Final answer:
To be 98% confident that the estimated proportion is within 3 percentage points, the network should survey at least 611 customers.
Step-by-step explanation:
To determine the number of customers the network should survey, we can use the formula:
n = (z^2 * p' * (1 - p')) / E^2
Where:
n is the sample size
z is the z-value corresponding to the desired level of confidence. For 98% confidence, z = 2.33
p' is the estimated sample proportion. In this case, p' = 0.34
E is the maximum error or the desired margin of error. In this case, E = 0.03
Substituting the given values into the formula:
n = (2.33^2 * 0.34 * (1 - 0.34)) / 0.03^2 = 610.6
Rounding up to the next higher value, the network should survey at least 611 customers.
The network should survey approximately 1354 customers to be 98% confident that the estimated proportion of customers under the age of twenty is within 3 percentage points of the true population proportion, using a sample proportion of 34%.
To determine how many customers should be surveyed to be 98% confident that the estimated sample proportion is within 3 percentage points of the true population proportion of customers under the age of twenty, we can use the formula for the margin of error as follows:
E = Z * sqrt[(p'(1 - p') / n)]
where:
E is the desired margin of error,
Z is the z-score corresponding to the confidence level,
p' is the estimated sample proportion, and
n is the sample size.
For a 98% confidence level, the z-score is approximately 2.33. If we use 0.34 as the estimate for the sample proportion (p'), and we want our margin of error (E) to be 0.03 (3 percentage points), we can rearrange the formula to solve for n:
n = p'(1 - p') * (Z / E)^2
n = 0.34(1 - 0.34) * (2.33 / 0.03)^2
n = 0.2244 * (77.67)^2
n = 0.2244 * 6033.6689
n ≈ 1353.8
Therefore, the network should survey approximately 1354 customers (rounded up to the next higher value) to maintain the desired confidence level and margin of error.