Question

A market research company wishes to know how many energy drinks teenagers drink each week. they want to construct a 99% 99 % confidence interval with an error of no more than 0.07 0.07 . a consultant has informed them that a previous study found the mean to be 3.4 3.4 energy drinks per week and found the variance to be 0.49 0.49 . what is the minimum sample size required to create the specified confidence interval? round your answer up to the next integer.

Answer 1

The minimum sample size required to construct a 99% confidence interval with an error of no more than 0.07, given a variance of 0.49, is 706 teenagers after rounding up.

Step-by-step explanation:

The student's question is about finding the minimum sample size required to construct a 99% confidence interval for the mean number of energy drinks teenagers drink each week, with a maximum error of 0.07 and known variance of 0.49. We must use the formula for the sample size (n) in a confidence interval which involves the Z-score associated with the confidence level, the standard deviation of the population (σ), and the desired error bound (E).

The formula is: n = (Z * σ / E)2

First, we need to find the Z-score for a 99% confidence level, which is the value from the Z-distribution such that 99% of the distribution is to the left of it. This value is typically around 2.576.

Substituting the known values into the formula we get: n = (2.576 * sqrt(0.49) / 0.07)2, then we calculate n and round up to the next integer.

By performing the calculation, we find that the minimum sample size required is:

n = (2.576 * 0.7 / 0.07)2 = (2.576 * 10)2 = 26.5762 ≈ 706

Therefore, the market research company needs a minimum sample size of 706 teenagers to create the specified confidence interval.