Find the minimum sample size needed to estimate the population proportion if the margin of error is 0.07 confidence level is 95% and from a prior study p hat is estimated by decimal equivalent of 94%
A) 39 b)45 c)43 d)44
To find the minimum sample size needed to estimate the population proportion, we can use the formula:
n = (Z^2 * p * (1-p)) / (E^2)
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level
p = estimated proportion from the prior study (p-hat)
E = margin of error
Given:
Margin of error (E) = 0.07
Confidence level = 95% (which corresponds to a Z-score of approximately 1.96)
Estimated proportion (p-hat) = 0.94
Plugging in these values into the formula, we get:
n = (1.96^2 * 0.94 * (1-0.94)) / (0.07^2)
n ≈ 43.434
Since we cannot have a fractional sample size, we round up to the nearest whole number.
Therefore, the minimum sample size needed is 44 (option d).
To find the minimum sample size needed to estimate the population proportion, you can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-value corresponding to the desired confidence level (in this case, Z-value for 95% confidence level is approximately 1.96)
p = estimated proportion from prior study (in this case, p = 0.94)
E = margin of error (in this case, E = 0.07)
Plug in the values into the formula:
n = (1.96^2 * 0.94 * (1-0.94)) / 0.07^2
n ≈ (3.8416 * 0.94 * 0.06) / 0.0049
n ≈ (3.2334904 * 0.06) / 0.0049
n ≈ 0.1940094 / 0.0049
n ≈ 39.593
Since the sample size should be a whole number, you need to round up the result to the nearest whole number.
Therefore, the minimum sample size needed to estimate the population proportion is 40.
Since none of the options given include 40, the closest option is 39 (option A).