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 (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)?
( , )
Stop now!!
Go back and add YOUR OWN THOUGHTS to each of your umpteen assignments. Then a tutor might be able to help you.
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 .29.
To answer these questions, we'll use the formula for sample size calculation and the formula for confidence interval calculation.
a. To determine the sample size needed to estimate the proportion of smokers with a desired margin of error, we can use the following formula:
n = (z^2 * p * (1-p)) / E^2
where:
- n is the sample size needed
- z is the z-score corresponding to the desired confidence level (95% confidence corresponds to a z-score of approximately 1.96)
- p is the preliminary estimate of the proportion who smoke (0.30)
- E is the desired margin of error (0.02)
Plugging in these values into the formula, we get:
n = (1.96^2 * 0.30 * (1-0.30)) / 0.02^2
Simplifying this calculation gives us:
n ≈ 1068.4444
Rounding up to the nearest whole number, the sample size needed is 1069.
b. The point estimate of the proportion of smokers in the population can be calculated as the number of smokers in the sample divided by the sample size. In this case, the number of smokers is given as 520, and the sample size is 1069. So, the point estimate is:
Point estimate = 520 / 1069 ≈ 0.4861 (rounded to 4 decimals)
c. To calculate the confidence interval for the proportion of smokers in the population, we can use the following formula:
CI = (p̂ - z * √((p̂ * (1-p̂)) / n), p̂ + z * √((p̂ * (1-p̂)) / n))
where:
- CI is the confidence interval
- p̂ is the point estimate of the proportion of smokers (0.4861)
- z is the z-score corresponding to the desired confidence level (1.96)
- n is the sample size (1069)
Plugging in these values into the formula, we get:
CI = (0.4861 - 1.96 * √((0.4861 * (1 - 0.4861)) / 1069), 0.4861 + 1.96 * √((0.4861 * (1 - 0.4861)) / 1069))
Simplifying this calculation gives us:
CI ≈ (0.4480, 0.5241) (rounded to 4 decimals)
Therefore, the 95% confidence interval for the proportion of smokers in the population is approximately (0.4480, 0.5241).