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Section 9.3: Confidence Intervals for a Population Standard Deviation

Objectives

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

  1. find critical values for the Χ2 distribution
  2. construct and interpret CIs about σ2 and σ

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

The last parameters we need to find confidence intervals for are the population variance (σ2) 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 (Χ2) distribution

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

The Chi-Square (Χ2) distribution

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

chi-square

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

chi-square distribution

Properties of the Χ2 distribution

  1. It is not symmetric.
  2. The shape depends on the degrees of freedom.
  3. As the number of degrees of freedom increases, the distribution becomes more symmetric.
  4. Χ2≥0

Finding Critical Values

Find critical values in the Χ2 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 your textbook. It should look something like the image below (trimmed to make it more viewable).

t-table

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 Χ2 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. Your textbook should also come with a copy you can use.

[ reveal answer ]

chi-square table

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

Finding Critical Values Using StatCrunch

Click on Stat > Calculators > Chi-Square

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

Example 2

Use the technology of your choice to find Χ20.01 with 20 degrees of freedom.

[ reveal answer ]

Χ20.01,20 ≈ 37.566

Constructing Confidence Intervals about σ2 and σ

Now that we have the basics of the distribution of the variable Χ2, we can work on constructing a formula for the confidence interval.

From the distribution shape on the previous page, we know that 1-alpha of the Χ2 values will be between the two critical values shown below.

chi-square distribution

This gives us the following inequality:

formula

If we solve the inequality for σ2, we get the formula for the confidence interval:

A (1-α)100% confidence interval for σ2 is

confidence interval

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:

critical value and critical value

Then the confidence interval is:

calculation

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

  1. Click on Stat > Variance > One sample
  2. Select with data if you have the data, or with summary if you only have the summary statistics.
  3. If you chose with data, click on the variable that you want for the confidence interval. Otherwise, enter the sample statistics.
  4. Click on Next.
  5. Click the Confidence Interval radio button
  6. Enter the desired level of confidence and press Calculate

The confidence interval should be displayed.

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.

[ reveal answer ]

Using StatCrunch, we get the following result:

StatCrunch calculation

So the standard deviation is between calculation and calculation.

 

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