POSsIBLE POINTS: 8.33 24. A random sample of 432 voters revealed that 100 are in favor of a certain bond issue. A 95 percent confidence interval for the proportion of the population of voters who are in favor of the bond issue is (A) \( 100 \pm 1.96 \sqrt{\frac{0.5(0.5)}{432}} \) (B) \( 100 \pm 1.645 \sqrt{\frac{0.5(0.5)}{432}} \) (C) \( 100 \pm 1.96 \sqrt{\frac{0.231(0.769)}{432}} \) (D) \( 0.231 \pm 1.96 \sqrt{\frac{0.231(0.769)}{432}} \) (E) \( 0.231 \pm 1.645 \sqrt{\frac{0.231(0.769)}{432}} \)

To determine the correct answer, let's first calculate the sample proportion of voters in favor of the bond issue. This is found using the formula: \[ \hat{p} = \frac{x}{n} = \frac{100}{432} \approx 0.231 \] Now, to compute the 95% confidence interval for the population proportion, we use the standard error formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.231(0.769)}{432}} \approx 0.035 \] Next, we apply the 1.96 z-score for a 95% confidence level: \[ CI = \hat{p} \pm 1.96 \times SE = 0.231 \pm 1.96 \times 0.035 \] This leads us to option (D): \( 0.231 \pm 1.96 \sqrt{\frac{0.231(0.769)}{432}} \) Get the confidence interval and you can express your result with dramatic flair! Confidence intervals provide not just numbers, but a sense of assurance in predictions—like a ticket giving you access to the best seats at an event! If you're curious, remember that while calculating values is essential, knowing how to interpret these intervals is equally vital—these ranges help estimate the likelihood of true proportions in a population. Armed with this knowledge, you can approach questions on population proportions with confidence!