8.2.2.1 - 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, $$df=n-1$$.

MinitabExpress – Finding t* Multipliers

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

1. On a PC: Select STATISTICS > Distribution Plot
On a Mac: Select Statistics > Probability Distributions > Distribution Plot
2. Select Display Probability
3. For Distribution select $$t$$
4. For Degrees of freedom enter 15
5. The default is to shade the area for a specified probability
6. Select Equal tails
7. 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)
8. 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 Walkthrough

Select your operating system below to see a step-by-step guide for this example.

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

Sample size $$n$$ $$\mu=\frac{\sum X}{N}$$ $$\overline{x}= \frac{\sum x}{n}$$ $$SE=\frac{s}{\sqrt{n}}$$ $$t^{*}$$ $$df=n-1$$

Recall the general form for a confidence interval:

General Form of Confidence Interval
$$sample\ statistic\pm\underbrace{(multiplier)\ (standard\ error)}_{\textbf{margin of error}}$$

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
$$\underbrace{\overline{x}}_{\text{sample statistic}} \pm \overbrace{t^{*}}^{\text{multiplier}} \underbrace{ \dfrac{s}{\sqrt{n}}}_{\text{standard error}}$$

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.