Question

Suppose the researcher wants to estimate the mean with a margin of error of no more than $20 with 99% confidence. This could be challenging, though, because the typical American tends to spend anywhere from $0 to $328. What is the minimum number of people he should survey

Answer 1

To estimate the mean with a margin of error of no more than $20 with 99% confidence, a minimum of 39 people should be surveyed.

Step-by-step explanation:

To estimate the mean with a margin of error of no more than $20 with 99% confidence, we need to consider the formula for sample size:

n = (Z * σ) / E

Where:

• n = sample size
• Z = z-score for the desired confidence level (z-score for 99% confidence is approximately 2.33)
• σ = standard deviation (in this case, $328)
• E = margin of error (in this case, $20)

By plugging in the given values into the formula, we can solve for n:

n = (2.33 * 328) / 20 ≈ 38.19

Since the number of people surveyed should be a whole number, we round up to the nearest whole number:

Minimum number of people to survey is 39.

Answer 2

To estimate the mean with a margin of error of no more than $20 with 99% confidence, the researcher needs to survey a minimum of 2667 people.

Step-by-step explanation:

To estimate the mean with a margin of error of no more than $20 with 99% confidence, the researcher needs to determine the minimum sample size. The formula to calculate the minimum sample size is:

n = (Z * σ / E)²

where n is the sample size, Z is the Z-score corresponding to the desired confidence level (in this case, 99% confidence corresponds to a Z-score of 2.576), σ is the population standard deviation, and E is the desired margin of error (in this case, $20).

Since we don't have the population standard deviation, we can use a conservative estimate based on the range of spending habits mentioned. The maximum range of spending is $328 - $0 = $328. Assuming a worst-case scenario where the population standard deviation is half the range, we can use σ = $328 / 2 = $164.

Substituting the values into the formula:

n = (2.576 * $164 / $20)² = 2666.39 ≈ 2667

Therefore, the minimum number of people the researcher should survey is 2667 to estimate the mean with a margin of error of no more than $20 with 99% confidence.