Construct the confidence interval for the population mean

Construct the confidence interval for the population mean .

c = 0.90, $\overline{x}$ s = 17.8, s = 12.0, and n = 110

Solution

Use the Formula E = $z_{c}\frac{\sigma }{\sqrt{n}}$

to find the margin of error, where $z_{c}$ is the z-score corresponding to an area of c,

$\sigma$ is the population standard deviation and

n is the sample size.

When n $\geq$ 30 you can use s in place of $\sigma$ , where s is the sample standard deviation.

$z_{c} = \frac{1}{2}(1-c)$ .

Substitute the value of c and evaluate to find the area in each tail.

$z_{c} = \frac{1}{2}(1-c)$

= $\frac{1}{2}(1-0.90)$ = 0.05

Use a standard normal table to find the positive critical value the corresponds to a tail area of 0.05

$z_{c}$ = 1.645

The population standard deviation $\sigma$ is unknown but because n $\geq$ 30, use s in place of $\sigma$ .

Substitute the value $z_{c} = 1.645, \sigma \approx s \approx$ 12.0, and n = 110 to find E.

E = $z_{c}\frac{\sigma }{\sqrt{n}}$

$1.645\times \frac{12}{\sqrt{110}}$

$\approx$ 1.9

Use the margin of error to find the left endpoint.

Left endpoint = $\overline{x}$ – E

= 17.8 – 1.9 = 15.9

Now find the right endpoint

Right endpoint = $\overline{x}$ + E

= 17.8 + 1.9 = 19.7

Therefore, a 90% confidence interval for $\mu$ is (15.9, 19.7).