Use the given information to find the minimum sample size required to estimate an unknown population mean μ.

How many weeks of data must be randomly sampled to estimate the mean weekly sales of a new line of athletic footwear? We want 99% confidence that the sample mean is within $200 of the population mean, and the population standard deviation is known to be $1400.

Question

Use the given information to find the minimum sample size required to estimate an unknown population mean μ.

How many weeks of data must be randomly sampled to estimate the mean weekly sales of a new line of athletic footwear? We want 99% confidence that the sample mean is within $200 of the population mean, and the population standard deviation is known to be $1400.

BlackTom AI · Accepted Answer

We need the minimum sample size n to estimate a population mean with known sigma at a given confidence level and margin of error.

First, identify the formula for known-variance case: n = (z * sigma / E)^2, where z is the z-score for the desired confidence level, sigma is the population standard deviation, and E is the desired margin of error.

Given data:
- Confidence level = 99% → two-tailed, z ≈ 2.576 (commonly used value around 2.58).
- sigma = $1400
- E = $200

Compute the numerator: z * sigma = 2.576 * 1400 ≈ 3606.4.
Then divide by E: (z * sigma) / E = 3606.4 / 200 ≈ 18.032.
Square to get n: n ≈ (18.032)^2 ≈ 325.15.
So the minimum integer n that guarantees the margin is not larger than $200 is the ceiling of 325.15, which is about 326.

Now evaluate the provided options against this calculation:
- 133: This is far below 326, so it would not achieve the desired margin of error.
- 325: This is very close to the computed 325.15. Interpreted strictly, it would yield a margin of error just slightly larger than $200 (since 325 < 326), so it technically may not meet the 99% confidence/margin requirement.
- 189: Also well below the 326 threshold, thus insufficient.
- 267: Still below the needed size; would not guarantee the specified precision.

A quick sense-making note: since the precise calculation yields about 326, the closest option is 325, which would not quite meet the requirement if you enforce n to be an integer that guarantees the margin. In many contexts people round up to the next integer, producing 326, which is not among the choices.

In summary, each smaller option fails to meet the required precision, and 325 is the closest approximation to the calculated minimum, though it may not strictly satisfy the 99%/$,200 margin criterion if you enforce the ceiling to guarantee the specification.

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