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-value for the desired confidence level (95% confidence level corresponds to Z-value of 1.96)
P = estimated proportion (94% as the decimal equivalent is 0.94)
E = margin of error (0.07)
Let's plug in the values and calculate:
n = (1.96^2 * 0.94 * (1 - 0.94)) / 0.07^2
n ≈ (3.8416 * 0.94 * 0.06) / 0.0049
n ≈ (0.2312) / 0.0049
n ≈ 47.1836
Since we are looking for the minimum sample size, we round up to the nearest whole number:
n ≈ 48
Therefore, the minimum sample size needed to estimate the population proportion is 48.
None of the given options (A, B, C, or D) match the correct answer of 48.
To find the minimum sample size needed to estimate the population proportion, we can use the formula:
n = (Z^2 * p * q) / E^2
Where:
n = sample size
Z = z-value corresponding to the desired confidence level (in this case, 95% confidence level corresponds to a z-value of approximately 1.96)
p = estimated proportion from the prior study (p hat = 94% = 0.94)
q = 1 - p (the complement of the estimated proportion)
E = margin of error (0.07)
Let's 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 ≈ (0.23035584) / 0.0049
n ≈ 47.01
Since the sample size needs to be a whole number, we round up to the nearest integer. Therefore, the minimum sample size needed is 48.
None of the answer choices provided (39, 45, 43, 44) match the correct minimum sample size of 48.