Stats Homework chapter 8

Question

Assume that a sample is used to estimate a population proportion p. Find the 90% confidence interval for a sample of size 322 with 177 successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places

90% C.I. = 

Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

Answer 1

confidence intervals for proportions are determined as p-hat -/+ z(alpha/2) x sqrt((p-hat x q-hat)/n), where p-hat=proportion of successes and q-hat=1-p-hat=proportion of failures.

For a 90% confidence interval z(alpha/2) = 1.645.

p-hat=177/322 and q-hat=1-p-hat=(322-177)/322=145/322

Let X = (p-hat x q-hat)/n = ((177/322) x (145/322)) / 322

z(alpha/2) x sqrt(X) = 1.645 x (0.027725974) = 0,045609228

Thus the 95% CI, to 3 decimal places, = 0.550 -/+ 0.046 = (0.504, 0.596)