Comcast wants to know the percent of their cable subscribers who use the internet at home. How large of a sample size should they survey to determine the proportion who use the internet within 2% and with 90% confidence interval.
Please show formula when computing sample size.
To compute the required sample size, we will use the formula for sample size estimation for proportion:
n = [Z^2 * p * (1 - p)] / E^2
Where:
n = required sample size
Z = critical value for the desired level of confidence (for 90% confidence interval, Z = 1.645)
p = estimated proportion of cable subscribers who use the internet (unknown in this case, but we will assume p = 0.5 for a conservative estimate, as this maximizes the required sample size)
E = margin of error (in this case, E = 0.02)
Now, let's substitute the values into the formula:
n = [1.645^2 * 0.5 * (1 - 0.5)] / 0.02^2
Simplifying the equation:
n = [2.705 * 0.5 * 0.5] / 0.0004
n = (2.705 * 0.25) / 0.0004
n = 0.67625 / 0.0004
n ≈ 1691.25
Rounding up to the nearest whole number:
n = 1692
Therefore, Comcast should survey a sample size of approximately 1692 cable subscribers to determine the proportion who use the internet with a 90% confidence level and a 2% margin of error.
To determine the sample size needed to estimate the proportion of cable subscribers who use the internet at home within a certain margin of error and confidence interval, you can use the formula:
n = (z^2 * p * (1-p)) / E^2
Where:
n = the sample size needed
z = the z-score corresponding to the desired confidence level
p = the estimated proportion of cable subscribers who use the internet at home (if unknown, use 0.5 for a conservative estimate)
E = the desired margin of error (expressed as a proportion)
In this case, the desired confidence level is 90% and the desired margin of error is 2%, which can be expressed as 0.02. The z-score corresponding to a 90% confidence level is approximately 1.645.
Using the formula, we can substitute the known values:
n = (1.645^2 * 0.5 * (1-0.5)) / 0.02^2
Simplifying the equation:
n = (2.705025 * 0.5 * 0.5) / 0.0004
n = 0.676256 / 0.0004
n ≈ 1690.64
Rounding up to the nearest whole number, Comcast would need a sample size of 1691 cable subscribers to estimate the proportion who use the internet at home within a 2% margin of error with a 90% confidence interval.