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 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

ns

ns

xx 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

ns

ns

xx


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

ns

nszxx

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

64.9771

1.9196.18417.78

)()(2

22

1

21

2/21)(2/21 21

ns

nszxxzxx xx





11

08.483.532

64.9771

1.9196.18417.78

)()(2

22

1

21

2/21)(2/21 21

ns

nszxxzxx xx

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



12

H0: (µ1 - µ2) = D0

Ha: (µ1 - µ2) > D0 (< D0)

Rejection region:z < -z (> z)

Test Statistic:

where

H0: (µ1 - µ2) = D0

Ha: (µ1 - µ2) ≠ D0

Rejection region:|z| > z/2

Two-Tailed TestOne-Tailed Test

)(

021

21

)(

xx

Dxxz

2

22

1

21

2

22

1

21

)( 21 ns

ns

nnxx


13


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.





14

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

64.9771

17.91

2

22

1

21

)( 21 n

sns

xx


799.208.283.50)(

05.0)(:0)(:

)(

21

21

210

21

xx

a

xxz

HH


15

Test statistic:

Reject the null hypothesis:

96.1799.2|| 2/

zz


16


For small samples, the t-distribution can be used with a pooled sample estimator of 2, sp2

2)1()1(

21

222

2112

nn

snsnsp


17


Small Sample Confidence Interval for (µ1 - µ2 )

21

22/21)(2/21

11)()(21 nn

stxxtxx pxx

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



18

H0: (µ1 - µ2) = D0

Ha: (µ1 - µ2) > D0 (< D0)

Rejection region:t < -t (> t)

Test Statistic:

H0: (µ1 - µ2) = D0*

Ha: (µ1 - µ2) ≠ D0

Rejection region:|t| > t/2

Two-Tailed TestOne-Tailed Test

21

2

021

11

)(

nns

Dxxt

p


19


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.


20

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 = .05, test H0 : µ1 = µ2



21


Variance: 196n: 21


Variance: 289n: 21

832.

211

2115.242

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

nnsnsns

HH

p

p

a



22


Variance: 196n: 21


Variance: 289n: 21

832.

211

2115.242

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

nnsnsns

HH

p

p

a

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

Do not reject the null hypothesis



23


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

HH

x xts sn n

(72 86) 0 3.16154 16313 21



24


Variance: 154n: 13


Variance: 163n: 21

056.22615.26

12121/163

11313/154

21/16313/154

1/

1/

// Freedom of Degrees

16.3

26,025.

22

2

2

22

22

1

21

21

22

221

21

dft

nns

nns

nsnsv

t



25


Variance: 154n: 13


Variance: 163n: 21

056.22615.26

12121/163

11313/154

21/16313/154

1/

1/

// Freedom of Degrees

16.3

26,025.

22

2

2

22

22

1

21

21

22

221

21

dft

nns

nns

nsnsv

t

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



26

9.3: Comparing Two Population Means: Paired Difference Experiments

observed pairs ofnumber n wheresdifference 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

dd

dd

d

dd

d

dd

x

nnstx

nszx

nzx


27


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?


28


74.14103833.14

confident 90%

34

1,2/21

d

dndfd

d

d

nstx

sx

d

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


29


dd

dα, n

dd

d

dd

d

nsDxt

nsDx

nDxz

d /

statistic test sample Small

//

statistic test sample Large

forTests Hypothesis Difference Paired

01

00

21d

ttzz

DHDH

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

ttzz

DHDH

da

d


30


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.


31


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

$1.75 Standard Deviation:

$10.35 = .05

07.2150/35.10075.1

/

0:0:

0

0

z

z

nDxz

HH

dd

d

da

d


32


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

$1.75 Standard Deviation:

$10.35 = .05

07.2150/35.10075.1

/

0:0:

0

0

z

z

nDxz

HH

dd

d

da

d

The critical value of z.05 is

1.96, so we would reject

this null hypothesis.


33

9.4: Comparing Two Population Proportions: Independent Sampling

.ˆ and ˆ examiningby and about inferences makecan We

21 21 pppp

stics.characteri particular regardingsproportionsimilar have notmay or may groups Two


34


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 nqp

nqp

nqp

nqp

ppppEpp

pp


35


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

nqp

nqpzpp

nqp

nqpzppzpp

pp

pp


36


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


37


Men Womenp .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) 1.96340 250

.05 .062

M Wp q p qp p zn n


38


Men Womenp .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) 1.96340 250

.05 .062

M Wp q p qp p zn 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.


39


21about Hypothesis ofTest Sample-Large pp

zzDorppH

DppH

a

|| :regionRejection )()(:

0)often very ()(:Test Tailed-One

021

0210

2/

021

0210

|| :regionRejection )(:)(:Test Tailed-Two

zzDppHDppH

a

21)ˆˆ(

21

21

)ˆˆ(

21

11ˆˆ andˆ where

)ˆˆ( :StatisticTest

21

21

nnqp

nnxxp

ppz

pp

pp


40

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 = .01 level?

1992 1998Survey n 21,484 23,199

Number who gambled 4.684 5,313

Proportion who gambled .218 .229


41


1992 1998Survey 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 DH 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 .22421, 484 23,199

1 1ˆ ˆ

1 1(.224)(.776) .0039521, 484 23,199

ˆ ˆ( ) (.218 .229)Test Statistic: .00395

2.786

p p

p p

x xpn n

pqn n

p pz


42


1992 1998Survey 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 DH 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 .22421, 484 23,199

1 1ˆ ˆ

1 1(.224)(.776) .0039521, 484 23,199

ˆ ˆ( ) (.218 .229)Test Statistic: .00395

2.786

p p

p p

x xpn 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 = .01 level.


43


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

15ˆ and 15ˆ15ˆ and 15ˆ

2222

1111

qnpnqnpn


44

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


45


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

required

Estimates of 1 and 2 will be needed

2

22

21

22/

21 SE)()(

znn


46


Suppose you need to estimate the difference between two population means to within 2.2 at the = 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(96.1SE

)()(

21

2

2

21

2

22

21

22/

21

nn

nn

znn


47


SE = 2.2 = 5% level. 1

2 = 22 =15.

n1 and n2 = ?


48


Sample size needed to estimate (p1 - p2) Given (1 - ) 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


49


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[(.645.1SE

)()(

21

2

2

21

22211

22/

21

nn

nn

qpqpznn


50

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?


51

9.6: Comparing Two Population Variances: Independent Sampling


52


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


53


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 numeratorand 1 denominator degrees of freed

sE s E s Es

s Fs

F nn

om.


54


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.


55


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 ss 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.

nn


56


Home Runs, 2003

American League

National League

µ 178.5 169.25

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.


57


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

212

,.025 2 ,.025

2122

1 1

1415.8 1 1415.8 1951.4 951.4

1415.8 1 1415.8951.4 2.92 951.

U L

U L

s ss F s F

F F

2122

14 .32

.5096 4.6504

Home Runs, 2003

American League

National League

µ 178.5 169.25

2 1415.8 951.4

n 14 16


58


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

212

,.025 2 ,.025

2122

1 1

1415.8 1 1415.8 1951.4 951.4

1415.8 1 1415.8951.4 2.92 951.

U L

U L

s ss F s F

F F

2122

14 .32

.5096 4.6504

Home Runs, 2003

American League

National League

µ 178.5 169.25

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 Hs

sF HsF F


59


2 20 1 2

2 21 2

/2

Two-Tailed Test

:

:Test Statistic:

Larger sample varianceSmaller 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 (=5%)?


60

* Source: euroekonom.com


61


2 20 1 2

2 21 2

.025

Two-Tailed Test

:

:Test Statistic:

Larger sample variance 2.89 8.5Smaller sample variance .34

Rejection Region: 4.03

a

H

H

F

F F

µfast-growing = 8.69%µslow-growing = 2.29%2

fast-growing = 2.892

slow-growing = .34


62


2 20 1 2

2 21 2

.025

Two-Tailed Test

:

:Test Statistic:

Larger sample variance 2.89 8.5Smaller sample variance .34

Rejection Region: 4.03

a

H

H

F

F F

µfast-growing = 8.69%µslow-growing = 2.29%2

fast-growing = 2.892

slow-growing = .34

Reject the null hypothesis at the =.05 level.

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