6.4 Confidence Intervals for Variance and Standard Deviation

Statistics

Mrs. Spitz

Spring 2009

Objectives/Assignment

How to interpret the c hi-square distribution and use a chi-square distribution table.

How to use the chi square distribution to construct a confidence interval for the variance and standard deviation.

ASSIGNMENT: pp. 288-289 #1-18 all

Schedule for coming weeks:

Today – Notes 6.3. Homework due BOC on Friday.

Friday, 1/16/09 – Notes 6.4. Assignment due Tuesday on our return.

Monday – 1/19/09 – No school Tuesday – 1/20/09 – Chapter Review Thursday-Chapter Review 6 DUE – Test –

Chapter 6 Friday – 1/23/09 – 7.1 Hypothesis Testing

Study Tip

The Greek letter Χ is pronounced “ki,” which

rhymes with the more familiar Greek letter .

As sample size increases and d.f. increases, the distribution for a chi-square becomes flatter.

Chi-square distributions are positively skewed.

The Chi Square Distribution

In many manufacturing processes, it is necessary to control the amount that the process varies. For instance, an automobile part manufacturer must produce thousands of parts that can be used in the manufacturing process. It is imperative that the parts vary little or not at all. How can you measure and consequently control, the amount of variation in the car parts? You can start with a point estimate.

Definition

The points estimate for 2 is s2 and the point estimate for is s. s2 is the most unbiased estimate for 2.

Reminder: is sigma for population standard deviation and

2 is population variance

You can use a chi-square distribution to construct a confidence interval for the variance and standard deviation.

Critical Values

There are two critical values for each level of confidence. The value of represents the right-tail critical value and repesents the left-tail critical value. Table 6 in Appendix B lists critical values of X2 for various degrees of freedom and areas. Each area in the table represents the region under the chi-square curve to the RIGHT of the critical value.

RX

2

LX

2

Study Tip

For chi-square critical values with a confidence level, the following value are what you look up in Table 6 in Appendix B.

Study Tip

For chi-square critical values with a confidence level, the following value are what you look up in Table 6 in Appendix B.

Study Tip

For chi-square critical values with a confidence level, the following value are what you look up in Table 6 in Appendix B.

Ex. 1: Finding Critical Values for X2

Find the critical values, and , for a

90% confidence interval when the sample

size is 20.

RX

2

LX

2

SOLUTION

Because the sample size is 20, there are d.f.=

n – 1 = 20 – 1 = 19 degrees of freedom. The

areas to the right of and are:RX

2

LX

2

Area to right of 05.02

90.01

2

12

c

RX

Area to left of 95.02

90.01

2

12

c

LX

Part of Table 6 is shown. Using d.f. = 19 and the areas 0.95 and 0.05, you can find the critical values, as shown by the highlighted areas in the table.

From the table, you c an see that and

So, 90% of the area under the curve lies between 10.117 and 30.144.

144.302

R

X 117.102

L

X

Confidence Intervals

You can use the critical values and

to construct confidence intervals for a population

variance and standard deviation. As you would

expect, the best point estimate for the variance is s2

and the best point estimate for the standard deviation

is s.

RX

2

LX

2

Definition

Ex. 2: Constructing a Confidence Interval You randomly select and weigh 30 samples

of an allergy medication. The sample standard deviation is 1.2 milligrams. Assuming the weights are normally distributed, construct 99% confidence intervals for the population variance and standard deviation.

SOLUTION

The areas to the right of and are: R

X2

LX

2

Area to the right of 005.02

99.01

2

12

c

RX

Area to the left of 995.02

99.01

2

12

c

LX

Using the values n = 30, d.f. = 29 and c = 0.99, the critical values for

RX

2

LX

2and are:

RX

2336.52

LX

2121.13

Using these critical values and s = 1.2, the confidence interval for 2 is as follows:

Solution:

The confidence interval for is :

183.3798.0

78.189.0

So, with 99% confidence, you can say that the population variance is between .798 and 3.183. The population standard deviation is between 0.89 and 1.78 milligrams.