A researcher would like to estimate the population proportion of adults living in a certain town who have at least a high school education. No information is available about its value. How large a sample size is needed to estimate it to within 0.19 with 99% confidence?
n=
(Round up to the nearest integer)
To determine the required sample size, we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = required sample size
Z = Z-value for the desired confidence level (99% confidence corresponds to a Z-value of 2.576)
p = estimated proportion (since no information is available, we can assume a conservative estimate of 0.5)
E = desired margin of error (0.19 in this case)
Plugging in the values, we have:
n = (2.576^2 * 0.5 * (1-0.5)) / 0.19^2
n = (6.656 * 0.25) / 0.0361
n = 1.664 / 0.0361
n ≈ 46.09
Rounding up to the nearest integer, the required sample size is 47. Therefore, a sample size of 47 is needed to estimate the population proportion of adults in the town with at least a high school education, with a margin of error of 0.19 at a 99% confidence level.
To determine the sample size needed to estimate a population proportion with a given level of confidence and margin of error, we can use the formula:
n = (Z^2 * p * (1 - p)) / E^2
Where:
- n is the required sample size
- Z is the z-score corresponding to the desired level of confidence
- p is the estimated population proportion (since we don't have any information, we can use 0.5 as a conservative estimate)
- E is the desired margin of error
In our case, the desired level of confidence is 99%, so the z-score corresponding to a 99% confidence level is approximately 2.58 (from standard normal distribution). The desired margin of error is 0.19.
Plugging in the values into the formula:
n = (2.58^2 * 0.5 * (1 - 0.5)) / 0.19^2
Calculating this expression:
n = (6.6564 * 0.25) / 0.0361
n = 1.6641 / 0.0361
n ≈ 46.07
Since the sample size must be a whole number, we round up to the nearest integer:
n = 47
Therefore, a sample size of 47 is needed to estimate the population proportion to within 0.19 with 99% confidence.
Formula to find sample size:
n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for 99% confidence, p = .50 (when no value is stated in the problem), q = 1 - p, ^2 means squared, * means to multiply, and E = 0.19.
Plug values into the formula and calculate n.
I hope this will help get you started.