Question

A fashion designer wants to know how many new dresses women buy each year. Assume a previous study found the variance to be 2.89. She thinks the mean is 5.6 dresses per year. What is the minimum sample size required to ensure that the estimate has an error of at most 0.11 at the 98% level of confidence?

Answer 1

The minimum sample required = 1296.65

Explanation:

Given that:

Variance
`\sigma^2 = 2.89`

Standard deviation
`\sigma = √(2.89)`

Standard deviation
`\sigma = 1.7`

Margin of error = 0.11

Confidence Interval = 98%

Level of significance = 1 - 0.98 = 0.02

The critical value =
`Z _(\alpha//2) = Z_(0.02/2) = Z_(0.01)`

= 2.33

Thus, the minimum sample size is given by the formula:

`n = \bigg ( (Z_(\alpha/2) * \sigma )/(E) \bigg)^2`

`n = \bigg ((2.33 * 1.7 )/(0.11) \bigg)^2`

n = 1296.65