Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. $n=56, x=30 ; 95 \%$ confidence
A. $0.425
Solution
Step 1 ：Given the sample size (n) as 56 and the number of successes (x) as 30, we can calculate the sample proportion (p̂) as \(\frac{x}{n} = \frac{30}{56} = 0.5357142857142857\).
Step 2 ：The Z-score for a 95% confidence level is approximately 1.96.
Step 3 ：We can calculate the standard error (SE) using the formula \(SE = \sqrt{\frac{p̂*(1-p̂)}{n}} = \sqrt{\frac{0.5357142857142857*(1-0.5357142857142857)}{56}} = 0.06664464529394455\).
Step 4 ：Substitute these values into the confidence interval formula to get the lower and upper bounds of the confidence interval: \(p̂ ± Z * SE\).
Step 5 ：For the lower bound, we get \(0.5357142857142857 - 1.96 * 0.06664464529394455 = 0.4050931811757076\).
Step 6 ：For the upper bound, we get \(0.5357142857142857 + 1.96 * 0.06664464529394455 = 0.6663353902528638\).
Step 7 ：\(\boxed{\text{Final Answer: The 95% confidence interval for the population proportion p is } 0.405
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Solution
Step 1 ：Given the sample size (n) as 56 and the number of successes (x) as 30, we can calculate the sample proportion (p̂) as \(\frac{x}{n} = \frac{30}{56} = 0.5357142857142857\).
Step 2 ：The Z-score for a 95% confidence level is approximately 1.96.
Step 3 ：We can calculate the standard error (SE) using the formula \(SE = \sqrt{\frac{p̂*(1-p̂)}{n}} = \sqrt{\frac{0.5357142857142857*(1-0.5357142857142857)}{56}} = 0.06664464529394455\).
Step 4 ：Substitute these values into the confidence interval formula to get the lower and upper bounds of the confidence interval: \(p̂ ± Z * SE\).
Step 5 ：For the lower bound, we get \(0.5357142857142857 - 1.96 * 0.06664464529394455 = 0.4050931811757076\).
Step 6 ：For the upper bound, we get \(0.5357142857142857 + 1.96 * 0.06664464529394455 = 0.6663353902528638\).
Step 7 ：\(\boxed{\text{Final Answer: The 95% confidence interval for the population proportion p is } 0.405
