A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if she wishes the estimate to be within five percentage points with 90% confidence, assuming that
(a) she uses the estimates of 22.3% male and 18.2% female from a previous year?
(b) she does not use any prior estimates?
To determine the sample size needed, we can use the formula for sample size estimation for a proportion. The formula is given as:
n = (z^2 * p * (1-p)) / E^2
Where:
n = sample size
z = z-score corresponding to the desired confidence level
p = estimated proportion
E = desired margin of error
Let's calculate the sample size for both scenarios:
(a) With prior estimates:
Given:
Estimated proportion for males (p1) = 22.3% = 0.223
Estimated proportion for females (p2) = 18.2% = 0.182
Desired margin of error (E) = 5% = 0.05
Confidence level = 90% --> z-score = 1.645 (taken from the standard normal distribution table)
Calculating the sample size:
n = (z^2 * p * (1-p)) / E^2
n = (1.645^2 * 0.223 * (1-0.223)) / (0.05^2)
n ≈ 116.2
Therefore, a sample size of at least 117 should be obtained if using prior estimates.
(b) Without prior estimates:
Since we don't have any prior estimates, we can assume a conservative proportion of 50%, which will give us the largest sample size.
Given:
p = 0.5 (assuming equal proportion for males and females)
E = 5% = 0.05
Confidence level = 90% --> z-score = 1.645
Calculating the sample size:
n = (z^2 * p * (1-p)) / E^2
n = (1.645^2 * 0.5 * (1-0.5)) / (0.05^2)
n ≈ 270.4
Therefore, a sample size of at least 271 should be obtained if not using any prior estimates.
To determine the required sample size in each scenario, we can use the formula for estimating sample size needed for a proportion:
n = (Z^2 * p * q) / E^2
Where:
n = required sample size
Z = Z-score (critical value) corresponding to the desired level of confidence
p = estimated proportion
q = 1 - p (complement of the estimated proportion)
E = desired margin of error
(a) Using the estimates of 22.3% male and 18.2% female from the previous year:
Since the previous year's proportions are available, we can use them for our estimation.
For males:
p1 = 0.223 (22.3% expressed as a decimal)
q1 = 1 - 0.223 = 0.777
E = 0.05 (5 percentage points expressed as a decimal)
For females:
p2 = 0.182 (18.2% expressed as a decimal)
q2 = 1 - 0.182 = 0.818
E = 0.05 (5 percentage points expressed as a decimal)
Since we want 90% confidence, the corresponding Z-score is approximately 1.645.
Using the formula, the required sample sizes can be calculated separately for males and females:
For males:
n1 = (1.645^2 * 0.223 * 0.777) / 0.05^2
For females:
n2 = (1.645^2 * 0.182 * 0.818) / 0.05^2
(b) Without any prior estimates:
In the absence of any prior estimates, we assume a conservative estimate of 0.5 for p since it maximizes the required sample size and ensures the largest sample will be collected.
For males:
p1 = 0.5
q1 = 0.5
E = 0.05 (5 percentage points expressed as a decimal)
For females:
p2 = 0.5
q2 = 0.5
E = 0.05 (5 percentage points expressed as a decimal)
Continuing with the formula, we can calculate the required sample sizes:
For males:
n1 = (1.645^2 * 0.5 * 0.5) / 0.05^2
For females:
n2 = (1.645^2 * 0.5 * 0.5) / 0.05^2
Using these formulas, you can calculate the specific sample sizes required for each case.