1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.

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IS 310 – Business StatisticsIS 310 – Business Statistics

IS 310

Business Statistic

sCSU

Long Beach

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Inferences About Population VariancesInferences About Population Variances

Previously, we have made inferences about population Previously, we have made inferences about population meansmeans and and proportionsproportions. We can make similar . We can make similar inferences about population inferences about population variancesvariances..

Examples:Examples:

A manufacturing plant produces a part whose length should be A manufacturing plant produces a part whose length should be 1.2”. Some 1.2”. Some variation of length variation of length is allowed. We want to is allowed. We want to determine that the variation of length is no more than 0.05”.determine that the variation of length is no more than 0.05”.

The amount of rainfall varies from state to state. A researcher The amount of rainfall varies from state to state. A researcher wants to know if the wants to know if the variation in the amount of rainfall variation in the amount of rainfall in a in a specific state is less than 3”.specific state is less than 3”.

In these cases, we are dealing with population variances (not In these cases, we are dealing with population variances (not means or proportions)means or proportions)

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Inferences About Population VariancesInferences About Population Variances

Inference about a Population VarianceInference about a Population Variance

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Inferences About a Population VarianceInferences About a Population Variance

Chi-Square DistributionChi-Square Distribution

Interval EstimationInterval Estimation

Hypothesis TestingHypothesis Testing

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Inferences About a Population VarianceInferences About a Population Variance

In studying population variances, the following In studying population variances, the following quantity is importantquantity is important

2 22 2

(n – 1) s / (n – 1) s / σσ

This quantity follows a distribution that is called chi-square This quantity follows a distribution that is called chi-square distribution (with n – 1 degree of freedom)distribution (with n – 1 degree of freedom)

As the degree of freedom increases with sample size, the shape of As the degree of freedom increases with sample size, the shape of the chi-square distribution resembles that of a normal the chi-square distribution resembles that of a normal

distributiondistribution..

See Figure 11.1 (10-page 437; 11-page 450)See Figure 11.1 (10-page 437; 11-page 450)

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Examples of Sampling Distribution of (Examples of Sampling Distribution of (nn - - 1)1)ss22//22

00

With 2 degreesWith 2 degrees of freedomof freedomWith 2 degreesWith 2 degrees of freedomof freedom

2

2

( 1)n s



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The chi-square distribution can be used The chi-square distribution can be used

o To estimate a population variance o To estimate a population variance

o Perform hypothesis tests about populationo Perform hypothesis tests about population

variancesvariances

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2 2 2.975 .025 2 2 2.975 .025


For example, there is a .95 probability of For example, there is a .95 probability of obtaining a obtaining a 22 (chi-square) value such that (chi-square) value such that

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

2 2

2 2

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95% of thepossible 2 values 95% of thepossible 2 values

22

00

.025.025

2.0252.025

.025.025

2.9752.975

Interval Estimation of Interval Estimation of 22

22 2.975 .0252

( 1)n s

2

2 2.975 .0252

( 1)n s

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( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

2 2 2(1 / 2) / 2 2 2 2(1 / 2) / 2

22 2(1 / 2) / 22

( 1)n s

2

2 2(1 / 2) / 22

( 1)n s

Substituting (Substituting (nn – 1) – 1)ss22//22 for the for the 22 we get we get

Performing algebraic manipulation we getPerforming algebraic manipulation we get

There is a (1 – There is a (1 – ) probability of obtaining a ) probability of obtaining a 22 valuevalue

such thatsuch that

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Interval Estimate of a Population VarianceInterval Estimate of a Population Variance


( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

where the where the values are based on a chi-squarevalues are based on a chi-square

distribution with distribution with nn - 1 degrees of freedom and - 1 degrees of freedom and

where 1 - where 1 - is the confidence coefficient. is the confidence coefficient.

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Interval Estimation of Interval Estimation of

Interval Estimate of a Population Standard DeviationInterval Estimate of a Population Standard Deviation

Taking the square root of the upper and lowerTaking the square root of the upper and lower

limits of the variance interval provides the confidencelimits of the variance interval provides the confidence

interval for the population standard deviation.interval for the population standard deviation.

2 2

2 2/ 2 (1 / 2)

( 1) ( 1)n s n s

2 2

2 2/ 2 (1 / 2)

( 1) ( 1)n s n s

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Buyer’s Digest rates thermostatsBuyer’s Digest rates thermostats

manufactured for home temperaturemanufactured for home temperature

control. In a recent test, 10 thermostatscontrol. In a recent test, 10 thermostats

manufactured by ThermoRite weremanufactured by ThermoRite were

selected and placed in a test room thatselected and placed in a test room that

was maintained at a temperature of 68was maintained at a temperature of 68ooF.F.

The temperature readings of the ten thermostats The temperature readings of the ten thermostats areare

shown on the next slide. shown on the next slide.


Example: Buyer’s Digest (A)Example: Buyer’s Digest (A)

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We will use the 10 readings below toWe will use the 10 readings below todevelop a 95% confidence intervaldevelop a 95% confidence intervalestimate of the population variance.estimate of the population variance.

Example: Buyer’s Digest (A)Example: Buyer’s Digest (A)

Temperature Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.267.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

ThermostatThermostat 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10

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Degrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper TailDegrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper Tail


Selected Values from the Chi-Square Distribution TableSelected Values from the Chi-Square Distribution Table

Our Our value value

Our Our value value

2.9752.975

For For nn - 1 = 10 - 1 = 9 d.f. and - 1 = 10 - 1 = 9 d.f. and = .05= .05

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22

00

.025.025

2

2.0252

( 1)2.700

n s

2

2.0252

( 1)2.700

n s

Area inArea inUpper TailUpper Tail

= .975= .975

2.7002.700


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Degrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209


of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper Tail


Selected Values from the Chi-Square Distribution TableSelected Values from the Chi-Square Distribution Table


Our Our value value

Our Our value value 2

.025 2

.025

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22

00

.025.025

2.7002.700


nn - 1 = 10 - 1 = 9 degrees of freedom and - 1 = 10 - 1 = 9 degrees of freedom and = .05= .05

2

2

( 1)2.700 19.023

n s

2

2

( 1)2.700 19.023

n s

19.02319.023

Area in UpperArea in UpperTail = .025Tail = .025Area in UpperArea in UpperTail = .025Tail = .025

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Sample variance Sample variance ss22 provides a point estimate of provides a point estimate of 22..

sx xni2

2

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70

( ) .

.sx xni2

2

16 39

70

( ) .

.

( )..

( )..

10 1 7019 02

10 1 702 70

2 ( ).

.( ).

.10 1 70

19 0210 1 70

2 702


.33 .33 << 2 2 << 2.33 2.33

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

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Left-Tailed TestLeft-Tailed Test

Hypothesis TestingHypothesis TestingAbout a Population VarianceAbout a Population Variance

22

02

1 ( )n s

22

02

1 ( )n s

where is the hypothesized valuewhere is the hypothesized valuefor the population variancefor the population variance

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•Test StatisticTest Statistic

•HypothesesHypotheses2 2

0 0: H 2 20 0: H

2 20: aH 2 20: aH

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Left-Tailed Test (continued)Left-Tailed Test (continued)


Reject Reject HH00 if if pp-value -value << pp-Value approach:-Value approach:

Critical value approach:Critical value approach:•Rejection RuleRejection Rule

Reject Reject HH00 if if 2 2(1 ) 2 2(1 )

where is based on a chi-squarewhere is based on a chi-squaredistribution with distribution with nn - 1 d.f. - 1 d.f.

2(1 ) 2(1 )

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Right-Tailed TestRight-Tailed Test


H02

02: H0

202:

Ha : 202Ha : 202

22

02

1 ( )n s

22

02

1 ( )n s


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•HypothesesHypotheses

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Right-Tailed Test (continued)Right-Tailed Test (continued)


Reject Reject HH00 if if 2 2 2 2

Reject Reject HH00 if if pp-value -value <<

22where is based on a chi-squarewhere is based on a chi-square

distribution with distribution with nn - 1 d.f. - 1 d.f.

pp-Value approach:-Value approach:


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Two-Tailed TestTwo-Tailed Test


22

02

1 ( )n s

22

02

1 ( )n s


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•HypothesesHypotheses

Ha : 202Ha : 202

H02

02: H0

202:

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Two-Tailed Test (continued)Two-Tailed Test (continued)


Reject Reject HH00 if if pp-value -value <<

pp-Value approach:-Value approach:


2 2 2 2(1 / 2) / 2 or 2 2 2 2(1 / 2) / 2 or Reject Reject HH00 if if

where are based on awhere are based on achi-square distribution with chi-square distribution with nn - 1 d.f. - 1 d.f.

2 2(1 / 2) / 2 and 2 2(1 / 2) / 2 and

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Recall that Buyer’s Digest is ratingRecall that Buyer’s Digest is rating

ThermoRite thermostats. Buyer’s DigestThermoRite thermostats. Buyer’s Digest

gives an “acceptable” rating to a thermo-gives an “acceptable” rating to a thermo-

stat with a temperature variance of 0.5stat with a temperature variance of 0.5

or less.or less.


Example: Buyer’s Digest (B)Example: Buyer’s Digest (B)

We will conduct a hypothesis test (withWe will conduct a hypothesis test (with

= .10) to determine whether the ThermoRite= .10) to determine whether the ThermoRite

thermostat’s temperature variance is “acceptable”.thermostat’s temperature variance is “acceptable”.

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Using the 10 readings, we willUsing the 10 readings, we will

conduct a hypothesis test (with conduct a hypothesis test (with = .10) = .10)

to determine whether the ThermoRiteto determine whether the ThermoRite

thermostat’s temperature variance isthermostat’s temperature variance is

““acceptable”.acceptable”.

Example: Buyer’s Digest (B)Example: Buyer’s Digest (B)

Temperature Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.267.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

ThermostatThermostat 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10

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HypothesesHypotheses2

0 : 0.5H 20 : 0.5H

2: 0.5aH 2: 0.5aH


Reject Reject HH00 if if 22 >> 14.684 14.684

Rejection RuleRejection Rule

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Degrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209


of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper TailSelected Values from the Chi-Square Distribution TableSelected Values from the Chi-Square Distribution Table



Our Our value value Our Our value value 2

.10 2

.10

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22

00 14.68414.684

Area in UpperArea in UpperTail = .10Tail = .10Area in UpperArea in UpperTail = .10Tail = .10


Rejection RegionRejection Region

2 22

2

( 1) 9.5

n s s

2 2

22

( 1) 9.5

n s s

Reject Reject HH00Reject Reject HH00

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Test StatisticTest Statistic

2 9(.7)12.6

.5 2 9(.7)

12.6.5


Because Because 22 = 12.6 is less than 14.684, we cannot = 12.6 is less than 14.684, we cannot

reject reject HH00. The sample variance . The sample variance ss22 = .7 is insufficient = .7 is insufficient

evidence to conclude that the temperature varianceevidence to conclude that the temperature variance

for ThermoRite thermostats is unacceptable.for ThermoRite thermostats is unacceptable.

ConclusionConclusion

The sample variance The sample variance s s 22 = 0.7 = 0.7

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Using the Using the pp-Value-Value

• The sample variance of The sample variance of s s 22 = .7 is = .7 is insufficient evidence to conclude that theinsufficient evidence to conclude that the temperature variance is unacceptable (>.5).temperature variance is unacceptable (>.5).

• Because the Because the pp –value > –value > = .10, we = .10, we cannot reject the null hypothesis.cannot reject the null hypothesis.

• The rejection region for the ThermoRite The rejection region for the ThermoRite thermostat example is in the upper tail; thus, thethermostat example is in the upper tail; thus, the appropriate appropriate pp-value is less than .90 (-value is less than .90 (22 = 4.168) = 4.168) and greater than .10 (and greater than .10 (22 = 14.684). = 14.684).


A precise A precise pp-value-valuecan be found usingcan be found usingMinitab or Excel.Minitab or Excel.

A precise A precise pp-value-valuecan be found usingcan be found usingMinitab or Excel.Minitab or Excel.

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Sample ProblemSample Problem

Problem # 4 (10-Page 443; 11-Page 457)Problem # 4 (10-Page 443; 11-Page 457)

Given: 2Given: 2

n = 18 s = 0.36n = 18 s = 0.36

The 90% confidence interval estimate for population variance is:The 90% confidence interval estimate for population variance is:

2 2 2 2 22 2 2 2 2

(n – 1) s / (n – 1) s / ΧΧ ≤ ≤ σσ ≤ (n – 1) s / ≤ (n – 1) s /ΧΧ

.05 .95.05 .95

[(18 – 1) (0.36)/27.587 < < [(18 – 1) (0.36)/8.672)[(18 – 1) (0.36)/27.587 < < [(18 – 1) (0.36)/8.672)

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Sample ProblemSample Problem

Problem # 9 (10-Page 445; 11-Page 459)Problem # 9 (10-Page 445; 11-Page 459)

22

Given: n = 30 s = 0.0005 = 0.05Given: n = 30 s = 0.0005 = 0.05

2 22 2

H : H : σσ ≤ 0.0004 H : ≤ 0.0004 H : σσ > 0.0004 > 0.0004

0 a0 a

2 2 22 2 2

Test statistic, Test statistic, ΧΧ = (n – 1) s / = (n – 1) s / σσ = (29) (.0005) / .0004 = 36.25 = (29) (.0005) / .0004 = 36.25

00

2 22 2

Reject Null Hypothesis if Reject Null Hypothesis if ΧΧ > > ΧΧ

2 2 0.05, 292 2 0.05, 29

Since Since ΧΧ (=36.25) < (=36.25) < ΧΧ (=42.557), we do not reject the Null (=42.557), we do not reject the Null HypothesisHypothesis

0. 05, 29 0. 05, 29

Test indicates that the population variance specification is not being violated.Test indicates that the population variance specification is not being violated.

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End of Chapter 11End of Chapter 11

1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.

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