# Section 9.3: Confidence Intervals for a Population Standard Deviation

## Objectives

By the end of this lesson, you will be able to...

- find critical values for the
*Χ*^{2}distribution - construct and interpret CIs about
*σ*^{2}and*σ*

For a quick overview of this section, watch this short video summary:

The last parameters we need to find confidence intervals for are the population variance (*σ*^{2}) and standard deviation (*σ*).

There are many instances where we might be interested in knowing something about the spread of a population based on a sample. For example, we might believe that a particular group of students seems to have a wider variation in their grades than those from the past. Or a part manufacturer may be concerned that one of the parts it's manufacturing is too inconsistent, even though the mean may be at specifications.

Before we can develop a confidence interval for the variance, we need another distribution.

## The Chi-Square (*Χ*^{2}) distribution

Note: "chi-square" is pronounced "kai" as in sky, not "chai" like the tea.

#### The Chi-Square (*Χ*^{2}) distribution

If a simple random sample size n is obtained from a normally distributed population with mean *μ* and standard deviation *σ*, then

has a **chi-square distribution** with n-1 degrees of freedom.

#### Properties of the *Χ*^{2} distribution

- It is
*not*symmetric. - The shape depends on the degrees of freedom.
- As the number of degrees of freedom increases, the distribution becomes more symmetric.
*Χ*^{2}≥0

## Finding Critical Values

Find critical values in the *Χ*^{2} distribution using a table is done in the same manner we found critical values for the t-distribution.

Before we start the section, you need a copy of the table. You can download a printable copy of this table, or use the table in the back of a textbook. It should look something like the image below (trimmed to make it more viewable).

Notice that the table is similar to the t-table, with probabilities along the top and critical values in the middle. This is because we primarily use the chi-square-table to find critical values.

Let's try an example.

Example 1

Find the critical values in the *Χ*^{2} distribution which separate the middle 95% from the 2.5% in each tail, assuming there are 12 degrees of freedom.

You can use the table above, or print one out yourself. Any textbook should also come with a copy you can use.

We can see form the table that the two critical values are 4.404 and 23.337.

### Finding Critical Values Using StatCrunch

Click on Enter the degrees of freedom, the direction of the inequality, and the probability (leave X blank). Then press |

Example 2

Use the technology of your choice to find *Χ*^{2}_{0.01} with 20 degrees of freedom.

*Χ*^{2}_{0.01,20} ≈ 37.566

## Constructing Confidence Intervals about *σ*^{2} and *σ*

Now that we have the basics of the distribution of the variable *Χ*^{2}, we can work on constructing a formula for the confidence interval.

From the distribution shape on the previous page, we know that of the *Χ*^{2} values will be between the two critical values shown below.

This gives us the following inequality:

If we solve the inequality for *σ*^{2}, we get the formula for the confidence interval:

A **(1-α)100% confidence interval for σ^{2}** is

Note: The sample *must* be taken from a normally distributed population.

Note #2: If a confidence interval for *σ* is desired, we can take the square root of each part.

Now that we have the confidence interval formula, let's try a couple examples.

Example 3

Suppose a sample of 30 ECC students are given an IQ test. If the sample has a standard deviation of 12.23 points, find a 90% confidence interval for the population standard deviation.

**Solution:** We first need to find the critical values:

and

Then the confidence interval is:

So we are 90% confident that the standard deviation of the IQ of ECC students is between 10.10 and 15.65 bpm.

### Finding Confidence Intervals Using StatCrunch

- Click on
**Stat**>**Variance**>**One Sample** - Select
**With Data**if you have the data, or**With Summary**if you only have the summary statistics. - If you chose With Data, click on the variable that you want for the confidence interval. Otherwise, enter the sample statistics.
- Click on the Confidence Interval radio button and enter your confidence level if it is not 95%.
- Click
**Compute**.
Note: If you need a confidence interval about the population standard deviation, take the square root of the values in the resulting confidence interval. |

Here's one for you to try:

Example 4

In Example 3 in Section 9.1, we assumed the standard deviation of the resting heart rates of students was 10 bpm.

heart rate |
||||

61 | 63 | 64 | 65 | 65 |

67 | 71 | 72 | 73 | 74 |

75 | 77 | 79 | 80 | 81 |

82 | 83 | 83 | 84 | 85 |

86 | 86 | 89 | 95 | 95 |

(Click here to view the data in a format more easily copied.)

Use StatCrunch to find a 95% confidence interval for the standard deviation of the resting heart rates for students in this particular class.

Using StatCrunch, we get the following result:

So the standard deviation is between and .