The confidence interval for the population mean when $LaTeX: \sigma$ is unknown.

X must be normally distributed or the sample size must be at least 30.

The confidence interval for the population mean when the population standard deviation is unknown is a symmetric type encountered in the previous unit.

estimate - E<parameter< estimate + E

The interval is symmetric about the estimate for the population mean $LaTeX: \bar{x}$ because the t- distribution is symmetric about zero, just like the standard normal distribution.

$LaTeX: E=t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}$ , where $LaTeX: t_{\frac{\alpha}{2}}$ is the critical value from the t-distribution with df=n-1, s is the sample standard deviation, and n is the sample size.

The interval is

$LaTeX: \bar{x}-t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}<\mu<\bar{x}+t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}$

Example

The website for UN Gender Statistics contains data on 52 quantitative and 11 qualitative indicators addressing relevant issues related to gender equality and /or women's empowerment.

A simple random sample of 41 women ages 15 and over were asked number of hours spent each day on domestic chores. The sample mean hours was 3.9 hours and the sample standard deviation was 1.3 hours. Construct a 90% confidence interval for the mean hours spend on domestic chores by women ages 15 and over.

Solution

n = 41, (greater than 30), population standard deviation is unknown. Use the t interval.

$LaTeX: \bar{x} = 3.9, s=1.3, \alpha=0.1$ .

Now we need the critical value, d.f.=n-1=41-1=40. We find critical values in our calculator, but here we will find it using a table. See below.

t table to find critical values for degrees of freedom equal to 40 and alpha equal to .1

$LaTeX: t_{\frac{\alpha}{2}}=1.684$

Plug in value into the formula

$LaTeX: \bar{x}-t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}<\mu<\bar{x}+t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}$

$LaTeX: 3.9-1.684\cdot\frac{1.3}{\sqrt{41}}<\mu<3.9+1.684\cdot\frac{1.3}{\sqrt{41}}$

$LaTeX: 3.6<\mu<4.2$

We are 90% confident that the number of hours spent on domestic chores by women ages 15 and older is between 3.6 and 4.2 hours.