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 chisquare 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
2
confidence. The value of X R represents
2
X
the right-tail critical value and L 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.
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, X
2
R
and X
2
L
, for a
90% confidence interval when the sample
size is 20.
SOLUTION
 Because the sample size is 20, there are d.f.=
n – 1 = 20 – 1 = 19 degrees of freedom. The
2
2
areas to the right of X
and X L are:
Area to right of
Area to left of
R
2 1  c 1  0.90
X 

 0.05
R
2
2
1  c 1  0.90
X 

 0.95
L
2
2
2
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 X
2
 30.144 and
X
2
 10.117
R
L
So, 90% of the area under the curve lies between 10.117 and
30.144.
Confidence Intervals
 You can use the critical values
X
2
R
and
X
2
L
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.
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 X
2
and X
Area to the right of
R
2 1  c 1  0.99
X 

 0.005
R
2
2
Area to the left of
1  c 1  0.99
X 

 0.995
L
2
2
2
L
2
Using the values n = 30, d.f. = 29 and c = 0.99, the critical values for
X
2
R
and
X
2
L
are:
X
2
R
 52.336
X
2
L
 13.121
are:
Using these critical values and s = 1.2, the
confidence interval for 2 is as follows:
Solution:
 The confidence interval for  is :
0.798    3.183
0.89    1.78
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.