Question

how many weeks of data must be randomly sampled to estimate the mean weekly sales of a new line of athletic footwear? We want 98% confidence that the sample mean is within $500 of the population mean, and the population standard deviation is known to be $1300

Answer 1

We need to sample at least 37 weeks of data.

Explanation:

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

`\alpha = (1-0.98)/(2) = 0.01`

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

So it is z with a pvalue of
`1-0.01 = 0.99`, so
`z = 2.327`

Now, find the margin of error M as such

`M = z*(\sigma)/(√(n))`

We want 98% confidence that the sample mean is within $500 of the population mean, and the population standard deviation is known to be $1300

This is at least n weeks, in which n is found when
`M = 500, \sigma = 1300`

So

`M = z*(\sigma)/(√(n))`

`500 = 2.327*(1300)/(√(n))`

`500√(n) = 2.327*1300`

`√(n) = (2.327*1300)/(500)`

`(√(n))^(2) = ((2.327*1300)/(500))^(2)`

`n = 36.6`

Rounding up

We need to sample at least 37 weeks of data.