# 8.2.1.1 - Formulas

Earlier in this lesson we considered confidence intervals for proportions and the multiplier in our intervals was a value from the standard normal (i.e., *z*) distribution. But, what if our variable of interest is a quantitative variable and we want to estimate a population mean?

We apply similar techniques when constructing a confidence interval for a mean, but now we are interested in estimating the population mean (\(\mu\)) by using the sample statistic (\(\overline{x}\)) and the multiplier is a *t* value. Similar to the *z* values that you used as the multiplier for constructing confidence intervals for population proportions, here you will use *t* values as the multipliers. Because *t* values vary depending on the number of degrees of freedom (df), you will need to use statistical software to look up the appropriate *t* value for each confidence interval that you construct. The degrees of freedom will be based on the sample size. Since we are working with one sample here, \(df=n-1\).

### Minitab Express - Finding t* Multipliers

To find the t* multiplier for a 98% confidence interval with 15 degrees of freedom:

- On a
**PC**: Select**STATISTICS > Distribution Plot**

On a**Mac**: Select**Statistics > Probability Distributions > Distribution Plot** - Select
*Display Probability* - For
*Distribution*select*t* - For
*Degrees of freedom*enter 15 - The default is to shade the area for
*a specified probability* - Select
*Equal tails* - For
*Probability*enter 0.02 (if there is 0.98 in the middle, then 0.02 is split equally between the left and right tails) - Click OK

This should result in output similar to the output below. Note that your results may be slightly different due to random sampling variation.

**Video Review- No sound**

Let’s review some of symbols and equations that we learned in previous lessons:

Sample size | \(n\) |

Population mean | \(\mu=\frac{\sum X}{N}\) |

Sample mean | \(\overline{x}= \frac{\sum x}{n}\) |

Standard error of the mean | \(SE=\frac{s}{\sqrt{n}}\) |

Multiplier | \(t^{*} \) |

Degrees of freedom (one group) | \(df=n-1\) |

Recall the general form for a confidence interval:

General Form of Confidence Interval

sample statistic \(\pm\) (multiplier) (standard error)

When constructing a confidence interval for a population mean the point estimate is the sample mean, \(\overline{x}\). The multiplier is taken from a *t* distribution. And, the standard error is equal to \(\frac{s}{\sqrt{n}}\).

**Confidence Interval for a Population Mean**

\[\overline{x} \pm t^{*} \frac{s}{\sqrt{n}}\]

On the following pages we will walk through examples of constructing confidence intervals for population means by hand. Then, you will learn how to compute confidence intervals using Minitab Express.