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 .25.
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. To estimate the sample size needed to estimate the proportion of smokers in the population with a margin of error of 0.02 and a 95% confidence level, we can use the formula for sample size calculation for proportions:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (in this case, 1.96 for a 95% confidence level)
p = preliminary estimate of the proportion who smoke (given as 0.25)
E = margin of error (given as 0.02)
Plugging in the values into the formula:
n = (1.96^2 * 0.25 * (1-0.25)) / 0.02^2
n = (3.8416 * 0.25 * 0.75) / 0.0004
n = 0.7206 / 0.0004
n ≈ 1801.5
Rounding up to the nearest whole number, a sample size of 1802 should be taken.
b. The point estimate of the proportion of smokers in the population would be the number of smokers in the sample divided by the sample size:
Point estimate = Number of smokers / Sample size
Point estimate ≈ 520 / 1802
Point estimate ≈ 0.2881 (rounded to 4 decimals)
c. To find the 95% confidence interval for the proportion of smokers in the population, we can use the formula:
Confidence interval = Point estimate ± (Z * standard error)
Where the standard error is calculated as:
Standard error = sqrt((point_estimate * (1 - point_estimate)) / sample_size)
Plugging in the values:
Standard error = sqrt((0.2881 * (1 - 0.2881)) / 1802)
Standard error ≈ sqrt(0.207924 / 1802)
Standard error ≈ sqrt(0.0001154)
Standard error ≈ 0.010747
Now, substituting the values into the confidence interval formula:
Confidence interval = 0.2881 ± (1.96 * 0.010747)
Confidence interval ≈ 0.2881 ± 0.021 (rounded to 4 decimals)
Therefore, the 95% confidence interval for the proportion of smokers in the population ranges from approximately 0.2671 to 0.3091 (rounded to 4 decimals).
a. To determine the sample size needed to estimate the proportion of smokers in the population with a margin of error of 0.02 and a confidence level of 95%, we can use the formula:
n = (Z^2 * p * q) / E^2
Where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (in this case, 95% or 1.96)
- p is the preliminary estimate of the proportion who smoke
- q is the complementary probability of p (1-p)
- E is the desired margin of error
Given:
- Preliminary estimate of the proportion who smoke (p) = 0.25
- Margin of error (E) = 0.02
Substituting these values into the formula:
n = (1.96^2 * 0.25 * 0.75) / 0.02^2
Solving this equation will give us the sample size required. Let's calculate it:
n = (3.8416 * 0.25 * 0.75) / 0.0004
n = 0.7194 / 0.0004
n = 1798.5
Since the sample size needs to be a whole number, we round up to the nearest whole number. Therefore, the required sample size is approximately 1799.
b. With a sample size of 1799 and 520 smokers, we can calculate the point estimate of the proportion of smokers in the population using the formula:
Point Estimate = (Number of Successes) / (Sample Size)
Point Estimate = 520 / 1799
Point Estimate ≈ 0.289
Therefore, the point estimate of the proportion of smokers in the population is approximately 0.289 (or 28.9%).
c. To calculate the 95% confidence interval for the proportion of smokers in the population, we can use the formula:
CI = Point Estimate ± (Z * SE)
Where:
- CI is the confidence interval
- Point Estimate is the proportion of smokers obtained from part (b)
- Z is the Z-score corresponding to the desired confidence level (in this case, 95% or 1.96)
- SE is the standard error, which can be calculated as SE = sqrt((p*q) / n)
Given:
- Point estimate of proportion of smokers = 0.289
- Z-score for 95% confidence level (Z) = 1.96
- Sample size (n) = 1799
Calculating the standard error:
SE = sqrt((0.25 * 0.75) / 1799)
SE ≈ 0.00912
Substituting these values into the confidence interval formula:
CI = 0.289 ± (1.96 * 0.00912)
CI ≈ 0.289 ± 0.0179
The 95% confidence interval for the proportion of smokers in the population is approximately 0.2711 to 0.3069.