To test the claim that the birth rate in Bonn (Population 1) is higher than the national average (Population 2) at a $0.05$ level of significance, we can set up a hypothesis test for two population proportions. Let's follow the steps to calculate the test statistic.

Step 1: Define Hypotheses

Let:

• $p_1$ = proportion of births in Bonn
• $p_2$ = proportion of births in Germany

Our hypotheses are:

• Null Hypothesis: $H_0: p_1 \leq p_2$
• Alternative Hypothesis: $H_1: p_1 > p_2$

Step 2: Calculate Sample Proportions

Given:

• For Population 1 (Bonn): $x_1 = 14$ births, $n_1 = 1400$
• For Population 2 (Germany): $x_2 = 24$ births, $n_2 = 4000$

The sample proportions are: $\hat{p}_1 = \frac{x_1}{n_1} = \frac{14}{1400} = 0.01$ $\hat{p}_2 = \frac{x_2}{n_2} = \frac{24}{4000} = 0.006$

Step 3: Calculate the Pooled Proportion

Since we assume the null hypothesis is true for the calculation, we use a pooled proportion $\hat{p}$: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{14 + 24}{1400 + 4000} = \frac{38}{5400} \approx 0.00704$

Step 4: Compute the Standard Error

The standard error (SE) of the difference between the two proportions is: $SE = \sqrt{\hat{p} \left(1 - \hat{p}\right) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$ Substitute the values: $SE = \sqrt{0.00704 \times (1 - 0.00704) \times \left(\frac{1}{1400} + \frac{1}{4000}\right)}$ $SE = \sqrt{0.00704 \times 0.99296 \times (0.0007143 + 0.00025)}$ $SE \approx \sqrt{0.00704 \times 0.99296 \times 0.0009643} \approx \sqrt{0.00000675} \approx 0.0026$

Step 5: Calculate the Test Statistic

The test statistic $Z$ for comparing two proportions is given by: $Z = \frac{\hat{p}_1 - \hat{p}_2}{SE}$ Substitute the values: $Z = \frac{0.01 - 0.006}{0.0026} \approx \frac{0.004}{0.0026} \approx 1.54$

Answer

The value of the test statistic is approximately: $Z \approx 1.54$

Would you like me to continue with interpretation or calculations for the p-value?

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Related Questions

1. What would be the next steps to determine if we reject the null hypothesis?
2. How would changing the significance level affect our decision?
3. Can this method be applied if the sample sizes were different?
4. What are the assumptions underlying this test?
5. How does the sample size impact the test statistic's sensitivity?

Tip

To strengthen your understanding, try solving a similar problem with different sample sizes or proportions.