Error (SE):
- The standard error SE of the difference in proportions is calculated using the pooled proportion:
- [tex]\( SE = \sqrt{\hat{p}_{\text{pool}} \times (1 - \hat{p}_{\text{pool}}) \times \left( \frac{1}{n1} + \frac{1}{n2} \right)} \)[/tex]
- Substituting the values: [tex]\( SE = \sqrt{0.85 \times (1 - 0.85) \times \left( \frac{1}{200} + \frac{1}{100} \right)} \)[/tex]
- After computation, we find that [tex]\( SE = 0.0437 \)[/tex] (rounded to 4 decimal places for intermediate calculation accuracy).
4. Calculate the Test Statistic (z):
- The z-score (test statistic) is calculated using the difference in sample proportions divided by the standard error:
- [tex]\( z = \frac{p1 - p2}{SE} \)[/tex]
- Substituting the values: [tex]\( z = \frac{0.87 - 0.81}{0.0437} \)[/tex]
- After computation, we find that [tex]\( z = 1.37 \)[/tex] (rounded to 2 decimal places).
### Conclusion:
The test statistic for comparing the population proportions of males over the age of 30 that have been married at least once between the two countries is [tex]\( \boxed{1.37} \)[/tex].
### Step-by-Step Solution:
1. Identify the Sample Proportions and Sample Sizes:
- For Country 1: [tex]\( n1 = 200 \)[/tex]
- Sample proportion [tex]\( p1 = 0.87 \)[/tex] (87% of 200 males have been married at least once)
- For Country 2: [tex]\( n2 = 100 \)[/tex]
- Sample proportion [tex]\( p2 = 0.81 \)[/tex] (81% of 100 males have been married at least once)
2. Calculate the Pooled Proportion:
- The pooled proportion [tex]\( \hat{p}_{\text{pool}} \)[/tex] is calculated by combining the two sample proportions weighted by their respective sample sizes.
- [tex]\( \hat{p}_{\text{pool}} = \frac{p1 \times n1 + p2 \times n2}{n1 + n2} \)[/tex]
- Substituting the values: [tex]\( \hat{p}_{\text{pool}} = \frac{0.87 \times 200 + 0.81 \times 100}{200 + 100} \)[/tex]
- After computation, we find that [tex]\( \hat{p}_{\text{pool}} = 0.85 \)[/tex]
3. Calculate the Standard Error (SE):
- The standard error SE of the difference in proportions is calculated using the pooled proportion:
- [tex]\( SE = \sqrt{\hat{p}_{\text{pool}} \times (1 - \hat{p}_{\text{pool}}) \times \left( \frac{1}{n1} + \frac{1}{n2} \right)} \)[/tex]
- Substituting the values: [tex]\( SE = \sqrt{0.85 \times (1 - 0.85) \times \left( \frac{1}{200} + \frac{1}{100} \right)} \)[/tex]
- After computation, we find that [tex]\( SE = 0.0437 \)[/tex] (rounded to 4 decimal places for intermediate calculation accuracy).
4. Calculate the Test Statistic (z):
- The z-score (test statistic) is calculated using the difference in sample proportions divided by the standard error:
- [tex]\( z = \frac{p1 - p2}{SE} \)[/tex]
- Substituting the values: [tex]\( z = \frac{0.87 - 0.81}{0.0437} \)[/tex]
- After computation, we find that [tex]\( z = 1.37 \)[/tex] (rounded to 2 decimal places).
### Conclusion:
The test statistic for comparing the population proportions of males over the age of 30 that have been married at least once between the two countries is [tex]\( \boxed{1.37} \)[/tex].
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