1 portions of these notes are adapted from statistics 7e © 1997 prentice- hall, inc. making...

11

Portions of these notes are adapted from Statistics 7e © 1997 Prentice-

Hall, Inc.

Making ComparisonsMaking Comparisons

Inferences Based on Two Samples: Inferences Based on Two Samples: Confidence Intervals & Tests of HypothesesConfidence Intervals & Tests of Hypotheses

PBAF 527 Winter 2005PBAF 527 Winter 2005

22


Hall, Inc.TodayToday

1.1. Scallops, Sampling and the LawScallops, Sampling and the Law• Confidence Intervals, Hypothesis Testing, and SamplingConfidence Intervals, Hypothesis Testing, and Sampling

2.2. Hypothesis TestingHypothesis Testing• Special Cases: Small samples, proportionsSpecial Cases: Small samples, proportions

3.3. Making ComparisonsMaking Comparisons• Solve Hypothesis Testing Problems for Two PopulationsSolve Hypothesis Testing Problems for Two Populations

MeanMean Proportion Proportion

• Distinguish Independent & Related PopulationsDistinguish Independent & Related Populations• Create Confidence Intervals for the DifferencesCreate Confidence Intervals for the Differences

33


Hall, Inc.

Scallops, Sampling, and Scallops, Sampling, and the Lawthe Law

• Read CaseRead Casea.a. Can a reliable estimate of the mean weight of all Can a reliable estimate of the mean weight of all

the scallops be obtained from a sample size of the scallops be obtained from a sample size of 18?18?

b.b. Do you see any flaws in the rule to confiscate a Do you see any flaws in the rule to confiscate a scallop catch if the sample mean weight is less scallop catch if the sample mean weight is less than 1/36 of a pound?than 1/36 of a pound?

c.c. Develop your own procedure for determining Develop your own procedure for determining whether a ship is in violation of the weight whether a ship is in violation of the weight restriction using the data provided.restriction using the data provided.

d.d. Apply your procedure to the data provided.Apply your procedure to the data provided.

44








55


Hall, Inc.

Hypothesis Testing When Hypothesis Testing When

n is Small and n is Small and σσ UnknownUnknown

Because the sample is smallBecause the sample is small Cannot assume normalityCannot assume normality Cannot assume s is a good approximation for Cannot assume s is a good approximation for

σσ

So, use t-distribution:So, use t-distribution:

with n-1 degrees of freedomwith n-1 degrees of freedom

n

sx

t

n

sx

t

66


Hall, Inc.

Small Sample t-test Small Sample t-test Example 1 (1)Example 1 (1)

Most water treatment facilities monitor the quality of their drinking Most water treatment facilities monitor the quality of their drinking water on hourly basis. One variable monitored it is pH, which water on hourly basis. One variable monitored it is pH, which measures the degree of alkalinity or acidity in the water. A pH measures the degree of alkalinity or acidity in the water. A pH below 7.0 is acidic, one above 7.0 is alkaline, and a pH of 7.0 is below 7.0 is acidic, one above 7.0 is alkaline, and a pH of 7.0 is neutral. One water treatment plant has a target pH of 8.5 (most neutral. One water treatment plant has a target pH of 8.5 (most try to maintain a slightly alkaline level). The mean and standard try to maintain a slightly alkaline level). The mean and standard deviation of 1 hour’s test results, based on 17 water samples at deviation of 1 hour’s test results, based on 17 water samples at this plant are:this plant are:

Does this sample provide sufficient evidence that the mean pH Does this sample provide sufficient evidence that the mean pH level in the water differs from 8.5?level in the water differs from 8.5?

s=.16s=.1624.8x 24.8x

77


Hall, Inc.


1. Establish hypotheses1. Establish hypotheses

2. Set the decision rule for the test: 2. Set the decision rule for the test:

pick pick find tfind t at n-1 dfat n-1 df

3.3. Find test statisticFind test statistic

4. Compare test statistic to critical value.4. Compare test statistic to critical value.

05.2039.

08.

17

16.5.842.8

n

sx

t

05.2039.

08.

17

16.5.842.8

n

sx

t

=.05 (for two-sided test this is .025 in each tail)

H 0:=8.5 Ha: 8.5

if |t|>t at n-1 df then reject the null hypothesis

t=2.12 with 16 degrees of freedom

Since |t|< t we cannot the null hypothesis at a 5% level. We cannot conclude that that the mean pH differs from the target based on the sample evidence.

88


Hall, Inc.


A major car manufacturer wants to test a new engine to A major car manufacturer wants to test a new engine to determine whether it meets new air-pollution standards. The determine whether it meets new air-pollution standards. The mean emission mean emission for all engines of this type must be less than for all engines of this type must be less than 20 parts per million of carbon. 10 engines are manufactured for 20 parts per million of carbon. 10 engines are manufactured for testing purposes, and the emission level for each is determined. testing purposes, and the emission level for each is determined. The mean and standard deviation for the tests are: The mean and standard deviation for the tests are:

Do the data supply enough evidence to allow the manufacturer Do the data supply enough evidence to allow the manufacturer to conclude that this type of engine meets the pollution to conclude that this type of engine meets the pollution standard? Assume the manufacturer is willing to risk a Type I standard? Assume the manufacturer is willing to risk a Type I error with probability error with probability =.01.=.01.

s=2.98s=2.9817.17x 17.17x

99


Hall, Inc.




pick pick find tfind t



00.3

10

98.22017.17

n

sx

t

00.3

10

98.22017.17

n

sx

t

=.01 (for one-sided test this is .01 in the tail)

H 0:≥20 Ha: <20

if t<t then reject the null hypothesis

t=-2.821 with 9 degrees of freedom

We can reject the null. The actual value is less than 20 ppm, and the new engine type meets the pollution standard.

1010


Hall, Inc.

Large Sample Test for Large Sample Test for the Population the Population

ProportionProportion

When the sample size is large (np and nq When the sample size is large (np and nq are greater than 5)are greater than 5) Assume is distributed normally with Assume is distributed normally with

mean p and standard deviationmean p and standard deviation where q=1-pwhere q=1-p

Test statistic: Test statistic:

2- or 1-tailed tests2- or 1-tailed tests

p̂p̂

n

pqn

pq

nqp

ppz

00

0ˆ

nqp

ppz

00

0ˆ

1111


Hall, Inc.

Large Sample Tests Large Sample Tests for Proportion for Proportion Example (1)Example (1)

In screening women for breast cancer, doctors use a In screening women for breast cancer, doctors use a method that fails to detect cancer in 20% of the women method that fails to detect cancer in 20% of the women who actually have the disease. Suppose a new method who actually have the disease. Suppose a new method has been developed that researchers hope will detect has been developed that researchers hope will detect cancer more accurately. This new method was used to cancer more accurately. This new method was used to screen a random sample of 140 women known to have screen a random sample of 140 women known to have breast cancer. Of these, the new method failed to detect breast cancer. Of these, the new method failed to detect cancer in 12 women. cancer in 12 women.

Does this sample provide evidence that the failure rate Does this sample provide evidence that the failure rate of the new method differs from the one currently in use?of the new method differs from the one currently in use?

1212


Hall, Inc.

Large Sample Tests Large Sample Tests for Proportion for Proportion Example (2)Example (2)



pick pick find tfind t



=.05 (for two-sided test this is .025 in each tail)

H 0:p=.2 Ha: p≠.2

if |z|>z then reject the null hypothesis

z=1.96 36.3

034.

114.

140)8)(.2(.

2.086.ˆ

00

0

nqp

ppz 36.3

034.

114.

140)8)(.2(.

2.086.ˆ

00

0

nqp

ppz

Since the test statistic falls in the rejection region, we can reject the null. The rate of detection for the new test differs from the old at a .05 level of significance.

1313








1414


Hall, Inc.

How Would You Try to How Would You Try to Answer These Answer These

Questions?Questions?1. Do house prices in two Seattle neighborhoods differ? 1. Do house prices in two Seattle neighborhoods differ?

By how much?By how much?

2. Does one method of teaching reading produce better 2. Does one method of teaching reading produce better results than another? results than another? Can I still have a reliable result with a small sample size? Can I still have a reliable result with a small sample size? How much better are the results of the method?How much better are the results of the method?

3. Do energy conservation efforts really reduce 3. Do energy conservation efforts really reduce consumption over time? consumption over time? How much of a reduction?How much of a reduction?

4. Is the proportion of subprime mortgages to low-income 4. Is the proportion of subprime mortgages to low-income households greater than that for moderate-income households greater than that for moderate-income households? households? How much greater?How much greater?

1515


Hall, Inc.Two Population TestsTwo Population Tests

TwoPopulations

Z Test(Large

sample)

t Test(Pairedsample)

Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

1616


Hall, Inc.

Comparison of Means for Comparison of Means for Independent SubsamplesIndependent Subsamples

Three scenarios:Three scenarios:

1. H1. H00: : 11--22=0=0 Ha: Ha: 11--2200

2. H2. H00: : 11--2200 Ha: Ha: 11--22>0>0

3.3. HH00: : 11--22DD Ha: Ha: 11--22>D >D

(not common, nor is 2-tailed test of D)(not common, nor is 2-tailed test of D)

Could be:Could be: Separate (unequal) Variances (Large Samples)Separate (unequal) Variances (Large Samples) Equal Population Variances (Small Samples)Equal Population Variances (Small Samples)

1717



TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

1818


Hall, Inc.

Comparison of MeansComparison of MeansSeparate (Unequal) Variances Separate (Unequal) Variances for 2 Independent Subsamplesfor 2 Independent Subsamples

1.1. AssumptionsAssumptions Independent, Random SamplesIndependent, Random Samples Populations Are Normally DistributedPopulations Are Normally Distributed If Not Normal, Can Be Approximated by Normal If Not Normal, Can Be Approximated by Normal

Distribution (Distribution (nn11 30 & 30 & nn22 30 ) 30 ) For n’s<30, use t with the smaller of nFor n’s<30, use t with the smaller of n11-1, n-1, n22-1 df-1 df

2.2. Two Independent Sample Z-Test StatisticTwo Independent Sample Z-Test Statistic

ZZXX XX

nn nn

11 22 11 22

1122

11

2222

22

XX XX

nn nn

11 22 11 22

1122

11

2222

22

ss ss

1919


Hall, Inc.

Example Example Separate (unequal) Separate (unequal)

VariancesVariances

You gather data on property value for a random sample You gather data on property value for a random sample of 32 properties in Sandpoint and find that =$345,650 of 32 properties in Sandpoint and find that =$345,650 and s=$48,500. Then you gather data on the value of a and s=$48,500. Then you gather data on the value of a random sample of 35 properties in Ravenna and find random sample of 35 properties in Ravenna and find that =$289,440 and s=$87,090. Is the average property that =$289,440 and s=$87,090. Is the average property value of all properties in both locations equal or not?value of all properties in both locations equal or not?

Do house prices in two Seattle Neighborhoods differ? Do house prices in two Seattle Neighborhoods differ?

By how much?By how much?Is the average price of a home of a certain size equal Is the average price of a home of a certain size equal in Sandpoint and Ravenna?in Sandpoint and Ravenna?

2020


Hall, Inc.

Separate (unequal) Separate (unequal) Variances Variances SolutionSolution

HH00::

HHaa::

nn11 = = , , nn22 = =

Critical Value(s) Critical Value(s) or or /2? /2?::

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

2121


Hall, Inc.

Separate (unequal) Separate (unequal) Variances Variances SolutionSolution

HH00::

HHaa::

nn11 = = , , nn22 = =

Critical Value(s) Critical Value(s) or or /2? /2?::


Decision:Decision:


µµ11 - µ - µ2 2 = 0 (µ= 0 (µ1 1 = µ= µ22))

µµ11 - µ - µ2 2 ≠≠ 0 (µ 0 (µ1 1 ≠ µ≠ µ22))

.05.05

35353232

zz00 1.961.96-1.96-1.96

.025.025

Reject HReject H00 Reject HReject H00

.025.025

3.3

35090,87

32650,48

)0(440,289650,34522

2

22

1

21

02121

nn

xxz

|z|>z|z|>z/2/2

Reject at Reject at = .05 = .05

There is Evidence of a There is Evidence of a Difference in MeansDifference in Means

2222


Hall, Inc.

Confidence Interval for Confidence Interval for the Differencethe Difference

A (1-A (1-)100% confidence interval for the difference )100% confidence interval for the difference between two population means between two population means 11--22 using independent using independent

random sampling:random sampling:



for n’s<30 use tfor n’s<30 use t/2/2 with the with the

lesser of nlesser of n11-1, n-1, n22-1 df-1 df 2

22

1

21

2/21 nnzxx

NB: are the variances of each of the two populations; when NB: are the variances of each of the two populations; when 1122 and and 22

22 are are

unknown, use sunknown, use s1122 and sand s

2222..

2323


Hall, Inc.

Confidence Interval for Confidence Interval for the Differencethe Difference

Construct a 95% confidence interval around the Construct a 95% confidence interval around the difference and interpret it in words.difference and interpret it in words.



2424



TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

2525


Hall, Inc.

Comparison of MeansComparison of MeansEqual Population Equal Population

Variances Variances

1.1.Tests Means of 2 Independent Tests Means of 2 Independent Populations Having Populations Having EqualEqual Variances Variances

2.2.AssumptionsAssumptions Independent, Random SamplesIndependent, Random Samples Both Populations Are Normally DistributedBoth Populations Are Normally Distributed Population Variances Are Population Variances Are UnknownUnknown But But

Assumed Assumed EqualEqual

3. 3. Usually small samplesUsually small samples

2626


Hall, Inc.


Variances Variances

2

2

11

11

21

21

222

2112

21

2

2121

nndf

nn

snsns

nns

xxt

P

P

2

2

11

11

21

21

222

2112

21

2

2121

nndf

nn

snsns

nns

xxt

P

P

Estimate of Estimate of Standard Standard ErrorError

We select two independent random samples: from population 1 of size n1 with mean 1x and variance s1

2

from population 2 of size n2 with mean 2x and variance s22

For large n’s, use zFor large n’s, use z

Pooled Pooled Estimate of Estimate of VarianceVariance

2727


Hall, Inc.


VariancesVariances

Example:Example:

Does one method of teaching reading Does one method of teaching reading produce better results than another? produce better results than another?

Can I still have a reliable result with a Can I still have a reliable result with a small sample size? small sample size?

How much better are the results of the How much better are the results of the method?method?

2828


Hall, Inc.

Comparison of MeansComparison of MeansEqual Population VariancesEqual Population Variances

ExampleExample

You decide to base this comparison on the results of a reading test You decide to base this comparison on the results of a reading test given at the end of a learning period of 6 months. given at the end of a learning period of 6 months.

Of a random sample of 22 slow learners, 10 are taught by the new Of a random sample of 22 slow learners, 10 are taught by the new method and 12 are taught by the standard method. Qualified method and 12 are taught by the standard method. Qualified instructors under similar conditions teach all 22 children for a 6-month instructors under similar conditions teach all 22 children for a 6-month period. The results of the reading test at the end of 6 months are as period. The results of the reading test at the end of 6 months are as follows:follows:

New Method: =76.4; sNew Method: =76.4; s1122=34.04=34.04

Standard Method: =72.33; sStandard Method: =72.33; s2222=40.24=40.24

Are the reading scores of children using the new method greater Are the reading scores of children using the new method greater than those of children using the standard method with than those of children using the standard method with alpha=.05?alpha=.05?

Compare a new method of teaching reading to “slow Compare a new method of teaching reading to “slow learners” to the current standard method.learners” to the current standard method.

1x1x

2x2x

2929


Hall, Inc.


SolutionSolution

HH00::

HHaa::

df df Critical Value(s):Critical Value(s):

Decision Rule:Decision Rule:


Decision:Decision:


n1+n2-2=10+12-2=20 df

3030


Hall, Inc.

Small-Sample t Test Small-Sample t Test SolutionSolution

45.37

21210

24.4011204.34110

2

11

35.0

121

101

45.37

033.724.76

11

21

222

2112

21

2

2121

nn

SnSnS

nnS

XXt

P

P

45.37

21210

24.4011204.34110

2

11

35.0

121

101

45.37

033.724.76

11

21

222

2112

21

2

2121

nn

SnSnS

nnS

XXt

P

P

3131


Hall, Inc.


SolutionSolution

HH00::

HHaa::

df df Critical Value(s):Critical Value(s):



Decision:Decision:


n1+n2-2=10+12-2=20 df

55.1

121

101

45.37

033.724.76

t

55.1

121

101

45.37

033.724.76

t

We do not have enough evidence to reject the null hypothesis and conclude that the new method of testing does not improve reading scores.

3232


Hall, Inc.

Confidence Interval for the Confidence Interval for the Difference Difference

(equal population variances)(equal population variances)

Does one method of teaching reading produce better results than Does one method of teaching reading produce better results than another? another?

How much better are the results of the method?How much better are the results of the method?

Construct a 95% confidence interval for the difference between the Construct a 95% confidence interval for the difference between the two means and interpret it.two means and interpret it.

]54.9,4.1[47.507.4)62.2)(086.2(07.4

12

1

10

145.37086.233.724.76

11

21

22/21

nn

stxx p

]54.9,4.1[47.507.4)62.2)(086.2(07.4

12

1

10

145.37086.233.724.76

11

21

22/21

nn

stxx p

3333



TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

3434


Hall, Inc.

Paired-Sample t Test Paired-Sample t Test for Mean Differencefor Mean Difference

1.1. Tests Means of 2 Related PopulationsTests Means of 2 Related Populations Paired or MatchedPaired or Matched Repeated Measures (Before/After)Repeated Measures (Before/After)

2.2. Eliminates Variation Among SubjectsEliminates Variation Among Subjects

3.3. AssumptionsAssumptions Both Population Are Normally DistributedBoth Population Are Normally Distributed If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by

Normal Distribution (Normal Distribution (nn11 30 & 30 & nn22 30 ) 30 )

Is a population parameter different over time or between groups?

3535


Hall, Inc.

Paired-Sample t Test Paired-Sample t Test HypothesesHypotheses

Note: Note: DDii = = XX11ii - - XX22ii for ith for ith observationobservation

Research QuestionsResearch Questions

HypothesisHypothesis No DifferenceNo DifferenceAny DifferenceAny Difference

Pop 1 Pop 1 Pop 2Pop 2Pop 1 < Pop 2Pop 1 < Pop 2

Pop 1 Pop 1 Pop 2Pop 2Pop 1 > Pop 2Pop 1 > Pop 2

HH00 DD = 0= 0 DD 00 DD 00

HH11 DD 00 DD < 0< 0 DD > 0> 0

3636


Hall, Inc.

Paired-Sample t Test Paired-Sample t Test Data Collection TableData Collection Table

ObservationObservation Group 1Group 1 Group 2Group 2 DifferenceDifference

11 xx1111 xx2121 DD11 = x= x1111-x-x2121

22 xx1212 xx2222 DD22 = x= x1212-x-x2222

ii xx1i1i xx2i2i DDii = x= x1i1i - x- x2i2i

nn xx1n1n xx2n2n DDnn = x= x1n1n - x- x2n2n

3737


Hall, Inc.

Paired-Sample t Test Paired-Sample t Test Test StatisticTest Statistic

Sample MeanSample Mean Sample Sample Standard Standard DeviationDeviation

z or tz or txxDD

ss

nn

dfdf nn

xxDD

nnss

(D(Dii - x - xDD))22

nn

00

DD

iiii

nn

DDii

nn

11 11

11

11

DD

DD

DD DD

DD

DD

When n>30,use z When n>30,use z When n<30 use t(n-1) dfWhen n<30 use t(n-1) df

DD00=0 when testing whether =0 when testing whether

there is any difference or not.there is any difference or not.

3838


Hall, Inc.

Paired-Sample t TestPaired-Sample t TestExampleExample

A study is undertaken to determine how consumers react to energy A study is undertaken to determine how consumers react to energy conservation efforts. A random group of 60 families is chosen. conservation efforts. A random group of 60 families is chosen. Each family’s rate of consumption of electricity is monitored in Each family’s rate of consumption of electricity is monitored in equal length time periods before and after they are offered financial equal length time periods before and after they are offered financial incentives to reduce their energy consumption rate. The difference incentives to reduce their energy consumption rate. The difference in electric consumption between the periods is recorded for each in electric consumption between the periods is recorded for each family. The average reduction in consumption is 0.2 kW and the family. The average reduction in consumption is 0.2 kW and the standard deviation of the differences sstandard deviation of the differences sDD=1.0 kW. At =1.0 kW. At =0.01, is =0.01, is

there evidence to conclude that the incentives reduce there evidence to conclude that the incentives reduce consumption?consumption?

Do energy conservation efforts really reduce consumption Do energy conservation efforts really reduce consumption over time? By how much?over time? By how much?

3939


Hall, Inc.


Decision:Decision:


Paired-Sample t Test Paired-Sample t Test SolutionSolution

HH00::

HHaa: :

==


Critical Value(s):Critical Value(s):

μD=0

μD<0

.01

n>30, so use z; z<z

zz00-2.326-2.326

.01.01

ttxx

ss

nn

00

DD

DD -0.2-0.2 00

11 00

6060

-1.55-1.55..

DD

DD

z is not less than z; we cannot reject the null hypothesis.

We do not have enough evidence to say that incentives reduce consumption.

4040


Hall, Inc.

Confidence Interval for Confidence Interval for Paired ObservationsPaired Observations

A (1-A (1-)100% confidence interval for the mean difference is )100% confidence interval for the mean difference is constructed using the t distribution for small sample sizes and z constructed using the t distribution for small sample sizes and z distribution for large sample sizes.distribution for large sample sizes.

n

szx D

D 2/n

szx D

D 2/

Do energy conservation efforts really reduce consumption Do energy conservation efforts really reduce consumption over time? over time? By how much?By how much?

When n>30,use z When n>30,use z When n<30 use t(n-1) df When n<30 use t(n-1) df

]1.0,5.0[60

0.1576.22.02/

n

szx D

D ]1.0,5.0[60

0.1576.22.02/

n

szx D

D

4141



TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

4242


Hall, Inc.

Z Test for Difference Z Test for Difference in Two Proportionsin Two Proportions

1.1. Can testCan test HH00: p: p11-p-p22=0=0 HHaa: p: p11-p-p2200

HH00: p: p11-p-p2200 HHaa: p: p11-p-p22>0>0

2. Assumptions2. Assumptions Populations Are IndependentPopulations Are Independent Normal Approximation Can Be UsedNormal Approximation Can Be Used

Does Not Contain 0 or nDoes Not Contain 0 or n

3.3. Z-Test Statistic for Two ProportionsZ-Test Statistic for Two Proportions

21

21

21

21 ˆ where11

ˆ1ˆ

0ˆˆ

nn

xxp

nnpp

ppz

21

21

21

21 ˆ where11

ˆ1ˆ

0ˆˆ

nn

xxp

nnpp

ppz

ˆ1ˆ3ˆ ppnpn ˆ1ˆ3ˆ ppnpn

4343


Hall, Inc.

Z Test for Difference Z Test for Difference in Two Proportionsin Two Proportions

In 1998, a sample of mortgages was taken from the over 1 In 1998, a sample of mortgages was taken from the over 1 million mortgages disclosed nationally under HMDA. Here is million mortgages disclosed nationally under HMDA. Here is a decription of the sample: a decription of the sample:

Income GroupIncome Group Percent SubprimePercent Subprime nn

Low-incomeLow-income 26%26% 400400

Moderate-incomeModerate-income 11%11% 600600

Is the proportion of sub-prime mortgages to low-income Is the proportion of sub-prime mortgages to low-income households greater than than for moderate-income households? households greater than than for moderate-income households? How much greater?How much greater?

Is there sufficient evidence to claim that the proportion of sub-Is there sufficient evidence to claim that the proportion of sub-prime mortgages to low-income households exceeds that prime mortgages to low-income households exceeds that among moderate income households? Test using among moderate income households? Test using ==.01.01

4444


Hall, Inc.


Decision: Decision:


Z Test for Two Z Test for Two Proportions SolutionProportions Solution

HH00: :

HHaa: :

= =

nn11 = = nn22 = =



p1-p2>0

p1-p2≤0

.01

4545


Hall, Inc.

19.6

6001

4001

17.117.

011.26.

11ˆ1ˆ

0ˆˆ

17.600400

66104ˆ

6611.600ˆ10426.400ˆ

21

21

21

21

2211

nnpp

ppZ

nn

XXp

pnpn

19.6

6001

4001

17.117.

011.26.

11ˆ1ˆ

0ˆˆ

17.600400

66104ˆ

6611.600ˆ10426.400ˆ

21

21

21

21

2211

nnpp

ppZ

nn

XXp

pnpn


4646


Hall, Inc.


Decision: Decision:



HH00: :

HHaa: :

= =

nn11 = = nn22 = =



p1-p2>0

p1-p2≤0

.01

4747


Hall, Inc.

Confidence Interval for Confidence Interval for the Difference in the Difference in

ProportionsProportions

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

2

22

1

112/21

ˆ1ˆˆ1ˆˆˆ

n

pp

n

ppzpp

2

22

1

112/21

ˆ1ˆˆ1ˆˆˆ

n

pp

n

ppzpp

Is the proportion of sub-prime mortgages to low-income Is the proportion of sub-prime mortgages to low-income households greater than than for moderate-income households? households greater than than for moderate-income households? How much greater?How much greater?

]18,.12[.)02538.0(575.215.

600

11.111.

400

26.126.575.211.26.

]18,.12[.)02538.0(575.215.600

11.111.

400

26.126.575.211.26.

How much greater is the proportion of subprime mortgages to How much greater is the proportion of subprime mortgages to low-income buyers compared to moderate-income buyers?low-income buyers compared to moderate-income buyers?

4848



TwoPopulations

Z Test(Large

sample)


Z Test

Proportion Variance

F Testt Test(Small

sample)

Paired

Indep.

Mean

4949



1.1. Scallops, Sampling and the LawScallops, Sampling and the Law• Confidence Intervals, Hypothesis Testing, and Confidence Intervals, Hypothesis Testing, and

SamplingSampling

2.2. Making ComparisonsMaking Comparisons• Solve Hypothesis Testing Problems for Two Solve Hypothesis Testing Problems for Two

PopulationsPopulations MeanMean Proportion Proportion


End of Chapter

Any blank slides that follow are blank intentionally.

1 portions of these notes are adapted from statistics 7e © 1997 prentice- hall, inc. making...

Documents