Mathematics: Suppose the number of square feet per house is normally distributed. If the population standard deviation is 155 square feet, what minimum sample size is needed to be 90% confident that the sample mean is within 47 square feet of the true population mean

Suppose the number of square feet per house is normally

normarismendoza

21.01.2022

The minimum sample size to be 90% confident that the sample mean is within 47 square feet of the true population mean is 30.

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:

$\alpha = \frac{1-0.9}{2} = 0.05$

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 1-0.05 = 0.95 , so z = 1.645

Now, find the margin of error M as such

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

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

What minimum sample size is needed to be 90% confident that the sample mean is within 47 square feet of the true population mean

This is n when $\sigma = 155, M = 47$ . So

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

$47 = 1.645*\frac{155}{\sqrt{n}}$

$47\sqrt{n} = 1.645*155$

$\sqrt{n} = \frac{1.645*155}{47}$

$(\sqrt{n})^{2} = (\frac{1.645*155}{47})^{2}$

n = 29.4

Rounding up

The minimum sample size to be 90% confident that the sample mean is within 47 square feet of the true population mean is 30.

0,0(0 оценок)