Out of 400 people sampled, 144 preferred Candidate A. Round to three decimals. Based on this estimate, what proportion (as a decimal) of the voting population ( $p$ ) prefers Candidate A?
.47
Compute a $99 \%$ confidence interval, and give your answers to 3 decimal places.
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Solution
Step 1 ：Given that 144 out of 400 people sampled preferred Candidate A, we can calculate the proportion of the voting population that prefers Candidate A by dividing the number of people who preferred Candidate A by the total number of people sampled. This gives us \(p = \frac{144}{400} = 0.36\).
Step 2 ：We are asked to compute a 99% confidence interval for this proportion. The formula for a confidence interval for a proportion is given by \(p \pm z \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion, \(n\) is the sample size, and \(z\) is the z-score corresponding to the desired level of confidence. For a 99% confidence interval, the z-score is approximately 2.576.
Step 3 ：Substituting the given values into the formula, we get \(0.36 \pm 2.576 \sqrt{\frac{0.36(1-0.36)}{400}}\).
Step 4 ：Solving this gives us a confidence interval of \([0.298, 0.422]\).
Step 5 ：Final Answer: The proportion of the voting population that prefers Candidate A is \(0.36\). The 99% confidence interval for this proportion is \(\boxed{[0.298, 0.422]}\).
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Solution
Step 1 ：Given that 144 out of 400 people sampled preferred Candidate A, we can calculate the proportion of the voting population that prefers Candidate A by dividing the number of people who preferred Candidate A by the total number of people sampled. This gives us \(p = \frac{144}{400} = 0.36\).
Step 2 ：We are asked to compute a 99% confidence interval for this proportion. The formula for a confidence interval for a proportion is given by \(p \pm z \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion, \(n\) is the sample size, and \(z\) is the z-score corresponding to the desired level of confidence. For a 99% confidence interval, the z-score is approximately 2.576.
Step 3 ：Substituting the given values into the formula, we get \(0.36 \pm 2.576 \sqrt{\frac{0.36(1-0.36)}{400}}\).
Step 4 ：Solving this gives us a confidence interval of \([0.298, 0.422]\).
Step 5 ：Final Answer: The proportion of the voting population that prefers Candidate A is \(0.36\). The 99% confidence interval for this proportion is \(\boxed{[0.298, 0.422]}\).
