Chapter 8: Problem 57

The 2003 Statistical Abstract of the United States reported the percentage of people 18 years of age and older who smoke. 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 \(.02 ?\) 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? c. What is the \(95 \%\) confidence interval for the proportion of smokers in the population?

Short Answer

Expert verified

The required sample size to estimate the proportion of smokers in the population with a margin of error of 0.02 and 95% confidence level is 2,207. The point estimate of the proportion of smokers in the population is 23.55%. The 95% confidence interval for the proportion of smokers in the population is approximately (0.2143, 0.2567) or (21.43%, 25.67%).

Step by step solution

Part a: Calculate sample size

In order to calculate the sample size needed to estimate the proportion of smokers in the population with a margin of error of 0.02 at 95% confidence level, we need to use the following formula for sample size calculation: \( n = \frac{Z^2 \times p \times (1-p)}{E^2} \) Where: - \(n\) = sample size - \(Z\) = Z-score, which corresponds to the 95% confidence level (1.96) - \(p\) = the preliminary proportion of smokers (0.30) - \(E\) = the margin of error (0.02)

Calculate the sample needed using the formula

\( n = \frac{1.96^2 \times 0.30 \times (1-0.30)}{0.02^2} \) \( n = \frac{3.8416 \times 0.30 \times 0.70}{0.0004} \) \( n = 2206.79 \) To have a whole number as the sample size, we need to round up the result: \( n = 2207 \) Therefore, the sample size required to estimate the proportion of smokers in the population with a margin of error of 0.02 and 95% confidence level is 2,207.

Part b: Point estimate of the proportion of smokers

The study used the recommended sample size calculated in part (a) which was 2,207, and it found 520 smokers. In order to find the point estimate of the proportion of smokers, we need to divide the number of smokers by the total sample size:

Calculate the point estimate

\( \hat{p} = \frac{520}{2207} \) \( \hat{p} = 0.2355 \) The point estimate of the proportion of smokers in the population is 0.2355, or 23.55%.

Part c: 95% confidence interval for the proportion of smokers

To calculate the 95% confidence interval for the proportion of smokers in the population, we will use the following formula: Confidence Interval = \( \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \) Where: - \( \hat{p} \) = point estimate of the proportion of smokers (0.2355) - \( Z \) = Z-score, which corresponds to the 95% confidence level (1.96) - \( n \) = sample size (2207)

Calculate the confidence interval

Confidence Interval = \( 0.2355 \pm 1.96 \times \sqrt{\frac{0.2355 \times (1-0.2355)}{2207}} \) Confidence Interval = \( 0.2355 \pm 1.96 \times \sqrt{\frac{0.2355 \times 0.7645}{2207}} \) Confidence Interval = \( 0.2355 \pm 0.0212 \) Therefore, the 95% confidence interval for the proportion of smokers in the population is approximately (0.2143, 0.2567) or (21.43%, 25.67%).