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

Answer:

We need to sample at least 37 weeks of data.

Step-by-step explanation:

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

$\alpha = \frac{1-0.98}{2} = 0.01$

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

$M = z*\frac{\sigma}{\sqrt{n}}$

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

$M = z*\frac{\sigma}{\sqrt{n}}$

$500 = 2.327*\frac{1300}{\sqrt{n}}$

$500\sqrt{n} = 2.327*1300$

$\sqrt{n} = \frac{2.327*1300}{500}$

$(\sqrt{n})^{2} = (\frac{2.327*1300}{500})^{2}$

$n = 36.6$

Rounding up

We need to sample at least 37 weeks of data.