While considering buying advertising spots during television programming on a certain nes-work, an online retailer asks the television network to estimate the percentage uf its viewers who purchase goods online. The television network has several million viewers. The online retailer requires the estimate to have a margin of error of at most three percentage points with a confidence level of 99%. Of the following, which is the smallest sample size that the television network can select to meet the online retailer's requirement? (A) 30 (B) 1070 (C) 1510 (D) 1850 (E) At least 10% of the network's viewers
Question
While considering buying advertising spots during television programming on a certain nes-work, an online retailer asks the television network to estimate the percentage uf its viewers who purchase goods online. The television network has several million viewers. The online retailer requires the estimate to have a margin of error of at most three percentage points with a confidence level of 99%. Of the following, which is the smallest sample size that the television network can select to meet the online retailer's requirement? (A) 30 (B) 1070 (C) 1510 (D) 1850 (E) At least 10% of the network's viewers
Solution
The sample size needed for a survey can be calculated using the formula for the sample size of a proportion:
n = (Z^2 * p * (1-p)) / E^2
where:
- n is the sample size
- Z is the Z-score, which corresponds to the desired confidence level (for a confidence level of 99%, the Z-score is approximately 2.57)
- p is the estimated proportion of the population that has the characteristic of interest (since we don't have this information, we'll use the value that maximizes the product p*(1-p), which is 0.5)
- E is the desired margin of error (in this case, 0.03)
Substituting the values into the formula, we get:
n = (2.57^2 * 0.5 * 0.5) / 0.03^2 = 1842.43
Since we can't have a fraction of a person, we round up to the nearest whole number, which is 1843.
Therefore, the smallest sample size that the television network can select to meet the online retailer's requirement is 1843. This is closest to option (D) 1850.
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