Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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].
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.