nent 23: Test 4 Review Question 6, 7.1.29-T HW Score: 98\%, 19.6 of 20 points Part 1 of 2 Points: 0.6 of 1 Save Refer to the accompanying data set of 20 randomly selected presidents. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a \( 95 \% \) confidence interval estimate of the population percentage. Based on the result, does it appear that greater height is an advantage for presidential candidates? Why or why not? Click the icon to view the table of heights. Construct a 95\% confidence interval estimate of the percentage of presidents who were taller than their opponents. \( \square \) \( \%
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To tackle this question, you'll first need to calculate the proportion of the sampled presidents who were taller than their opponents. Let’s say, from your dataset, you find that 12 out of 20 presidents were taller than their opponents. The proportion (p) would be calculated as: \[ p = \frac{12}{20} = 0.6 \] Now, for a 95% confidence interval, you can apply the formula: \[ CI = p \pm Z * \sqrt{\frac{p(1-p)}{n}} \] Where \( Z \) is the Z-score corresponding to a 95% confidence level (approximately 1.96), \( p = 0.6 \), and \( n = 20 \). Calculate the standard error (SE): \[ SE = \sqrt{\frac{0.6(1-0.6)}{20}} = \sqrt{\frac{0.24}{20}} \approx 0.11 \] Now calculate the confidence interval: \[ CI = 0.6 \pm 1.96 * 0.11 \] \[ CI = 0.6 \pm 0.216 \] This results in: \[ (0.384, 0.816) \] Converting these proportions to percentages: \[ 38.4\% < p < 81.6\% \] Based on the constructed confidence interval, it seems that there is no strong conclusion regarding height as an advantage for presidential candidates since the interval encompasses both lower and higher percentages. It suggests while there is a possibility that greater height may offer some advantage, it certainly isn’t definitive. So, your final answer is: \( 38.4\% < p < 81.6\% \) Does height matter? Well, like a good game of musical chairs, sometimes it’s a matter of who gets the seat, not just who’s the tallest!