QMS 202 BUSINESS STATISTICS II
1/9/2011
Dr.Chua
QMS202 BUSINESSSTATISTICSII InferentialStatistics allowsyoutoestimateunknown populationcharacteristics,i.e.population meanorpopulationproportion
WhatarerequiredforQMS202? Requiredtextbook BusinessStatistics,ATworSemesterTextforBusiness Management,SeventhCustomEdition forRyerson University,CompiledbyDarrylSmithandClareChua PrenticeHall,2010 2 R 2.RequiredCalculator i dC l l t CASIOFXr9750GII Casio FXr9750Gplusisacceptable TIr83calculatormaybeusedinthecourse.Howeverthe professorwillnotbesupportingitinclass.Youare responsibleforknowinghowtousetheTIr83functions. 3. Statisticalsoftwarer SPSS
OFFICE:TRS1r030 EMAIL:[email protected] TheSubjectfieldofeachemailshouldbecarefullycompleted,e.g.QMS202:sick. Itisrequestedthatemailsbesenttotheprofessoronlyifthequestion(s)being askedcannotwaittobeansweredinthenextscheduledlecture.Ortheanswers cannotbefoundinthecourseoutline. f
OfficeHours: Byappointmentonly
MethodofEvaluation Component
Percent/ Weight
1.
Number of Crib sheet allowed
Date
8.5x11inchcrib sheet(twor sided)
Quiz #1
12%
one
Week 4*
Test
20%
one
Week 8*
13%
one
W k 11* Week
Q i #2 Quiz
5%
Group project (SPSS) Final Exam
50%
TOTAL
Week 10*
three
100%
TBA * Please refer to handout for the test dates
TOPICS– SEQUENCE&SCHEDULE Week
Topic Standardized normal random variable Z. Confidence Intervals
Chapter 6 Handouts and chapter 9 in the textbook.
Answer questions in handouts and questions in ch.9 of the textbook.
3 to 4
Fundamentals of Hypothesis Testing: One-Sample Tests
Handouts and chapter 10 in the textbook.
Answer questions in handouts and questions in ch.10 of the textbook.
5 to 7
Two-Sample Tests Handouts and and d One-Way O W ANOVA. ANOVA chapter h t 11 iin the th textbook.
Answer questions in h d t and handouts d questions in ch.11 of the textbook.
8 to 9
Chi-Square Tests
Handouts and chapter 12 in the textbook.
Answer questions in handouts and questions in ch.12 of the textbook.
Simple Linear Regression
Handouts and chapter 13 in the textbook.
Answer questions in handouts and questions in ch.13 of the textbook.
12
I ntroduction to Multiple Regression
Chapter 14
Answer questions in handouts.
13
Review
10 to 11
Statisticscanbe….
Activities & Assignments
1 to 2
• Descriptive QMS102
QMS202
_collectinganddescribingthedata r CollectDatae.g.survey<SourcesofData> r PresentDatae.g.TablesandGraphs r CharacterizeData
•Inferential istheprocessofusingsampleresultstodrawconclusion aboutthecharacteristicsofapopulation. •Estimation isusedtoestimatepopulationparameters • Enablesustoestimateunknownpopulationcharacteristicssuchasa populationmeanorapopulationproportion
•HypothesisTesting
1
www.notesolution.com
1/9/2011
WhatIwillbecoveringforthenext 1½weeks?
Chapter9 ConfidenceIntervalEstimation Topicscovered: ConfidenceIntervalEstimationfortheMean(known) ConfidenceIntervalEstimationfortheMean(unknown) ConfidenceIntervalEstimationfortheProportion DeterminingSampleSize
Confidence Interval Estimation
Qualitative Data
Quantitative Data Population Mean P
Population Proportion E
X Normal and/or nt30? If no stop.
np t 5 and n(1-p) t 5? If no stop.
V known?
ESTIMATION TWOtypesofestimatesusedto estimatepopulationparameters: (1) Pointestimate(example:Cx,p) (2) Intervalestimate
V unknown? Limits: prz
p=x/n Limits: x r z
1 n
p(1 p) n
Limits: x r t s n df=n-1
Pointestimater MarginofError
Two things you need to find: (1) Confidence Interval and (2) Sample size
Estimations Population Parameters S
• Whatisapointestimate? isavalueofasingle samplestatistics. Example:samplemean,,isapoint x estimate ofthepopulationmean,P sampleproportion,p,isapoint Sample estimateofthepopulationproportion, Estimators E. sampling Cx • Whatisintervalestimate? p isarangeofnumbers,calledaninterval, s constructedaroundthepoint t t d d th i t estimate. Theconfidence intervalisconstructed suchthattheprobabilitythe populationparameterislocated somewherewithintheintervalis Inference known.
Pointestimater MarginError ConfidenceIntervalEstimationfortheMean Sample Estimators
Population Parameters S
sampling
Cx s p
x r confidence * s tan dard error x r zD / 2 * s tan dard error x r zD / 2 *
Pointestimate
V n MarginError
Inference
2
www.notesolution.com
1/9/2011
ConfidenceIntervalEstimation Sample Estimators
Population Parameters sampling
S
Cx p s
Inference
EXAMPLE Supposeyouareaskedtoestimate themeanheightofALL the studentswhotaketheQMS202 courseinWinter2011. Themeanheightofallthe studentsisanunknown populationmean,denotedbyP . Yourandomlyselectasampleof10 studentsandfoundthatthe samplemean,CX ,is___,whichis thepointestimateofthe populationmean,P. Howaccurateismyestimateof _____? Toanswerthisquestion,youmust constructaconfidenceinterval estimate.
ConfidenceIntervalEstimation Sample Estimators
Population Parameters sampling
S
Inference
Cx p s
EXAMPLE Population:ALLthestudentsin theclass(150students) POPULATIONDATA:heightmeasurement are______________________. Thepopulationmeantestscore,µ,= _____ Samplesize=10students Data: SampleMean=CX=<meanwillbe calculatedinclass>
Objective Learnhowtoconstruct andinterpret confidenceintervalestimates x r
Construct 95%CI
z
* D /2
V n
42.1 d P d 62.5
Beforelearningtoconstructthe confidenceinterval,youneedto know; 1. CentralLimitTheorem 2. Zrvalue
Interpretation: A95%confidenceintervalforthepopulationaverage (mean) testscoreis(42.1,62.5). Note:DefineXor Note:DefineXor
Note: WeusedCx toestimate .Therefore,ourobjectiveistousethesampling distributionofCx tosaysomethingabout .
CentralLimitTheorem Assumethatthepopulationfromwhichwewillrandomlyselecta sampleofnmeasurementshasameanP andstandard deviationV.Thenthepopulationofallpossiblesamplemeans has:
Px P 1. Mean: V 2. standarddeviation: V x n 3a.Ifpopulationofindividualmeasurementsis x normal,isnormal 3b.Ifpopulationofindividualmeasurements x isnotnormal,isnormalifsamplesizeis largeenough(nt 30)
Beforelearningtoconstructthe confidenceinterval,youneedto know; 1. CentralLimitTheorem 2. Zrvalue Note: Weusedxtoestimate .Therefore,ourobjectiveistousethesampling distributionofxtosaysomethingabout .
3
www.notesolution.com
1/9/2011
Zrvalue StandardNormalDistribution CharacteristicsoftheStandardNormal WhydowehavetolearnaboutStandard Normal? Example: l X=Amountoftimeauniversitystudentspendssleepingdaily,followinga normaldistributionwithmean =8hoursandstandarddeviation=1.5 hours SupposeSamspends9.5hourssleepingonenight. Howdoeshissleeptimecomparetotheaverage?
Zvalue StandardNormalDistribution CharacteristicsoftheStandardNormal WhydowewanttolearnaboutStandardNormal? Example: X=Amountoftimeauniversitystudentspendssleepingdaily,followinganormal distribution with mean =8 distributionwithmean 8hoursandstandarddeviation hours and standard deviation =1 1.5hours 5 hours SupposeSamspends9.5hourssleepingonenight. Didhesleepa“lotmore”thantheaverage? 1.Howmuchmoreis9.5hoursthantheaverageof8hours?1.5hours 2.Calculatehowmanystandarddeviationfromtheaverage? Asyoucansee1.5hoursequals1standarddeviation WecansaythatSam’ssleeptimewas1stddevmorethantheaverage. WHICHISABETTERMEASURE?1or2
TransformationFormula WHICHISABETTERMEASURE?1or2 2.Calculatehowmanystandarddeviationfromtheaverage? Asyoucansee1.5hoursequals1standarddeviation WecansaythatSam’ssleeptimewas1stddevmorethantheaverage. Mathematically, X=9.5,µ=8,=1.5
Z
ConvertanynormalrandomvariableXtostandardizednormal randomvariableZ.
Z
X P V
IfXisnormallydistributedN( ,),thestandardizedvariableZ hasastandardnormaldistributionwithmean=0(or =0)and standard deviation 1 (or =1), standarddeviation1(or 1),denotedN(0,1) denoted N(0,1)
X P V
Thisiscalledthe“relativestanding”r knownaszrvalue “transformationofeachxrvaluetoazrvalue” r3
r2
r1
+1
Example
+2
+3
=2
ThetimetodownloadtheWebpageisnormally distributedwithamean =7secondsanda standarddeviation=2seconds What is the Z value for a download time of 9 WhatistheZvalueforadownloadtimeof9 seconds? Z
97 2
r3
1
r1
+1
1
r2 3
5
7
9
+2 11
+3 13
r3
r2
r1
0
+1
+2
+3
Xscale(µ=7, =2) Zscale(µ=0, =1)
Answer:adownloadtimeof9secondsisequivalentto1standardizedunit
4
www.notesolution.com
1/9/2011
Example
=2
ThetimetodownloadtheWebpageisnormally distributedwithamean =7secondsanda standarddeviation=2seconds What is the Z value for a download time of 1 WhatistheZvalueforadownloadtimeof1 second? Z
1 7 2
r3
3
r1
+1
1
r2 3
5
7
9
+2 11
+3 13
r3
r2
r1
0
+1
+2
+3
Xscale(µ=7, =2) Zscale(µ=0, =1)
Answer:adownloadtimeof1secondisequivalentto3standardizedunits
FindingAreasUsingStandardizedvariables
FindingAreasUsingStandardizedvariables
=7
=7
Area=?
Area=?
X =75
X=86 Z=?
=75 Z=? X=86 Solution: Z=(xr )/ Z=(86r75)/7=1.5714
Example: SamtookanEconomicsexamandscored 86points.Theclassmeanwas75withasstandard deviationof7.WhatpercentileisSamin?
FindingAreasUsingStandardizedvariables
GivenAreafindZvalue
=1
=1
Area=?
Area==0.95
Z =0
Usingthecalculatortofindthearea: (1) Usingzrscoretofindarea: P(Z
Dr.Chua
QMS202 BUSINESSSTATISTICSII InferentialStatistics allowsyoutoestimateunknown populationcharacteristics,i.e.population meanorpopulationproportion
WhatarerequiredforQMS202? Requiredtextbook BusinessStatistics,ATworSemesterTextforBusiness Management,SeventhCustomEdition forRyerson University,CompiledbyDarrylSmithandClareChua PrenticeHall,2010 2 R 2.RequiredCalculator i dC l l t CASIOFXr9750GII Casio FXr9750Gplusisacceptable TIr83calculatormaybeusedinthecourse.Howeverthe professorwillnotbesupportingitinclass.Youare responsibleforknowinghowtousetheTIr83functions. 3. Statisticalsoftwarer SPSS
OFFICE:TRS1r030 EMAIL:[email protected] TheSubjectfieldofeachemailshouldbecarefullycompleted,e.g.QMS202:sick. Itisrequestedthatemailsbesenttotheprofessoronlyifthequestion(s)being askedcannotwaittobeansweredinthenextscheduledlecture.Ortheanswers cannotbefoundinthecourseoutline. f
OfficeHours: Byappointmentonly
MethodofEvaluation Component
Percent/ Weight
1.
Number of Crib sheet allowed
Date
8.5x11inchcrib sheet(twor sided)
Quiz #1
12%
one
Week 4*
Test
20%
one
Week 8*
13%
one
W k 11* Week
Q i #2 Quiz
5%
Group project (SPSS) Final Exam
50%
TOTAL
Week 10*
three
100%
TBA * Please refer to handout for the test dates
TOPICS– SEQUENCE&SCHEDULE Week
Topic Standardized normal random variable Z. Confidence Intervals
Chapter 6 Handouts and chapter 9 in the textbook.
Answer questions in handouts and questions in ch.9 of the textbook.
3 to 4
Fundamentals of Hypothesis Testing: One-Sample Tests
Handouts and chapter 10 in the textbook.
Answer questions in handouts and questions in ch.10 of the textbook.
5 to 7
Two-Sample Tests Handouts and and d One-Way O W ANOVA. ANOVA chapter h t 11 iin the th textbook.
Answer questions in h d t and handouts d questions in ch.11 of the textbook.
8 to 9
Chi-Square Tests
Handouts and chapter 12 in the textbook.
Answer questions in handouts and questions in ch.12 of the textbook.
Simple Linear Regression
Handouts and chapter 13 in the textbook.
Answer questions in handouts and questions in ch.13 of the textbook.
12
I ntroduction to Multiple Regression
Chapter 14
Answer questions in handouts.
13
Review
10 to 11
Statisticscanbe….
Activities & Assignments
1 to 2
• Descriptive QMS102
QMS202
_collectinganddescribingthedata r CollectDatae.g.survey<SourcesofData> r PresentDatae.g.TablesandGraphs r CharacterizeData
•Inferential istheprocessofusingsampleresultstodrawconclusion aboutthecharacteristicsofapopulation. •Estimation isusedtoestimatepopulationparameters • Enablesustoestimateunknownpopulationcharacteristicssuchasa populationmeanorapopulationproportion
•HypothesisTesting
1
www.notesolution.com
1/9/2011
WhatIwillbecoveringforthenext 1½weeks?
Chapter9 ConfidenceIntervalEstimation Topicscovered: ConfidenceIntervalEstimationfortheMean(known) ConfidenceIntervalEstimationfortheMean(unknown) ConfidenceIntervalEstimationfortheProportion DeterminingSampleSize
Confidence Interval Estimation
Qualitative Data
Quantitative Data Population Mean P
Population Proportion E
X Normal and/or nt30? If no stop.
np t 5 and n(1-p) t 5? If no stop.
V known?
ESTIMATION TWOtypesofestimatesusedto estimatepopulationparameters: (1) Pointestimate(example:Cx,p) (2) Intervalestimate
V unknown? Limits: prz
p=x/n Limits: x r z
1 n
p(1 p) n
Limits: x r t s n df=n-1
Pointestimater MarginofError
Two things you need to find: (1) Confidence Interval and (2) Sample size
Estimations Population Parameters S
• Whatisapointestimate? isavalueofasingle samplestatistics. Example:samplemean,,isapoint x estimate ofthepopulationmean,P sampleproportion,p,isapoint Sample estimateofthepopulationproportion, Estimators E. sampling Cx • Whatisintervalestimate? p isarangeofnumbers,calledaninterval, s constructedaroundthepoint t t d d th i t estimate. Theconfidence intervalisconstructed suchthattheprobabilitythe populationparameterislocated somewherewithintheintervalis Inference known.
Pointestimater MarginError ConfidenceIntervalEstimationfortheMean Sample Estimators
Population Parameters S
sampling
Cx s p
x r confidence * s tan dard error x r zD / 2 * s tan dard error x r zD / 2 *
Pointestimate
V n MarginError
Inference
2
www.notesolution.com
1/9/2011
ConfidenceIntervalEstimation Sample Estimators
Population Parameters sampling
S
Cx p s
Inference
EXAMPLE Supposeyouareaskedtoestimate themeanheightofALL the studentswhotaketheQMS202 courseinWinter2011. Themeanheightofallthe studentsisanunknown populationmean,denotedbyP . Yourandomlyselectasampleof10 studentsandfoundthatthe samplemean,CX ,is___,whichis thepointestimateofthe populationmean,P. Howaccurateismyestimateof _____? Toanswerthisquestion,youmust constructaconfidenceinterval estimate.
ConfidenceIntervalEstimation Sample Estimators
Population Parameters sampling
S
Inference
Cx p s
EXAMPLE Population:ALLthestudentsin theclass(150students) POPULATIONDATA:heightmeasurement are______________________. Thepopulationmeantestscore,µ,= _____ Samplesize=10students Data: SampleMean=CX=<meanwillbe calculatedinclass>
Objective Learnhowtoconstruct andinterpret confidenceintervalestimates x r
Construct 95%CI
z
* D /2
V n
42.1 d P d 62.5
Beforelearningtoconstructthe confidenceinterval,youneedto know; 1. CentralLimitTheorem 2. Zrvalue
Interpretation: A95%confidenceintervalforthepopulationaverage (mean) testscoreis(42.1,62.5). Note:DefineXor Note:DefineXor
Note: WeusedCx toestimate .Therefore,ourobjectiveistousethesampling distributionofCx tosaysomethingabout .
CentralLimitTheorem Assumethatthepopulationfromwhichwewillrandomlyselecta sampleofnmeasurementshasameanP andstandard deviationV.Thenthepopulationofallpossiblesamplemeans has:
Px P 1. Mean: V 2. standarddeviation: V x n 3a.Ifpopulationofindividualmeasurementsis x normal,isnormal 3b.Ifpopulationofindividualmeasurements x isnotnormal,isnormalifsamplesizeis largeenough(nt 30)
Beforelearningtoconstructthe confidenceinterval,youneedto know; 1. CentralLimitTheorem 2. Zrvalue Note: Weusedxtoestimate .Therefore,ourobjectiveistousethesampling distributionofxtosaysomethingabout .
3
www.notesolution.com
1/9/2011
Zrvalue StandardNormalDistribution CharacteristicsoftheStandardNormal WhydowehavetolearnaboutStandard Normal? Example: l X=Amountoftimeauniversitystudentspendssleepingdaily,followinga normaldistributionwithmean =8hoursandstandarddeviation=1.5 hours SupposeSamspends9.5hourssleepingonenight. Howdoeshissleeptimecomparetotheaverage?
Zvalue StandardNormalDistribution CharacteristicsoftheStandardNormal WhydowewanttolearnaboutStandardNormal? Example: X=Amountoftimeauniversitystudentspendssleepingdaily,followinganormal distribution with mean =8 distributionwithmean 8hoursandstandarddeviation hours and standard deviation =1 1.5hours 5 hours SupposeSamspends9.5hourssleepingonenight. Didhesleepa“lotmore”thantheaverage? 1.Howmuchmoreis9.5hoursthantheaverageof8hours?1.5hours 2.Calculatehowmanystandarddeviationfromtheaverage? Asyoucansee1.5hoursequals1standarddeviation WecansaythatSam’ssleeptimewas1stddevmorethantheaverage. WHICHISABETTERMEASURE?1or2
TransformationFormula WHICHISABETTERMEASURE?1or2 2.Calculatehowmanystandarddeviationfromtheaverage? Asyoucansee1.5hoursequals1standarddeviation WecansaythatSam’ssleeptimewas1stddevmorethantheaverage. Mathematically, X=9.5,µ=8,=1.5
Z
ConvertanynormalrandomvariableXtostandardizednormal randomvariableZ.
Z
X P V
IfXisnormallydistributedN( ,),thestandardizedvariableZ hasastandardnormaldistributionwithmean=0(or =0)and standard deviation 1 (or =1), standarddeviation1(or 1),denotedN(0,1) denoted N(0,1)
X P V
Thisiscalledthe“relativestanding”r knownaszrvalue “transformationofeachxrvaluetoazrvalue” r3
r2
r1
+1
Example
+2
+3
=2
ThetimetodownloadtheWebpageisnormally distributedwithamean =7secondsanda standarddeviation=2seconds What is the Z value for a download time of 9 WhatistheZvalueforadownloadtimeof9 seconds? Z
97 2
r3
1
r1
+1
1
r2 3
5
7
9
+2 11
+3 13
r3
r2
r1
0
+1
+2
+3
Xscale(µ=7, =2) Zscale(µ=0, =1)
Answer:adownloadtimeof9secondsisequivalentto1standardizedunit
4
www.notesolution.com
1/9/2011
Example
=2
ThetimetodownloadtheWebpageisnormally distributedwithamean =7secondsanda standarddeviation=2seconds What is the Z value for a download time of 1 WhatistheZvalueforadownloadtimeof1 second? Z
1 7 2
r3
3
r1
+1
1
r2 3
5
7
9
+2 11
+3 13
r3
r2
r1
0
+1
+2
+3
Xscale(µ=7, =2) Zscale(µ=0, =1)
Answer:adownloadtimeof1secondisequivalentto3standardizedunits
FindingAreasUsingStandardizedvariables
FindingAreasUsingStandardizedvariables
=7
=7
Area=?
Area=?
X =75
X=86 Z=?
=75 Z=? X=86 Solution: Z=(xr )/ Z=(86r75)/7=1.5714
Example: SamtookanEconomicsexamandscored 86points.Theclassmeanwas75withasstandard deviationof7.WhatpercentileisSamin?
FindingAreasUsingStandardizedvariables
GivenAreafindZvalue
=1
=1
Area=?
Area==0.95
Z =0
Usingthecalculatortofindthearea: (1) Usingzrscoretofindarea: P(Z
