Step 1 :Given data: number of male students (n1) = 915, number of male students with blue eyes (x1) = 294, number of female students (n2) = 1062, number of female students with blue eyes (x2) = 349.
Step 2 :Calculate the sample proportions of blue eyes in males (\(\hat{p}_1\)) and females (\(\hat{p}_2\)) using the formulas \(\hat{p}_1 = \frac{x1}{n1}\) and \(\hat{p}_2 = \frac{x2}{n2}\).
Step 3 :\(\hat{p}_1 = \frac{294}{915} = 0.32131147540983607\) and \(\hat{p}_2 = \frac{349}{1062} = 0.3286252354048964\).
Step 4 :Calculate the combined sample proportion of blue eyes (\(\hat{p}\)) using the formula \(\hat{p} = \frac{x1 + x2}{n1 + n2}\).
Step 5 :\(\hat{p} = \frac{294 + 349}{915 + 1062} = 0.32524026302478504\).
Step 6 :Calculate the test statistic (z-score) using the formula \(z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n1} + \frac{1}{n2})}}\).
Step 7 :Substitute the calculated values into the formula: \(z = \frac{(0.32131147540983607 - 0.3286252354048964) - 0}{\sqrt{0.32524026302478504(1-0.32524026302478504)(\frac{1}{915} + \frac{1}{1062})}} = -0.3461253783175717\).
Step 8 :The test statistic (z-score) is approximately -0.35. This value represents how many standard deviations the observed difference in proportions is from the expected difference under the null hypothesis (which is 0 in this case). A negative z-score indicates that the observed difference is less than the expected difference.
Step 9 :Final Answer: The test statistic is \(\boxed{-0.35}\).