Question

The number of square feet per house are normally distributed with a population standard deviation of 197 square feet and an unknown population mean. If a random sample of 25 houses is taken and results in a sample mean of 1820 square feet, find a 99% confidence interval for the population mean.

Answer 1

Explanation:

(1718.51, 1921.49)

Answer 2

The 99% confidence interval for the population mean is betwen 1718.55 square feet and 1921.45 square feet.

Explanation:

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

`\alpha = (1-0.99)/(2) = 0.005`

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
`1-0.005 = 0.995`, so
`z = 2.575`

Now, find 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 = 2.575*(197)/(√(25)) = 101.45`

The lower end of the interval is the sample mean subtracted by M. So it is 1820 - 101.45 = 1718.55 square feet

The upper end of the interval is the sample mean added to M. So it is 1820 + 101.45 = 1921.45 square feet.

The 99% confidence interval for the population mean is betwen 1718.55 square feet and 1921.45 square feet.