Out of 500 people sampled, 70 preferred Candidate A. Based on this, estimate what proportion of the entire voting population $(p)$ prefers Candidate $A$.
Use a $90 \%$ confidence level, and give your answers as decimals, to three places.
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Solution
Step 1 ：We are given that out of 500 people sampled, 70 preferred Candidate A. We are asked to estimate what proportion of the entire voting population prefers Candidate A with a 90% confidence level.
Step 2 ：We can solve this problem using the formula for the confidence interval of a proportion, which is \(p \pm z \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion, \(n\) is the sample size, and \(z\) is the z-score corresponding to the desired confidence level.
Step 3 ：In this case, \(p = \frac{70}{500} = 0.14\), \(n = 500\), and \(z\) is the z-score for a 90% confidence level, which is approximately 1.645.
Step 4 ：Substituting these values into the formula, we get the margin of error as 0.025526668799512403.
Step 5 ：We then subtract this margin of error from \(p\) to get the lower bound of the confidence interval, which is 0.11447333120048761, and add it to \(p\) to get the upper bound, which is 0.16552666879951242.
Step 6 ：Thus, the estimated proportion of the entire voting population that prefers Candidate A, with a 90% confidence level, is between 0.114 and 0.166. So, \(0.114 < p < 0.166\).
Step 7 ：\(\boxed{0.114 < p < 0.166}\) is the final answer.
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Solution
Step 1 ：We are given that out of 500 people sampled, 70 preferred Candidate A. We are asked to estimate what proportion of the entire voting population prefers Candidate A with a 90% confidence level.
Step 2 ：We can solve this problem using the formula for the confidence interval of a proportion, which is \(p \pm z \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion, \(n\) is the sample size, and \(z\) is the z-score corresponding to the desired confidence level.
Step 3 ：In this case, \(p = \frac{70}{500} = 0.14\), \(n = 500\), and \(z\) is the z-score for a 90% confidence level, which is approximately 1.645.
Step 4 ：Substituting these values into the formula, we get the margin of error as 0.025526668799512403.
Step 5 ：We then subtract this margin of error from \(p\) to get the lower bound of the confidence interval, which is 0.11447333120048761, and add it to \(p\) to get the upper bound, which is 0.16552666879951242.
Step 6 ：Thus, the estimated proportion of the entire voting population that prefers Candidate A, with a 90% confidence level, is between 0.114 and 0.166. So, \(0.114 < p < 0.166\).
Step 7 ：\(\boxed{0.114 < p < 0.166}\) is the final answer.
