Question

While writing a computer programming code, it was found that Team A incurred 250 errors in a code of 1000 lines while Team B incurred 300 errors in a code of 800 lines. In order to determine whether Team A has performed better than Team B, a statistical hypothesis was set and tested. Then the value of the test statistic is:

• A. -5.72
• B. -2.96
• C. 1.99
• D. 3.42

Hint:

For a hypothesis test of two proportions, \(p_1\) and \(p_2\), the test statistic is a Z-score. Always use the pooled proportion \(\hat{p}\) in the denominator's standard error calculation when the null hypothesis is \(p_1 = p_2\).

Answer 1

The Correct Option is A

Step 1: Define the parameters and hypotheses. This is a two-sample test for proportions. Let \(p_A\) and \(p_B\) be the true error proportions for Team A and Team B. - Sample proportion for A: \( \hat{p}_A = \frac{250}{1000} = 0.25 \) with \( n_A = 1000 \). - Sample proportion for B: \( \hat{p}_B = \frac{300}{800} = 0.375 \) with \( n_B = 800 \). The hypothesis is to test if Team A performed better, meaning they have a lower error rate. So, \( H_1: p_A < p_B \) and \( H_0: p_A \ge p_B \).

Step 2: Calculate the pooled proportion (\(\hat{p}\)). Under \(H_0\), we assume the proportions are equal. We estimate this common proportion by pooling the data. \[ \hat{p} = \frac{\text{Total Errors}}{\text{Total Lines}} = \frac{250+300}{1000+800} = \frac{550}{1800} = \frac{11}{36} \]

Step 3: State the formula for the test statistic (Z). \[ Z = \frac{(\hat{p}_A - \hat{p}_B)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_A} + \frac{1}{n_B})}} \]

Step 4: Calculate the values and the final test statistic. - Numerator: \( \hat{p}_A - \hat{p}_B = 0.25 - 0.375 = -0.125 \). - Pooled proportion: \( \hat{p} = 11/36 \approx 0.3056 \). So, \( 1-\hat{p} = 25/36 \approx 0.6944 \). - Denominator: \[ \sqrt{\frac{11}{36} \cdot \frac{25}{36} \left(\frac{1}{1000} + \frac{1}{800}\right)} = \sqrt{\frac{275}{1296} \left(\frac{1}{1000} + \frac{1.25}{1000}\right)} \] \[ = \sqrt{0.2122 (0.00225)} = \sqrt{0.00047745} \approx 0.02185 \] - Test Statistic Z: \[ Z = \frac{-0.125}{0.02185} \approx -5.7208 \]