Question

A fashion designer wants to know how many new dresses women buy each year. A sample of 208 women was taken to study their purchasing habits. Construct the 85% confidence interval for the mean number of dresses purchased each year if the sample mean was found to be 6.2. Assume that the population standard deviation is 1.1.

Answer 1

The 85% confidence interval for the mean number of dresses purchased each year is (6.1, 6.3).

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.85)/(2) = 0.075`

Now, we have to find z in the Ztable as such z has a pvalue of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.075 = 0.925`, so Z = 1.44.

Now, find the margin of error M as such

`M = z(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

`M = 1.44(1.1)/(√(208)) = 0.1`

The lower end of the interval is the sample mean subtracted by M. So it is 6.2 - 0.1 = 6.1

The upper end of the interval is the sample mean added to M. So it is 6.2 + 0.1 = 6.3

The 85% confidence interval for the mean number of dresses purchased each year is (6.1, 6.3).