chapter 9: inferences based on two samples: confidence intervals and tests of hypotheses statistics

Chapter 9: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Statistics

McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

2

Where We’ve Been

Made inferences based on confidence intervals and tests of hypotheses

Studied confidence intervals and hypothesis tests for µ, p and σ 2

Selected the necessary sample size for a given margin of error


3

Where We’re Going

Learn to use confidence intervals and hypothesis tests to compare two populations

Learn how to use these tools to compare two population means, proportions and variances

Select the necessary sample size for a given margin of error when comparing parameters from two populations


4

9.1: Identifying the Target Parameter


5

9.2: Comparing Two Population Means: Independent Sampling

Point Estimators → µ 1 - 2 → µ1 - µ2To construct a confidence interval or conduct a hypothesis test, we need the standard deviation:

Singe sample nsx ̂

2

22

1

21

21ˆ

n

s

n

sxx Two

samples


6


The Sampling Distribution for (1 - 2)

1. The mean of the sampling distribution is (µ1-µ2).

2. If the two samples are independent, the standard deviation of the sampling distribution (the standard error) is

3. The sampling distribution for (1 - 2) is approximately normal for large samples.

2

22

1

21

21ˆ

n

s

n

sxx


7


The Sampling Distribution for (1 - 2)


8


Large Sample Confidence Interval for (µ1 - µ2 )

2

22

1

21

2/21

2

22

1

21

2/21)(2/21

)(

)()(21

n

s

n

szxx

nnzxxzxx xx


Private Colleges n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17

Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64


9

Two samples concerning retention rates for first-year students at private and public institutions were obtained from the Department of Education’s data

base to see if there was a significant difference in the two types of colleges.

What does a 95% confidence interval tell us about retention rates?

Source: National Center for Education Statistics





10

08.483.532

64.97

71

1.9196.18417.78

)()(2

22

1

21

2/21)(2/21 21

n

s

n

szxxzxx xx





11

08.483.532

64.97

71

1.9196.18417.78

)()(2

22

1

21

2/21)(2/21 21

n

s

n

szxxzxx xx

Since 0 is not in the confidence interval, the difference in the sample means appears to indicate a real difference in retention.


12


Small Sample Confidence Interval for (µ1 - µ2 )

21

22/21)(2/21

11)()(

21 nnstxxtxx pxx

The value of t is based on (n1 + n2 -2) degrees of freedom.


13


For small samples, the t-distribution can be used with a pooled sample estimator of σ 2, σ p

2

2

)1()1(

21

222

2112

nn

snsnsp



14

H0: (µ1 - µ2) = D0

Ha: (µ1 - µ2) > D0 (< D0)

Rejection region:

z < -za (> za)

Test Statistic:

where

H0: (µ1 - µ2) = D0

Ha: (µ1 - µ2) ≠ D0

Rejection region:

|z| > za/2

Two-Tailed TestOne-Tailed Test

)(

021

21

)(

xx

Dxxz

2

22

1

21

2

22

1

21

)( 21 n

s

n

s

nnxx


15


Conditions Required for Valid Large-Sample Inferences about (µ1 - µ2)

1. The two samples are randomly and independently selected from the target populations.

2. The sample sizes are both ≥ 30.





16

Let’s go back to the retention data and test the hypothesis that there is no significant difference in retention at privates and publics.

08.232

64.97

71

17.91

2

22

1

21

)( 21 n

s

n

sxx


799.208.2

83.50)(

05.

0)(:

0)(:

)(

21

21

210

21

xx

a

xxz

H

H


17

Test statistic:

Reject the null hypothesis:

96.1799.2

|| 2/

zz


18


For small samples, the t-distribution can be used with a pooled sample estimator of σ 2, σ p

2

2

)1()1(

21

222

2112

nn

snsnsp



19

H0: (µ1 - µ2) = D0

Ha: (µ1 - µ2) > D0 (< D0)

Rejection region:

t < -ta (> ta)

Test Statistic:

H0: (µ1 - µ2) = D0*

Ha: (µ1 - µ2) ≠ D0

Rejection region:

|t| > ta/2

Two-Tailed TestOne-Tailed Test

21

2

021

11

)(

nns

Dxxt

p


20


Conditions Required for Valid Small-Sample Inferences about (µ1 - µ2)

1. The two samples are randomly and independently selected from the target populations.

2. Both sampled populations have distributions that are approximately normal.

3. The population variances are equal.


21

Does class time affect performance? The test performance of students in two

sections of international trade, meeting at different times, were compared.

8:00 a.m. ClassMean: 78

Standard Deviation: 14Variance: 196

n: 21


Standard Deviation: 17Variance: 289

n: 21

With a = .05, test H0 : µ1 = µ2



22


Variance: 196n: 21


Variance: 289n: 21

832.

211

211

5.242

0)8278(

11

0)( :StatisticTest

5.24222121

289)121(196)121(

2

)1()1(

05.

0:

0:

21

2

21

21

222

2112

21

210

nns

xxt

nn

snsns

H

H

p

p

a



23


Variance: 196n: 21


Variance: 289n: 21

832.

211

211

5.242

0)8278(

11

0)( :StatisticTest

5.24222121

289)121(196)121(

2

)1()1(

05.

0:

0:

21

2

21

21

222

2112

21

210

nns

xxt

nn

snsns

H

H

p

p

a

With df = 18 + 24 – 2 = 40, ta/2 = t.025 = 2.021. Since out test statistic t = -.812. |t| < t.025.

Do not reject the null hypothesis



24


Variance: 154n: 13


Variance: 163n: 21

0 1 2

1 2

1 2

2 21 2

1 2

Several students in the 8:00 a.m. section sleep through

the next exam. We can still compare the results,

with some modifications.

: 0

: 0

.05

( ) 0Test Statistic:

a

H

H

x xt

s s

n n

(72 86) 03.16

154 16313 21



25


Variance: 154n: 13


Variance: 163n: 21

056.2

2615.26

12121/163

11313/154

21/16313/154

1/

1/

// Freedom of Degrees

16.3

26,025.

22

2

2

2

222

1

2

121

2

2221

21

dft

nns

nns

nsnsv

t



26


Variance: 154n: 13


Variance: 163n: 21

056.2

2615.26

12121/163

11313/154

21/16313/154

1/

1/

// Freedom of Degrees

16.3

26,025.

22

2

2

2

222

1

2

121

2

2221

21

dft

nns

nns

nsnsv

t

Since |t| > t.025,df=26 ,, reject the null hypothesis.



27

9.3: Comparing Two Population Means: Paired Difference Experiments

observed pairs ofnumber n where

sdifference ofdeviation standard sample s where

differencemean sample where

freedom, of degrees )1(with :Sample Small

:Sample Large

for Interval Confidence Difference Paired

2/

2/2/

21d

d

d

d

d

d

dd

d

dd

d

dd

x

nn

stx

n

szx

nzx


28


Suppose ten pairs of puppies were housetrained using two different methods: one puppy from each pair was paper-trained, with the paper gradually moved outside, and the other was taken out every three hours and twenty minutes after each meal. The number of days until the puppies were considered housetrained (three days straight without an accident) were compared. Nine of the ten paper-trained dogs took longer than the other paired dog to complete training, with the average difference equal to 4 days, with a standard deviation of 3 days.

What is a 90% confidence interval on the difference in successful training?


29


74.1410

3833.14

confident 90%

3

4

1,2/21

d

dndfd

d

d

n

stx

s

x

d

Since 0 is not in the interval, one program does seem to work more effectively.


30


dd

dα, n

dd

d

dd

d

ns

Dxt

ns

Dx

n

Dxz

d /

statistic test sample Small

//

statistic test sample Large

forTests Hypothesis

Difference Paired

01

00

21d

tt

zz

DH

DH

da

d

)or ( :region rejection sample Small

)or ( :regionrejection sample Large

)or (:

:

Test Tailed-One

0

00

2/

2/

0

00

|| :region rejection sample Small

|| :regionrejection Sample Large

:

:

Test Tailed-Two

tt

zz

DH

DH

da

d


31


Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference: $1.75 Standard Deviation: $10.35 Test at the 95% level that the difference in

the two stores is zero.


32


Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference:

$1.75 Standard Deviation:

$10.35 a = .05

07.2150/35.10

075.1

/

0:

0:

0

0

z

z

n

Dxz

H

H

dd

d

da

d


33


Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference:

$1.75 Standard Deviation:

$10.35 a = .05

07.2150/35.10

075.1

/

0:

0:

0

0

z

z

n

Dxz

H

H

dd

d

da

d

The critical value of z.05 is

1.96, so we would reject

this null hypothesis.


34

9.4: Comparing Two Population Proportions: Independent Sampling

.ˆ and ˆ examiningby and

about inferences makecan We

21 21 pppp

stics.characteri particular regarding

sproportionsimilar have

notmay or may groups Two


35


normal.ely approximat ison distributi

sampling thelarge, are samples theIf .3

ˆˆˆˆ .2

)( )ˆˆ( .1

)ˆˆ( ofon Distributi Sampling theof Properties

2

22

1

11

2

22

1

11)ˆˆ(

2121

21

21 n

qp

n

qp

n

qp

n

qp

ppppE

pp

pp


36


2

22

1

112/21

2

22

1

112/21)ˆˆ(2/21

21

ˆˆˆˆ)ˆˆ(

)ˆˆ()ˆˆ(

)(for Interval Confidence )%-100(1 Sample-Large

21

n

qp

n

qpzpp

n

qp

n

qpzppzpp

pp

pp


37


A group of men and women were asked their opinions on the following important issue:

Are the Three Stooges funny?

The results are as follow:

Men Women

Yes 290 200

No 50 50

n 340 250


38


Men Women

p .85 .80

q .15 .20

n 340 250

Calculate a 95% confidence interval on the difference in the opinions of men and women.

1 1 2 2/2

1 2

ˆ ˆ( )

.85 .15 .80 .20(.85 .80) 1.96

340 250.05 .062

M W

p q p qp p z

n n


39


Men Women

p .85 .80

q .15 .20

n 340 250

Calculate a 95% confidence interval on the difference in the opinions of men and women.

1 1 2 2/2

1 2

ˆ ˆ( )

.85 .15 .80 .20(.85 .80) 1.96

340 250.05 .062

M W

p q p qp p z

n n

Since 0 is in the confidence interval, we cannot rule out

the possibility that both genders find the Stooges

equally funny. Nyuk nyuk nyuk.


40


21about Hypothesis ofTest Sample-Large pp

zz

DorppH

DppH

a

|| :regionRejection

)()(:

0)often very ()(:

Test Tailed-One

021

0210

2/

021

0210

|| :regionRejection

)(:

)(:

Test Tailed-Two

zz

DppH

DppH

a

21)ˆˆ(

21

21

)ˆˆ(

21

11ˆˆ andˆ where

)ˆˆ( :StatisticTest

21

21

nnqp

nn

xxp

ppz

pp

pp


41

9.4: Comparing Two Population Proportions: Independent Sampling Randy Stinchfield of the University of Minnesota studied the gambling

activities of public school students in 1992 and 1998 (Journal of Gambling Studies, Winter 2001). His results are reported below:

Do these results represent a statistically significant difference at the a = .01 level?

1992 1998

Survey n 21,484 23,199

Number who gambled 4.684 5,313

Proportion who gambled .218 .229


42


1992 1998

Survey n 21,484 23,199

Number who gambled

4.684 5,313

Proportion who gambled

.218 .229

0 1 2 0

1 2 0

/2

Two-Tailed Test

: ( )

: ( )

Rejection region:

| | 2.576

a

H p p D

H p p D

z z

1 2

1 2

1 2

1 2

ˆ ˆ( )1 2

1 2

ˆ ˆ( )

4,684 5,313ˆ .224

21,484 23,199

1 1ˆ ˆ

1 1(.224)(.776) .00395

21,484 23,199

ˆ ˆ( ) (.218 .229)Test Statistic:

.00395

2.786

p p

p p

x xp

n n

pqn n

p pz


43


1992 1998

Survey n 21,484 23,199

Number who gambled

4.684 5,313

Proportion who gambled

.218 .229

0 1 2 0

1 2 0

/2

Two-Tailed Test

: ( )

: ( )

Rejection region:

| | 2.576

a

H p p D

H p p D

z z

1 2

1 2

1 2

1 2

ˆ ˆ( )1 2

1 2

ˆ ˆ( )

4,684 5,313ˆ .224

21,484 23,199

1 1ˆ ˆ

1 1(.224)(.776) .00395

21,484 23,199

ˆ ˆ( ) (.218 .229)Test Statistic:

.00395

2.786

p p

p p

x xp

n n

pqn n

p pz

Since the computed value of z, -2.786, is of greater magnitude than the critical value, 2.576, we can reject the null hypothesis at the a = .01 level.


44


For valid inferences The two samples must be independent The two sample sizes must be large:

15ˆ and 15ˆ

15ˆ and 15ˆ

2222

1111

qnpn

qnpn


45

9.5: Determining the Sample Size

With a given level of confidence, and a specified sampling error, it is possible to calculate the required sample size Typically, n1 = n2


46


Sample size needed to estimate (µ1 - µ1) Given (1 - a ) and the sampling error (SE)

required

Estimates of σ1 and σ2 will be needed

2

22

21

22/

21 SE

)()(

znn


47


Suppose you need to estimate the difference between two population means to within 2.2 at the a = 5% level. You have good reason to believe the two variances are equal to each other, and equal 15.

How large must n1 and n2 be?

248.232.2

)1515(96.1

SE

)()(

21

2

2

21

2

22

21

22/

21

nn

nn

znn


48


SE = 2.2 a = 5% level. σ1

2 = σ22 =15.

n1 and n2 = ?


49


Sample size needed to estimate (p1 - p2) Given (1 - a ) and the sampling error (SE)

required

Estimates of p1 and p2 will be needed The most conservative values are p1 = p2 =.5

22211

22/

21 SE

)()( qpqpznn


50


Suppose you need to calculate a 90% confidence interval of width .05, with no information about possible values of p1 and p2.

What size do n1 and n2 need to be?

5422.541.05

)]5)(.5(.)5)(.5[(.645.1

SE

)()(

21

2

2

21

22211

22/

21

nn

nn

qpqpznn


51

9.5: Determining the Sample Size Suppose you

need to calculate a 90% confidence interval of width .05, with no information about possible values of p1 and p2.

What size do n1 and n2 need to be?


52

9.6: Comparing Two Population Variances: Independent Sampling


53


Since variances do not follow a normal distribution (due to the squaring involved) we use a different technique for inferences about variances.


54


2 22 2 2 2 1 1

1 1 2 2 2 22 2

22 2 1

1 2 22

1

2

If ( ) and ( ) , then .

If , the random variable .

The random variable is a ratio with 1 numerator

and 1 denominator degrees of freed

sE s E s E

s

sF

s

F n

n

om.


55


If the ratio of variances gets too far from 1 in value, in either direction the likelihood that both samples share a population variance drops. For convenience the larger sample variance is usually put in the numerator.


56


2 21 2

2 2 21 1 12 2 22 , /2 2 2 , /2

, /2

, /2

A (1 ) 100% Confidence Interval for /

1 1

where leaves ( /2)% of the distribution in a lower tail

and leaves ( /2)% o

U L

L

U

s s

s F s F

F

F

1

2

f the distribution in an upper tail.

Degrees of freedom are 1 in the numerator and

1 in the denominator.

n

n


57


Home Runs, 2003

American League

National League

µ 178.5 169.25

σ 2 1415.8 951.4

n 14 16

Has the designated hitter produced a significant difference in homeruns in the two major leagues?

To use the two-sample t-test, we should check the variance ratio to see if the assumption of equal variances is acceptable.


58


2 2 21 1 12 2 22 , /2 2 2 , /2

212

,.025 2 ,.025

2122

1 1

1415.8 1 1415.8 1

951.4 951.4

1415.8 1 1415.8

951.4 2.92 951.

U L

U L

s s

s F s F

F F

2122

1

4 .32

.5096 4.6504

Home Runs, 2003

American League

National League

µ 178.5 169.25

σ2 1415.8 951.4

n 14 16


59


2 2 21 1 12 2 22 , /2 2 2 , /2

212

,.025 2 ,.025

2122

1 1

1415.8 1 1415.8 1

951.4 951.4

1415.8 1 1415.8

951.4 2.92 951.

U L

U L

s s

s F s F

F F

2122

1

4 .32

.5096 4.6504

Home Runs, 2003

American League

National League

µ 178.5 169.25

σ 2 1415.8 951.4

n 14 16

Since 1 is in the confidence interval, the assumption of equal variances needed for the t-test is not rejected.

-Test for Equal Population VariancesF

2 20 1 2

2 21 2

22 221 22

1

22 211 22

2

One-Tailed Test

:

: (or )

Test Statistic: if :

or if :

Rejection Region:

a

a

a

H

H

sF H

s

sF H

s

F F


60


2 20 1 2

2 21 2

/2

Two-Tailed Test

:

:

Test Statistic:

Larger sample variance

Smaller sample variance

Rejection Region:

a

H

H

F

F F


In 2006, ten fast-growing economies had an average GDP growth rate of 8.69%, with a standard deviation of 1.7. Ten slow-growing economies had an average GDP growth rate of 2.29%, with a standard deviation of .58.* Do slow- and fast-growing economies have similar variability (a=5%)?


61

* Source: euroekonom.com


62


2 20 1 2

2 21 2

.025

Two-Tailed Test

:

:

Test Statistic:

Larger sample variance 2.898.5

Smaller sample variance .34

Rejection Region: 4.03

a

H

H

F

F F

µfast-growing = 8.69%µslow-growing = 2.29%σ 2fast-growing = 2.89σ 2slow-growing = .34


63


2 20 1 2

2 21 2

.025

Two-Tailed Test

:

:

Test Statistic:

Larger sample variance 2.898.5

Smaller sample variance .34

Rejection Region: 4.03

a

H

H

F

F F

µfast-growing = 8.69%µslow-growing = 2.29%σ 2fast-growing = 2.89σ 2slow-growing = .34

Reject the null hypothesis at the a=.05 level.

chapter 9: inferences based on two samples: confidence intervals and tests of hypotheses statistics

Documents

weve beenmade inferences

large samples

standard deviation

hypothesis tests

necessary sample size

singe sample

retention rates

given margin of errormcclave