Assume that a sample is used to estimate a population proportion p. Find the 95% confidence interval for a sample of size 342 with 41 successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places.
95% C.I. =
The formula for the 95% confidence interval for a proportion is:
p̂ ± Z * √(p̂(1-p̂)/n)
where p̂ is the sample proportion, Z is the Z-score corresponding to the desired level of confidence (in this case 95%), and n is the sample size.
In this case, the sample proportion (p̂) is given as 41 successes out of a sample size of 342, so p̂ = 41/342 = 0.1201.
We can calculate the Z-score using a table or a calculator. The Z-score for a 95% confidence level is approximately 1.96.
Plugging in the values:
95% C.I. = 0.1201 ± 1.96 * √(0.1201(1-0.1201)/342)
Calculating the square root term:
√(0.1201(1-0.1201)/342) ≈ 0.0234
Plugging in the values:
95% C.I. = 0.1201 ± 1.96 * 0.0234
Calculating the interval:
Lower bound = 0.1201 - 1.96 * 0.0234 ≈ 0.0750
Upper bound = 0.1201 + 1.96 * 0.0234 ≈ 0.1652
Therefore, the 95% confidence interval is (0.075, 0.165).