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In one country, a survey of 200 males over the age of 30 found that 87% had been married at least once. In another country, a survey of 100 males over the age of 30 found that 81% had been married at least once.

Compute the test statistic for a hypothesis test to compare the population proportions of males over the age of 30 that have been married at least once between the two countries. Round your answer to two decimal places.

Provide your answer below:

Sagot :

GinnyAnswer
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].
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