The comparison of two populations

Slide 1

The Comparison of Two Populations By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Shakeel NoumanM.Phil Statistics

The Comparison of Two Populations

Slide 2

• Using Statistics• Paired-Observation Comparisons• A Test for the Difference between Two

Population Means Using Independent Random Samples

• A Large-Sample Test for the Difference between Two Population Proportions

• The F Distribution and a Test for the Equality of Two Population Variances

• Summary and Review of Terms

The Comparison of Two Populations8


Slide 3

• Inferences about differences between parameters of two populationsPaired-ObservationsObserve the same group of persons or things

• At two different times: “before” and “after”• Under two different sets of circumstances or “treatments”

Independent Samples» Observe different groups of persons or things

• At different times or under different sets of circumstances

8-1 Using Statistics


Slide 4

• Population parameters may differ at two different times or under two different sets of circumstances or treatments because:The circumstances differ between times or

treatmentsThe people or things in the different groups are

themselves different• By looking at paired-observations, we are able to

minimize the “between group” , extraneous variation.

8-2 Paired-Observation Comparisons


Slide 5

freedom. of degrees 1)-(on with distributi t a has statistic the

, is differencemean population theand trueis hypothesis

null When the.hypothesis null under the differencemean

population theis symbol The ns.observatio of pairs of

number theis , size, sample theand s,difference theseof

deviation standard sample theis s ns,observatio ofpair

eachbetween difference average sample theis D where

: t testnsobservatio-paired for the statisticTest

0

0

0

D

n

n

n

s

Dt

D

D

D

D

Paired-Observation Comparisons of Means


Slide 6

A random sample of 16 viewers of Home Shopping Network was selected for an experiment. All viewers in the sample had recorded the amount of money they spent shopping during the holiday season of the previous year.

The next year, these people were given access to the cable network and were asked to keep a record of their total purchases during the holiday season. Home Shopping Network managers want to test the null hypothesis that

their service does not increase shopping volume, versus the alternative hypothesis that it does.

A random sample of 16 viewers of Home Shopping Network was selected for an experiment. All viewers in the sample had recorded the amount of money they spent shopping during the holiday season of the previous year.

The next year, these people were given access to the cable network and were asked to keep a record of their total purchases during the holiday season. Home Shopping Network managers want to test the null hypothesis that

their service does not increase shopping volume, versus the alternative hypothesis that it does.

Shopper Previous Current Diff 1 334 405 71 2 150 125 -25 3 520 540 20 4 95 100 5 5 212 200 -12 6 30 30 0 7 1055 1200 145 8 300 265 -35 9 85 90 510 129 206 7711 40 18 -2212 440 489 4913 610 590 -2014 208 310 10215 880 995 11516 25 75 50

Shopper Previous Current Diff 1 334 405 71 2 150 125 -25 3 520 540 20 4 95 100 5 5 212 200 -12 6 30 30 0 7 1055 1200 145 8 300 265 -35 9 85 90 510 129 206 7711 40 18 -2212 440 489 4913 610 590 -2014 208 310 10215 880 995 11516 25 75 50

H0: D 0H1: D > 0

df = (n-1) = (16-1) = 15

Test Statistic:

Critical Value: t0.05 = 1.753

Do not reject H0 if : t 1.753 Reject H0 if: t > 1.753

H0: D 0H1: D > 0

df = (n-1) = (16-1) = 15

Test Statistic:

Critical Value: t0.05 = 1.753

Do not reject H0 if : t 1.753 Reject H0 if: t > 1.753

t

D D

sD

n

0

Example 8-1


Slide 7

t

D D

sD

n

0 32 81 0

55 75

16

2 354.

..

2.131= t0.025

2.602= t0.01

1.753= t0.05

2.354=test

statistic

50-5

0.4

0.3

0.2

0.1

0.0t

f(t)

t Distribution: df=15

Nonrejection Region

Rejection Region

t = 2.354 > 1.753, so H0 is rejected and we conclude that there is evidence that shopping volume by network

viewers has increased, with a p-value between 0.01 an 0.025. The Template output gives a more exact p-value of 0.0163. See the next slide for the output.

t = 2.354 > 1.753, so H0 is rejected and we conclude that there is evidence that shopping volume by network

viewers has increased, with a p-value between 0.01 an 0.025. The Template output gives a more exact p-value of 0.0163. See the next slide for the output.

Example 8-1: Solution


Slide 8Example 8-1: Template for Testing Paired Differences


Slide 9

It has recently been asserted that returns on stocks may change once a story about a company appears in The Wall Street Journal column “Heard on the Street.” An investments analyst collects a random sample of 50 stocks that were recommended as winners by the editor of “Heard on the Street,” and proceeds to conduct a two-tailed test of whether or not the annualized return on stocks recommended in the column differs between the month before and the month after the recommendation. For each stock the analysts computes the return before and the return after the event, and computes the difference in the two return figures. He then computes the average and standard deviation of the differences.

It has recently been asserted that returns on stocks may change once a story about a company appears in The Wall Street Journal column “Heard on the Street.” An investments analyst collects a random sample of 50 stocks that were recommended as winners by the editor of “Heard on the Street,” and proceeds to conduct a two-tailed test of whether or not the annualized return on stocks recommended in the column differs between the month before and the month after the recommendation. For each stock the analysts computes the return before and the return after the event, and computes the difference in the two return figures. He then computes the average and standard deviation of the differences.

H0: D 0H1: D > 0

n = 50D = 0.1%sD = 0.05%

Test Statistic:

z

D D

sD

n

0

This test result is highly significant,and H0 may be rejected at any reasonable

level of significance.

p - value:

z

D D

sD

n

p z

0 0 1 0

0 05

50

14 14

14 14 0

.

..

( . )

Example 8-2


Slide 10

instead. .

2z usemay welarge, is size sample When theright, its to

2 of area

an off cuts that freedom of degrees 1)-(non with distributi t theof value theis 2

twhere

2tD

:D

differencemean for the interval confidence 100% )-(1A

n

Ds

Confidence Intervals for Paired Observations


Slide 11

0. value theincludenot does interval confidence that thisNote

]114.0,086.0[014.01.0

)0071)(.96.1(1.050

0.051.960.12

zD

: 28 Examplein data for the interval confidence 95%

nDs

Confidence Intervals for Paired Observations – Example 8-2


Slide 12

Confidence Intervals for Paired Observations – Example 8-2 Using

the Template


Slide 13

• When paired data cannot be obtained, use independent random samples drawn at different times or under different circumstances.Large sample test if:

» Both n1 30 and n2 30 (Central Limit Theorem), or» Both populations are normal and 1 and 2 are both

known

Small sample test if:» Both populations are normal and 1 and 2 are

unknown

8-3 A Test for the Difference between Two Population Means Using Independent Random

Samples


Slide 14

• I: Difference between two population means is 0 1= 2

» H0: 1 -2 = 0» H1: 1 -2 0

• II: Difference between two population means is less than 0 1 2

» H0: 1 -2 0» H1: 1 -2 0

• III: Difference between two population means is less than D 1 2+D

» H0: 1 -2 D» H1: 1 -2 D

Comparisons of Two Population Means: Testing Situations


Slide 15

Large-sample test statistic for the difference between two population means:

The term (1- 2)0 is the difference between 1 an 2 under the null hypothesis. Is is equal to zero in situations I and II, and it is equal to the prespecified value D in situation III. The term in the denominator is the standard deviation of the difference between the two sample means (it relies on the assumption that the two

samples are independent).

Large-sample test statistic for the difference between two population means:

The term (1- 2)0 is the difference between 1 an 2 under the null hypothesis. Is is equal to zero in situations I and II, and it is equal to the prespecified value D in situation III. The term in the denominator is the standard deviation of the difference between the two sample means (it relies on the assumption that the two

samples are independent).

2

2

2

1

2

1

02121)()(

nn

xxz

Comparisons of Two Population Means: Test Statistic


Slide 16

212 =

452 =x

1200=n

Visa Preferred :1 Population

1

1

1

185 =

523 =x

800=n

Card Gold :2 Population

2

2

2

Is there evidence to conclude that the average monthly charge in the entire population of American Express Gold Card members is different from the average monthly charge in the entire population of Preferred Visa

cardholders?

cesignifican of levelcommon any at rejected is 0

H

0 -7.926)<p(z :value-p

926.796.8

71

2346.80

71

800

2185

1200

2212

0)523452(

2

22

1

21

0)

21()

21(

021

:1

H

021

:0

H

nn

xxz

Two-Tailed Test for Equality of Two Population Means: Example

8-3


Slide 17

0.4

0.3

0.2

0.1

0.0z

f( z)

Standard Normal Distribution

NonrejectionRegion

RejectionRegion

-z0.01=-2.576 z0.01=2.576

Test Statistic=-7.926

RejectionRegion

0

Since the value of the test statistic is far below the lower critical

, point the null hypothesis may be

, rejected and we may conclude that there is

a statistically significant difference

between the average monthly charges of

Gold Card and Preferred Visa

.cardholders

Since the value of the test statistic is far below the lower critical

, point the null hypothesis may be

, rejected and we may conclude that there is

a statistically significant difference

between the average monthly charges of

Gold Card and Preferred Visa

.cardholders

Example 8-3: Carrying Out the Test


Slide 18Example 8-3: Using the Template


Slide 19

84=

308=x

100=n

Duracell :1 Population

1

1

1

67=

254=x

100=n

Energizer :2 Population

2

2

2

’ , , Is there evidence to substantiate Duracell s claim that their batteries last on average 45 ?at least minutes longer than Energizer batteries of the same size

cesignifican of level

common any at rejected benot may 0

H

0.201=0.838)>p(z :value-p

838.075.10

9

45.115

9

100

267

100

284

45)254308(

2

22

1

21

0)

21()

21(

4521

:1

H

4521

:0

H

nn

xxz

Two-Tailed Test for Difference Between Two Population Means:

Example 8-4


Slide 20

Is there evidence to substantiate Duracell’s claim that their batteries last, on average, at least 45 minutes longer than Energizer batteries of the same size?

Two-Tailed Test for Difference Between Two Population Means:

Example 8-4 – Using the Template


Slide 21

A large-sample (1-)100% confidence interval for the difference between two population means, 1- 2 , using independent

random samples:

A large-sample (1-)100% confidence interval for the difference between two population means, 1- 2 , using independent

random samples:

2

22

1

21

2

)21

(nn

zxx

A 95% confidence interval using the data in example 8-3:


]56.88,44.53[800

2185

1200

221296.1)452523(

2

22

1

21

2

)21

( nn

zxx

Confidence Intervals for the Difference between Two

Population Means


Slide 22

• If we might assume that the population variances 12 and 2

2 are equal (even though unknown), then the two sample variances, s1

2 and s22, provide two separate estimators of

the common population variance. Combining the two separate estimates into a pooled estimate should give us a better estimate than either sample variance by itself.

x1

** ** ** * *** ** * *

}Deviation from the mean. One for each sample data point.

Sample 1

From sample 1 we get the estimate s12 with

(n1-1) degrees of freedom.

Deviation from the mean. One for each sample data point.

* * ** ** * * * * ** * *

x2

}

Sample 2

From sample 2 we get the estimate s22 with

(n2-1) degrees of freedom.

From both samples together we get a pooled estimate, sp2 , with (n1-1) + (n2-1) = (n1+ n2 -2)

total degrees of freedom.

8-4 A Test for the Difference between Two Population Means: Assuming Equal Population

Variances


Slide 23

A pooled estimate of the common population variance, based on a sample variance s1

2 from a sample of size n1 and a sample variance s22 from a sample

of size n2 is given by:

The degrees of freedom associated with this estimator is:

df = (n1+ n2-2)

A pooled estimate of the common population variance, based on a sample variance s1

2 from a sample of size n1 and a sample variance s22 from a sample

of size n2 is given by:

The degrees of freedom associated with this estimator is:

df = (n1+ n2-2)

sn s n s

n np

2 1 1

2

2 2

2

1 2

1 12

( ) ( )

The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two

samples. That is, larger weight is given to the variance from the larger sample.

The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two

samples. That is, larger weight is given to the variance from the larger sample.

Pooled Estimate of the Population Variance


Slide 24

The estimate of the standard deviation of (x1 x2 is given by: sp2

)1

1

1

2n n

Test statistic for the difference between two population means, assuming equal population variances:

t =(x1 x2 1 2

sp2

where 1 2 is the difference between the two population means under the null

hypothesis (zero or some other number D).

The number of degrees of freedom of the test statistic is df = ( 1 (the

number of degrees of freedom associated with sp2

, the pooled estimate of the

population variance.

) ( )

( )

)

0

1

1

1

2

0

2 2

n n

n n

Using the Pooled Estimate of the Population Variance


Slide 25

Population 1: Oil price = $27.50

n1 = 14

x1 = 0.317%

s1 = 0.12%

Do the data provide sufficient evidence to conclude that average percentage increase in the CPI differs when oil sells at these two different prices?

H

H

Critical point: t0.025

= 2.080

H0 may be rejected at the 5% level of significance

0 1 2 0

1 1 2 0

1 2 1 2 0

1 1 12

2 1 22

1 2 2

1

1

1

2

0 107

0 00247

0 107

0 04972 154

:

:

( ) ( )

( ) ( )

.

.

.

..

tx x

n s n s

n n n nPopulation 2: Oil price = $20.00

n = 9

x = 0.21%

s = 0.11%

df = (n1

2

2

2

n2

2 14 9 2 21) ( )

Example 8-5


Slide 26

Do the data provide sufficient evidence to conclude that average percentage increase in the CPI differs when oil sells at these two different prices?

Example 8-5: Using the Template

P-value = 0.0430, so

reject H0 at the 5%

significance level.


Slide 27

Population 1: Before Reduction

n1 = 15

x1 = $6598

s1 = $844

The manufacturers of compact disk players want to test whether a small price reduction is enough

to increase sales of their product. Is there evidence that the small price reduction is enough

to increase sales of compact disk players?

cesignifican of level 10% at theeven rejected benot may 0

H

1.316=0.10

t:point Critical

91.096.298

272

25.89375

272

12

1

15

1

21215

2669)11(

2844)14(

0)65986870(

2

1

1

1

221

22

)12

(21

)11

(

0)

12()

12(

012

:1

H

012

:0

H

nnnn

snsn

xxt

Population 2: After Reduction

n = 12

x = $6870

s = $669

df = (n1

2

2

2

n2

2 15 12 2 25) ( )

Example 8-6



P-value = 0.1858, so

do not reject H0 at

the 5% significance

level.


Slide 29

543210-1-2-3-4-5

0.4

0.3

0.2

0.1

0.0t

f(t)

t Distribution: df = 25

NonrejectionRegion

RejectionRegion

t0.10=1.316

Test Statistic=0.91

Since the test statistic is less than t0.10, the null hypothesis

cannot be rejected at any reasonable level of

significance. We conclude that the price reduction does not significantly affect sales.

Since the test statistic is less than t0.10, the null hypothesis

cannot be rejected at any reasonable level of

significance. We conclude that the price reduction does not significantly affect sales.

Example 8-6: Continued


Slide 30

A (1-) 100% confidence interval for the difference between two population means, 1- 2 , using independent random samples and

assuming equal population variances:

A (1-) 100% confidence interval for the difference between two population means, 1- 2 , using independent random samples and

assuming equal population variances:

( )x x t sn np1 2

2

2 1

1

1

2

A 95% confidence interval using the data in Example 8-6:

A 95% confidence interval using the data in Example 8-6:

( ) ( ) . ( )( . ) [ . , . ]x x t s pn n

1 2

2

2 1

1

1

2

6870 6598 2 06 595835 0 15 343 85 887 85

Confidence Intervals Using the Pooled Variance


Slide 31

Confidence Intervals Using the Pooled Variance and the Template-

Example 8-6

Confidence IntervalThe Comparison of Two Populations By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Slide 32

• Hypothesized difference is zeroI: Difference between two population proportions is 0

• p1= p2 »H0: p1 -p2 = 0»H1: p1 -p20

II: Difference between two population proportions is less than 0 • p1 p2

»H0: p1 -p2 0»H1: p1 -p2 > 0

• Hypothesized difference is other than zero:III: Difference between two population proportions is less than

D• p1 p2+D»H0:p-p2 D»H1: p1 -p2 > D

8-5 A Large-Sample Test for the Difference between Two Population

Proportions


Slide 33

- A large sample test statistic for the difference between two , population proportions when the hypothesized difference is

:zero

1 where is the sample proportion in sample and is the sample

2. proportion in sample The symbol stands for the , combined sample proportion in both samples considered as

. :a single sample That is

- A large sample test statistic for the difference between two , population proportions when the hypothesized difference is

:zero

1 where is the sample proportion in sample and is the sample

2. proportion in sample The symbol stands for the , combined sample proportion in both samples considered as

. :a single sample That is

zp p

p pn n

( )

( )

1 2

1 2

0

11 1

pxn1

1

1

When the population proportions are hypothesized , to be equal then a pooled estimator of the

( ) proportion may be used in calculating the . test statistic

pxn1

1

1

p

21

11ˆnn

xxp

p

Comparisons of Two Population Proportions When the Hypothesized

Difference Is Zero: Test Statistic


Slide 34

Carry out a two-tailed test of the equality of banks’ share of the car loan market in 1980 and 1995.

Population 1: 1980

n1 = 100

x1 = 53

p1 = 0.53

H

H

Critical point: z0.05

= 1.645

H0 may not be rejected even at a 10%

level of significance.

0 1 2 0

1 1 2 0

1 2 0

11

1

1

2

0 53 0 43

48 521

100

1

100

0 10

0 004992

0 10

0 070651 415

:

:

( )

( )

. .

(. )(. )

.

.

.

..

p p

p p

zp p

p pn n

Population 2: 1995

n = 100

x = 43

p = 0.43

x1 + x2

n1 n2

2

2

2

.p

53 43

100 1000 48

Comparisons of Two Population Proportions When the Hypothesized

Difference Is Zero: Example 8-8


Slide 35

0.4

0.3

0.2

0.1

0.0z

f(z)


NonrejectionRegion

RejectionRegion

-z0.05=-1.645 z0.05=1.645

Test Statistic=1.415

RejectionRegion

0

Since the value of the test statistic is within the

nonrejection region, even at a 10% level of significance, we may conclude that there is no

statistically significant difference between banks’ shares of car loans in 1980

and 1995.

Since the value of the test statistic is within the

nonrejection region, even at a 10% level of significance, we may conclude that there is no

statistically significant difference between banks’ shares of car loans in 1980

and 1995.




P-value = 0.157, so do not reject H0 at the

5% significance

level.


Slide 37

Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such

buyers when no sweepstakes are on.

Population 1: With Sweepstakes

n1 = 300

x1 = 120

p1 = 0.40

H

H

Critical point: z0.001

= 3.09

H0 may be rejected at any common level of significance.

0 1 2 0 10

1 1 2 0 10

1 2

11

1

1

21

2

2

0 40 0 20 0 10

0 40 0 60

300

0 20 80

700

0 10

0 032073 118

: .

: .

( )

( ) ( )

( . . ) .

( . )( . ) ( . )(. )

.

..

p p

p p

zp p D

p p

n

p p

n

Population 2: No Sweepstakes

n = 700

x = 140

p = 0.20

2

2

2

Comparisons of Two Population Proportions When the Hypothesized Difference Is Not Zero:

Example 8-9


Slide 38

0.4

0.3

0.2

0.1

0.0z

f( z)


NonrejectionRegion

RejectionRegion

z0.001=3.09

Test Statistic=3.118

0

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on.

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on.




P-value = 0.0009, so reject H0 at

the 5% significance

level.


Slide 40

A (1-) 100% large-sample confidence interval for the difference between two population proportions:

A (1-) 100% large-sample confidence interval for the difference between two population proportions:



( )

( ) ( )

p p z

p p

n

p p

n1 2

2

11

1

1

21

2

2

( )

( ) ( )

( . . ) .( . )( . ) ( . )( . )

. ( . )( . ) . . [ . , . ]

p p z

p p

n

p p

n1 2

2

11

1

1

21

2

20 4 0 2 1 96

0 4 0 6

300

0 2 0 8

700

0 2 1 96 0 0321 0 2 0 063 0 137 0 263

Confidence Intervals for the Difference between Two Population Proportions


Slide 41

Confidence Intervals for the Difference between Two Population Proportions –

Using the Template – Using the Data from Example 8-9


Slide 42

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each

of which is divided by its own degrees of freedom.

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each

of which is divided by its own degrees of freedom.

An F random variable with k1 and k2 degrees of freedom:An F random variable with k1 and k2 degrees of freedom:

Fk

k

k k1 2

1

2

1

2

2

2

,

8-6 The F Distribution and a Test for Equality of Two Population

Variances


Slide 43

• The F random variable cannot be negative, so it is bound by zero on the left.

• The F distribution is skewed to the right.

• The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.

• The F random variable cannot be negative, so it is bound by zero on the left.

• The F distribution is skewed to the right.

• The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.

543210

1.0

0.5

0.0

F

F Distributions with different Degrees of Freedom

f(F)

F(5,6)

F(10,15)

F(25,30)

The F Distribution


Slide 44

Critical Points of the F Distribution Cutting Off a Right-Tail Area of 0.05

k1 1 2 3 4 5 6 7 8 9

k2

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.1810 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.0211 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.9012 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.8013 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.7114 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.6515 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.593.01

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F0.05=3.01

f(F

)

F Distribution with 7 and 11 Degre e s of Fre edom

F

The left-hand critical point to go along with F(k1,k2) is given by:

Where F(k1,k2) is the right-hand critical point for an F random variable with the reverse number of degrees of freedom.

1

2 1F k k,

Using the Table of the F Distribution


Slide 45

The right-hand critical point read directly from the table of

the F distribution is:

F(6,9) =3.37

The corresponding left-hand critical point is given by:

The right-hand critical point read directly from the table of

the F distribution is:

F(6,9) =3.37

The corresponding left-hand critical point is given by:

1 1410

0 24399 6F , .

.

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F

f(F)

F Distribution with 6 and 9 Degrees of Freedom

F0.05=3.37F0.95=(1/4.10)=0.2439

0.05

0.05

0.90

Critical Points of the F Distribution: F(6, 9), = 0.10


Slide 46

Test statistic for the equality of the variances of two normallydistributed populations:

Fs

sn n1 21 1

1

2

2

2 ,

I: Two-Tailed Test• 1 = 2

H0: 1 = 2 H1: 2

II: One-Tailed Test• 12

H0: 1 2 H1: 1 2

I: Two-Tailed Test• 1 = 2

H0: 1 = 2 H1: 2

II: One-Tailed Test• 12

H0: 1 2 H1: 1 2

Test Statistic for the Equality of Two Population Variances


Slide 47

The economist wants to test whether or not the event (interceptions and prosecution of insider traders) has decreased the variance of prices of stocks.

70.223,24

01.0

01.223,24

05.0

0.322

s

24=2

n

After :2 Population

3.921

s

25=1

n

Before :1 Population

F

F

H

H

H0 may be rejected at a 1% level of significance.

0 1

2

2

2

1

2

1 1

2

2

2

1 1 2 1 24 23

12

22

9 3

3 031

:

:

, ,

.

..

F

n nF

s

s

Example 8-10


Slide 48

Distribution with 24 and 23 Degrees of Freedom

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F0.01=2.7

f(F

)

F

Test Statistic=3.1

Since the value of the test statistic is above the critical

point, even for a level of significance as small as 0.01, the null hypothesis may be

rejected, and we may conclude that the variance of stock prices is reduced after

the interception and prosecution of inside traders.

Since the value of the test statistic is above the critical

point, even for a level of significance as small as 0.01, the null hypothesis may be

rejected, and we may conclude that the variance of stock prices is reduced after

the interception and prosecution of inside traders.



Slide 49Example 8-10: Solution Using

the Template

Observe that the p-value for the test is 0.0042 which is less

than 0.01. Thus the null hypothesis must be

rejected at this level of significance of 0.01.


Slide 50

Population 1 Population 2

n1

= 14 n2

= 9

s12 s

22

0 122 0 112

0 05

13 83 28

0 10

13 82 50

. .

.

,.

.

,.

F

F

H

H

H0

may not be rejected at the 10% level of significance.

0 12

22

1 12

22

1 1 2 1 13 812

22

0122

0112119

:

:

, ,

.

..

F

n nF

s

s

Example 8-11: Testing the Equality of Variances for

Example 8-5


Slide 51

Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject

the null hypothesis. We conclude the two population

variances are equal.

Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject

the null hypothesis. We conclude the two population

variances are equal.

F Distribution with 13 and 8 Degrees of Freedom

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F

f(F)

F0.10=3.28F0.90=(1/2.20)=0.4545

0.10

0.10

0.80

Test Statistic=1.19



Slide 52

Template to test for the Difference between Two Population Variances: Example 8-11

Observe that the p-value for the test is 0.8304 which is larger than 0.05. Thus the null hypothesis cannot be rejected at this level of significance of 0.05. That is, one can assume equal variance.


Slide 53

The F Distribution Template to


Slide 54

The Template for Testing Equality of Variances


Slide 55

M.Phil (Statistics)

GC University, . (Degree awarded by GC University)

M.Sc (Statistics) GC University, . (Degree awarded by GC University)

Statitical Officer(BS-17)(Economics & Marketing Division)

Livestock Production Research Institute Bahadurnagar (Okara), Livestock & Dairy Development

Department, Govt. of Punjab

Name Shakeel NoumanReligion ChristianDomicile Punjab (Lahore)Contact # 0332-4462527. 0321-9898767E.Mail [email protected] [email protected]


mailto:[email protected]

mailto:[email protected]

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The comparison of two populations

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