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Inferences About Population Variances

- Inference about a Population Variance

- Inferences about the Variances of Two Populations

- The U.S. General Accounting
Office (GAO) evaluators studied

a Department of Interior program

established to help clean up the

nation’s rivers and lakes.

- The audits reviewed sample data on the oxygen content, the pH level, and the amount of suspended solids in the effluent.

- The hypothesis test was conducted about the variance in pH level for the population of effluent. The population variance in pH level expected at a properly functioning plant.
- In this chapter you will learn how to conduct statistical inferences about the variances of one and two populations.

- Chi-Square Distribution(2)
- Interval Estimation of 2
- Hypothesis Testing

- The chi-square distribution is the sum of
- squared standardized normal random
- variables such as

- The chi-square distribution is based on
- samplingfrom a normal population.

- Probability density function

- where
- Mean: k
- Variance: 2k

- The sampling distribution of (n - 1)s2/ 2
- has a chi-square distribution whenever a simple
- random sample of sizenis selected from a
- normal population.

- We can use the chi-square distribution to
- developinterval estimates and conduct hypothesis
- tests about a population variance.

Examples of Sampling Distribution of (n - 1)s2/ 2

With 2 degrees

of freedom

With 5 degrees

of freedom

With 10 degrees

of freedom

0

- We will use the notation to denote the value for the chi-square distribution that provides an area of a to the right of the stated value.

- For example, there is a .95 probability of obtaining a (chi-square) value such that

Chi-Square Distribution

- A Chi-Square Distribution with 19 Degrees of Freedom

- Selected Values form the Chi-Square Distribution Table

Interval Estimation of 2

.025

.025

95% of the

possible 2 values

2

0

Interval Estimation of 2

- There is a (1 –a) probability of obtaining a c2value such that

- Substituting (n– 1)s2/s 2 for the c2 we get

- Performing algebraic manipulation we get

- Interval Estimate of a Population Variance

where thevalues are based on a chi-square

distribution withn - 1 degrees of freedom and

where 1 -is the confidence coefficient.

- Interval Estimate of a Population Standard Deviation

Taking the square root of the upper and lower

limits of the variance interval provides the

confidenceinterval for the population standard

deviation.

- Example: Buyer’s Digest (A)

Buyer’s Digest rates thermostats manufactured

for home temperature control. In a recent test, 10

thermostats manufactured by The rmoRitewere

selected and placed in a test room that

was maintained at a temperature of 68oF.

The temperature readings of the ten thermostats

are shown on the next slide.

Interval Estimation of 2

- Example: Buyer’s Digest (A)

We will use the 10 readings below to

develop a 95% confidence interval

estimate of the population variance.

Thermostat1 2 3 4 5 6 7 8 9 10

Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

Our value

Interval Estimation of 2

For n - 1 = 10 - 1 = 9 d.f. and a = .05

Selected Values from the Chi-Square Distribution Table

Interval Estimation of 2

For n - 1 = 10 - 1 = 9 d.f. and a =.05

.025

Area in

Upper Tail

= .975

2

0

2.700

Our value

Interval Estimation of 2

For n - 1 = 10 - 1 = 9 d.f. and a = .05

Selected Values from the Chi-Square Distribution Table

Interval Estimation of 2

n - 1 = 10 - 1 = 9 degrees of freedom and a = .05

Area in Upper

Tail = .025

.025

2

0

19.023

2.700

- Sample variance s2 provides a point estimate of 2.

- A 95% confidence interval for the population variance is given by:

.33 <2 < 2.33

whereis the hypothesized value

for the population variance

- Left-Tailed Test

- Hypotheses

- Test Statistic

RejectH0if

whereis based on a chi-square

distribution withn- 1 d.f.

Hypothesis TestingAbout a Population Variance

- Left-Tailed Test (continued)

- Rejection Rule

Critical value approach:

p-Value approach:

Reject H0if p-value<a

whereis the hypothesized value

for the population variance

Hypothesis TestingAbout a Population Variance

- Right-Tailed Test

- Hypotheses

- Test Statistic

RejectH0if

whereis based on a chi-square

distribution withn - 1 d.f.

Hypothesis TestingAbout a Population Variance

- Right-Tailed Test (continued)

- Rejection Rule

Critical value approach:

RejectH0if p-value<a

p-Value approach:

whereis the hypothesized value

for the population variance

Hypothesis TestingAbout a Population Variance

- Two-Tailed Test

- Hypotheses

- Test Statistic

RejectH0if

whereandare based on a

chi-square distribution withn - 1 d.f.

Hypothesis TestingAbout a Population Variance

- Two-Tailed Test (continued)

- Rejection Rule

Critical value approach:

p-Valueapproach:

RejectH0ifp-value<a

- Example: Buyer’s Digest (B)

Recall that Buyer’s Digest is rating

ThermoRite thermostats. Buyer’s Digest

gives an “acceptable” rating to a thermo-

stat with a temperature variance of 0.5

or less.

We will conduct a hypothesis test (with

a= .10) to determine whether the ThermoRite

thermostat’s temperature variance is “acceptable”.

Hypothesis TestingAbout a Population Variance

- Example: Buyer’s Digest (B)

Using the 10 readings, we will

conduct a hypothesis test (witha= .10)

to determine whether the ThermoRite

thermostat’s temperature variance is

“acceptable”.

Thermostat1 2 3 4 5 6 7 8 9 10

Temperature67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

- Hypotheses

- Rejection Rule

RejectH0ifc 2>14.684

Our value

Hypothesis TestingAbout a Population Variance

For n - 1 = 10 - 1 = 9 d.f. and a = .10

Selected Values from the Chi-Square

Distribution Table

Hypothesis TestingAbout a Population Variance

- Rejection Region

Area in Upper

Tail = .10

2

14.684

0

Reject H0

- Test Statistic

The sample variances2= 0.7

- Conclusion

Becausec2= 12.6 is less than 14.684, we cannotrejectH0. The sample variances2= .7 is insufficientevidence to conclude that the temperature variancefor ThermoRitethermostats is unacceptable.

- Using the p-Value

- The rejection region for the ThermoRite
- thermostat example is in the upper tail; thus, the
- appropriate p-value is less than .90 (c2 = 4.168)
- and greater than .10 (c2 = 14.684).

- Because the p –value > a = .10, we cannot
- reject the null hypothesis.

- The sample variance of s2 = .7 is insufficient
- evidence to conclude that the temperature
- variance is unacceptable (>.5).

- Summary

Hypothesis Testing About theVariances of Two Populations

- One-Tailed Test

- Hypotheses

Denote the population providing the

larger sample variance as population 1.

- Test Statistic

Hypothesis Testing About theVariances of Two Populations

- One-Tailed Test (continued)

- Rejection Rule

Critical value approach:

Reject H0if F>F

where the value ofFis based on an

Fdistribution withn1- 1 (numerator)

andn2 - 1 (denominator) d.f.

p-Valueapproach:

RejectH0 ifp-value<a

Hypothesis Testing About theVariances of Two Populations

- Two-Tailed Test

- Hypotheses

Denote the population providing the

larger sample variance as population 1.

- Test Statistic

Hypothesis Testing About theVariances of Two Populations

- Two-Tailed Test (continued)

- Rejection Rule

Critical value approach:

RejectH0if F>F/2

where the value ofF/2 is based on an

Fdistribution withn1- 1 (numerator)

andn2 - 1 (denominator) d.f.

p-Valueapproach:

RejectH0ifp-value<a

F Distribution

F Distribution

- Probability density function
- where

F Distribution

- mean
- for d2 > 2
- Variance
- ford2 > 2

- Selected Values From the F Distribution Table

Hypothesis Testing About theVariances of Two Populations

- Example: Buyer’s Digest (C)

Buyer’s Digest has conducted the

same test, as was described earlier, on

another 10 thermostats, this time

manufactured by TempKing. The

temperature readings of the ten

thermostats are listed on the next slide.

Hypothesis Testing About theVariances of Two Populations

- Example: Buyer’s Digest (C)

We will conduct a hypothesis test with= .10 to see if the variances are equal for ThermoRite’s thermostats and TempKing’s thermostats.

Hypothesis Testing About theVariances of Two Populations

- Example: Buyer’s Digest (C)

ThermoRite Sample

Thermostat 1 2 3 4 5 6 7 8 9 10

Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

TempKing Sample

Thermostat 1 2 3 4 5 6 7 8 9 10

Temperature 67.7 66.4 69.2 70.1 69.5 69.7 68.1 66.6 67.3 67.5

Hypothesis Testing About theVariances of Two Populations

- Hypotheses

(TempKing and ThermoRite

thermostats have thesame

temperature variance)

(Their variances are not equal)

- Rejection Rule

- TheFdistribution table (on next slide) shows that withwith= .10, 9 d.f. (numerator), and 9 d.f. (denominator),F.05= 3.18.

Reject H0 if F> 3.18

Hypothesis Testing About theVariances of Two Populations

Selected Values from the F Distribution Table

= 1.768/.700 = 2.53

Hypothesis Testing About theVariances of Two Populations

- Test Statistic

- TempKing’s sample variance is 1.768
- ThermoRite’s sample variance is .700

Hypothesis Testing About theVariances of Two Populations

- Conclusion

- We cannot rejectH0. F= 2.53 < F.05= 3.18.
- There is insufficient evidence to conclude that
- the population variances differ for the two
- thermostat brands.

Hypothesis Testing About theVariances of Two Populations

- Determining and Using the p-Value

Area in Upper Tail .10 .05 .025 .01

FValue (df1 = 9, df2 = 9) 2.44 3.18 4.03 5.35

- BecauseF= 2.53 is between 2.44 and 3.18, the area
- in the upper tail of the distribution is between .10
- and .05.

- But this is a two-tailed test; after doubling the
- upper-tail area, thep-valueis between .20 and .10.

- Becausea= .10, we have p-value > aand therefore
- we cannot reject the null hypothesis.

- Summary