Question

A psychologist is interested in constructing a 95% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 72 of the 835 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible.

a. With 95% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between __and __

b. If many groups of 835 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About __ percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about __ percent will not contain the true population proportion.

Answer 1

a)

sample proportion, = 0.0862
sample size, n = 835
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.0862 * (1 - 0.0862)/835) = 0.0097

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0097
ME = 0.019

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.0862 - 1.96 * 0.0097 , 0.0862 + 1.96 * 0.0097)
CI = (0.0672 , 0.1052)

With 95% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between 0.0672 and 0.1052

b. If many groups of 835 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About95 percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about 5 percent will not contain the true population proportion.