Tony's Pizza guarantees all pizza deliveries within 30 minutes of the placement of orders. An agency wants to estimate the proportion of all pizzas delivered within 30 minutes by Tony's. What is the most conservative estimate of the sample size that would limit the margin of error to within .02 of the population proportion for a 99% confidence interval?
Margin of error = 0.02
z = 2.58
Unknown p
We are going use 50%
n = (za/2 /E ) ^2 *p(1-p)
n = (2.58/0.02)^2*.5* .5 = 4160.25
n = 4161
4065
This is not correct
To determine the most conservative estimate of the sample size for estimating the proportion of pizzas delivered within 30 minutes by Tony's Pizza, we can use the formula for sample size calculation for estimating proportions.
The formula for sample size calculation is:
n = (Z^2 * p * (1-p)) / (E^2)
Where:
- n is the required sample size
- Z is the Z-score corresponding to the desired confidence level
- p is the estimated proportion of interest (population proportion)
- E is the desired margin of error
In this case, we want to limit the margin of error to within 0.02 (E = 0.02) with a 99% confidence interval.
The most conservative approach for estimating the sample size is to assume the worst case scenario, which is that the population proportion is 0.5 (p = 0.5). This is because having p = 0.5 yields the largest sample size, making it more conservative.
Now, let's solve for the required sample size:
n = (Z^2 * p * (1-p)) / (E^2)
n = (Z^2 * 0.5 * (1-0.5)) / (0.02^2)
n = (Z^2 * 0.5 * 0.5) / 0.0004
n = (Z^2 * 0.25) / 0.0004
To find the Z-score corresponding to a 99% confidence level, we can refer to a standard normal distribution table or use a statistical software. The Z-score for a 99% confidence interval is approximately 2.57.
Now substituting the values into the formula:
n = (2.57^2 * 0.25) / 0.0004
n = 6.6065 / 0.0004
n = 16516.25
Therefore, the most conservative estimate of the sample size that would limit the margin of error to within 0.02 of the population proportion for a 99% confidence interval is approximately 16517.