Question

In a simple random sample of 600 people taken from a city A, 400 smoke. In another sample of 900 people taken from a city B, 450 smoke. Then, the value of the test statistic to test the difference between the proportions of smokers in the two samples, is:

• A. 5.72
• B. 6.42
• C. 5.92
• D. 6.05

Hint:

When testing the hypothesis that two population proportions are equal (\(H_0: p_1 = p_2\)), it is crucial to use the pooled proportion \( \hat{p} \) to estimate the common population proportion and calculate the standard error. Using separate proportions in the standard error formula is for constructing confidence intervals for the difference \(p_1 - p_2\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The problem requires a two-sample z-test for the difference between two population proportions. The null hypothesis is that the proportions are equal (\(H_0: p_A = p_B\)). The test statistic measures how many standard errors the observed difference in sample proportions is from the hypothesized difference of zero.

Step 2: Key Formula or Approach:
The formula for the z-test statistic for two proportions is: \[ z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \] where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, and \( \hat{p} \) is the pooled proportion, calculated as: \[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \]
Step 3: Detailed Explanation:
First, define the parameters for each city. City A (Sample 1): - \( n_1 = 600 \), \( x_1 = 400 \) - \( \hat{p}_1 = \frac{400}{600} = \frac{2}{3} \) City B (Sample 2): - \( n_2 = 900 \), \( x_2 = 450 \) - \( \hat{p}_2 = \frac{450}{900} = \frac{1}{2} \) Next, calculate the pooled proportion \( \hat{p} \): \[ \hat{p} = \frac{400 + 450}{600 + 900} = \frac{850}{1500} = \frac{17}{30} \] Then \( 1 - \hat{p} = 1 - \frac{17}{30} = \frac{13}{30} \). Now, calculate the z-statistic: \[ z = \frac{\frac{2}{3} - \frac{1}{2}}{\sqrt{\left(\frac{17}{30}\right)\left(\frac{13}{30}\right)\left(\frac{1}{600} + \frac{1}{900}\right)}} \] Numerator: \( \frac{2}{3} - \frac{1}{2} = \frac{4-3}{6} = \frac{1}{6} \). Denominator term \( \left(\frac{1}{600} + \frac{1}{900}\right) = \frac{3+2}{1800} = \frac{5}{1800} = \frac{1}{360} \). Denominator: \( \sqrt{\frac{17 \times 13}{30 \times 30} \times \frac{1}{360}} = \sqrt{\frac{221}{900 \times 360}} = \sqrt{\frac{221}{324000}} \) \[ z = \frac{1/6}{\sqrt{221/324000}} = \frac{1}{6} \sqrt{\frac{324000}{221}} = \frac{1}{6} \frac{\sqrt{324000}}{\sqrt{221}} \approx \frac{1}{6} \frac{569.21}{14.866} \approx \frac{38.29}{6} \approx 6.38 \] The calculated value is approximately 6.38. This is closest to option (B) 6.42. The minor difference may be due to rounding in the problem's expected answer.
Step 4: Final Answer:
The value of the test statistic is approximately 6.38, which is best represented by the option 6.42.