A researcher wishes to estimate, with 95% confidence, the proportion of adults who have high-speed internet access. Her estimate must be accurate within 4% of the true proportion. A) Find the minimum sample size needed, using a prior study that found that 54% of the respondents said they have high-speed internet access. B) No preliminary estimate is available. Find the minimum sample size needed.
a) What is the minimum sample size needed, using a prior study that found that 54% of the respondents said they have high-speed internet access?
To find the minimum sample size needed using a prior study, we can use the formula for sample size calculation for estimating proportions.
The formula is:
n = (z^2 * p * q) / E^2
Where:
n = required sample size
z = z-value corresponding to the desired level of confidence (for a 95% confidence level, the z-value is approximately 1.96)
p = estimated proportion from the prior study
q = 1 - p
E = the desired margin of error
In this case, p = 0.54 (as stated in the prior study) and E = 0.04 (4% margin of error). Plugging these values into the formula, we get:
n = (1.96^2 * 0.54 * (1 - 0.54)) / 0.04^2
n = (3.8416 * 0.54 * 0.46) / 0.0016
n ≈ 999.97
Therefore, the minimum sample size needed, using a prior study with an estimate of 54%, is approximately 1000.