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: Concept Overview:
This problem utilizes a two-sample z-test to compare two population proportions. The null hypothesis assumes equal proportions (\(H_0: p_A = p_B\)). The z-test statistic quantifies the difference between observed sample proportions relative to the hypothesized zero difference, measured in standard errors.

Step 2: Core Formula:
The z-test statistic for comparing two proportions is calculated as follows: \[ z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]where \( \hat{p}_1 \) and \( \hat{p}_2 \) represent the sample proportions, and \( \hat{p} \) is the pooled proportion, determined by: \[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \]
Step 3: Step-by-Step Calculation:
First, let's 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, we compute the pooled proportion \( \hat{p} \):\[ \hat{p} = \frac{400 + 450}{600 + 900} = \frac{850}{1500} = \frac{17}{30} \]Therefore, \( 1 - \hat{p} = 1 - \frac{17}{30} = \frac{13}{30} \).Now, we 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 resulting z-value is approximately 6.38, closely aligning with option (B) 6.42. The slight discrepancy may be attributed to rounding differences in the expected solution.
Step 4: Conclusion:
The computed test statistic is roughly 6.38, making 6.42 the most appropriate answer.