High School

How many weeks of data must be randomly sampled to estimate the mean weekly sales of a new line of athletic footwear? We want 90% confidence that the sample mean is within $400 of the population mean, and the population standard deviation is known to be $1,200.

Answer :

24 weeks of data must be randomly sampled to estimate the mean weekly sales.

The data is ∝=0.10

[tex]\sigma=\$ 1200, E=\$ 400$[/tex]

The margin of error is given by this formula:

[tex]$$M E=z_{\alpha / 2} \frac{\sigma}{\sqrt{n}}$$[/tex]

And on this case we have that ME =400 and we are interested in order to find the value of n, if we solve n from equation.

[tex]$$\begin{aligned}n & =\left(z_\alpha\right)^2 \times\left(\frac{\sigma}{E}\right)^2 . \\& =(1.645)^2 \times\left(\frac{1200}{400}\right)^2 \\& =2.706025 \times \frac{1440000}{160000} \\& =24.354225 \approx 24 \\n & =24 \\\end{aligned}$$[/tex]

There are 24 weeks of data must be randomly sampled to estimate the mean weekly sales of a new line of athletic footwear.

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