Many high school students take the AP tests in different subject areas. In one year, of the 144,470 students who took the biology exam 84,801 of them were female. In that same year, of the 213,697 students who took the calculus $A B$ exam 124,859 of them were female. Estimate the difference in the proportion of female students taking the biology exam and female students taking the calculus AB exam using a $95 \%$ confidence level. Round to four decimal places. With $\%$ confidence, it can be concluded that the true difference in the proportion of female students taking the biology exam and the proportion of female students taking the calculus $A B$ exam $\left(p_{1}-p_{2}\right)$ is between and
Solution
Step 1 ：First, calculate the proportions of female students who took the biology and calculus AB exams. The proportion of female students who took the biology exam is \( \frac{84801}{144470} = 0.5869799958468886 \) and the proportion of female students who took the calculus AB exam is \( \frac{124859}{213697} = 0.5842805467554528 \).
Step 2 ：Next, calculate the standard error of the difference in proportions. The standard error is \( \sqrt{\left(\frac{0.5869799958468886 \times (1 - 0.5869799958468886)}{144470}\right) + \left(\frac{0.5842805467554528 \times (1 - 0.5842805467554528)}{213697}\right)} = 0.001677717719069156 \).
Step 3 ：Then, calculate the confidence interval. The z-score for a 95% confidence level is 1.96. The lower limit of the confidence interval is \( (0.5869799958468886 - 0.5842805467554528) - 1.96 \times 0.001677717719069156 = -0.0005888776379397221 \) and the upper limit of the confidence interval is \( (0.5869799958468886 - 0.5842805467554528) + 1.96 \times 0.001677717719069156 = 0.005987775820811369 \).
Step 4 ：Finally, the difference in the proportion of female students taking the biology exam and female students taking the calculus AB exam, with a 95% confidence level, is between -0.0006 and 0.0060. In other words, we can be 95% confident that the true difference in proportions is between these two values. This is represented as \(\boxed{[-0.0006, 0.0060]}\).
