6.4 Confidence Intervals for Variance and Standard Deviation

• Key Concepts:– Point Estimates for the Population Variance and

Standard Deviation– Chi-Square Distribution– Building and Interpreting Confidence Intervals for

the Population Variance and Standard Deviation

6.4 Confidence Intervals for Variance and Standard Deviation

• How do we estimate the population variance or the population standard deviation using sample data?– The variation we see in the sample will be our best

guess.• the sample variance, s2, is used to estimate σ2

• the sample standard deviation, s, is used to estimate σ

• To build confidence intervals for σ2 and σ, we start with the sampling distribution of a modified version of s2.

6.4 Confidence Intervals for Variance and Standard Deviation

• If we find all possible samples of size n from a normal population of size N and then record the value of

for each sample, it can be shown that follows a chi-square distribution with n – 1 degrees of freedom.

2 22

1ns

2

6.4 Confidence Intervals for Variance and Standard Deviation

• Properties of the chi-square distribution:– All chi-square vales are greater than or equal to zero.– The shape of a chi-square curve is determined by the number

of degrees of freedom.– The area below a chi-square curve is 1.– All chi-square curves are positively skewed.

• Practice working with chi-square curves#4 p. 334#6

6.4 Confidence Intervals for Variance and Standard Deviation

• How do we build confidence intervals using this information?

We can start with:

and use algebra to get to:

2 2 22

1L R

ns

2 22

2 2

1 1

R L

n s n s

6.4 Confidence Intervals for Variance and Standard Deviation

• Fortunately, we can use the previous result for both confidence intervals.– To build a confidence interval for the population variance, we

use:

– To build a confidence interval for the population standard deviation, we use:

2 22

2 2

1 1

R L

n s n s

2 2

2 2

1 1

R L

n s n s

6.4 Confidence Intervals for Variance and Standard Deviation

• Guidelines for constructing these confidence intervals are provided on page 332.– Remember the population must be normal for us to

apply these techniques.– When building our confidence intervals, we need

the chi-square curve with n – 1 degrees of freedom.• Practice:

#14 p. 334 (Cough Syrup)#16 p. 334 (Washers)#23 p. 335 (Waiting Times)