Question

# The Centers for Disease Control reported the percentage of people 18 years of age and older...

The Centers for Disease Control reported the percentage of people 18 years of age and older who smoke (CDC website, December 14, 2014). Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of .30.

a. How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02 (to the nearest whole number)? Use 95% confidence.

b. Assume that the study uses your sample size recommendation in part (a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population (to 4 decimals)?

c. What is the 95% confidence interval for the proportion of smokers in the population (to 4 decimals)?

a)
The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.02

The provided estimate of proportion p is, p = 0.3
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.3*(1 - 0.3)*(1.96/0.02)^2
n = 2016.84

Ans : Sample size, n = 2017

b)
point estimate, pcap = 520/2017 = 0.2578

c)
sample proportion, pcap = 0.2578
sample size, n = 2017
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.2578 * (1 - 0.2578)/2017) = 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

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.2578 - 1.96 * 0.0097 , 0.2578 + 1.96 * 0.0097)
CI = (0.2388 , 0.2768)