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 .32.
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)?
To answer these questions, we need to use the formula for sample size calculation and confidence interval for proportions. Let's break down the steps to determine the answers:
a. To determine the sample size required to estimate the proportion of smokers with a margin of error of 0.02 and a 95% confidence level, we can use the following formula:
n = (Z^2 * P * (1 - P)) / E^2
where:
n = sample size
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96)
P = preliminary estimate of the proportion who smoke
E = margin of error
Plugging in the values:
n = (1.96^2 * 0.32 * (1 - 0.32)) / 0.02^2
Calculating this expression will give us the required sample size. Let's do the calculations:
n = (3.8416 * 0.32 * 0.68) / 0.0004
n ≈ 8318.72
Since the sample size must be a whole number, we round up to the nearest whole number:
n ≈ 8319
Therefore, a sample size of approximately 8319 should be taken to estimate the proportion of smokers in the population with a margin of error of 0.02.
b. Now, let's calculate the point estimate of the proportion of smokers in the population using the given sample size of 520. The point estimate can be calculated by dividing the number of smokers (520) by the sample size (8319):
Point estimate = 520 / 8319 ≈ 0.0625 (rounded to 4 decimals)
Therefore, the point estimate of the proportion of smokers in the population is approximately 0.0625.
c. To calculate the 95% confidence interval for the proportion of smokers in the population, we can use the following formula:
CI = Point estimate ± (Z * SE)
where:
CI = Confidence interval
Point estimate = Proportion of smokers calculated in part (b)
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96)
SE = Standard error
First, let's calculate the standard error using the formula:
SE = sqrt((P * (1 - P)) / n)
Plugging in the values:
SE = sqrt((0.32 * 0.68) / 8319)
SE ≈ sqrt(0.2176 / 8319)
SE ≈ sqrt(0.000026159)
SE ≈ 0.005115
Now, we can calculate the confidence interval:
CI = 0.0625 ± (1.96 * 0.005115)
Calculating the expression will give us the lower and upper bounds of the confidence interval. Let's do the calculations:
CI ≈ 0.0625 ± 0.0100284
Rounded to 4 decimals:
Lower bound ≈ 0.0525
Upper bound ≈ 0.0725
Therefore, the 95% confidence interval for the proportion of smokers in the population is approximately 0.0525 to 0.0725.