Chapter 10 Business Analytics

1 1 Slide

Slide

IS 310 – Business StatisticsIS 310 – Business Statistics

IS 310

Business Statistic

sCSU

Long Beach

2 2 Slide

Slide


Inferences on Two PopulationsInferences on Two Populations

In the past, we dealt with one population mean In the past, we dealt with one population mean and one population proportion. However, and one population proportion. However, there are situations where two populations are there are situations where two populations are involved dealing with two means. involved dealing with two means.

Examples are the following:Examples are the following:

O We want to compare the mean salaries of male and O We want to compare the mean salaries of male and female graduates (two populations and two means).female graduates (two populations and two means).

O We want to compare the mean miles per gallon(MPG) of O We want to compare the mean miles per gallon(MPG) of two comparable automobile makes (two populations two comparable automobile makes (two populations and two means)and two means)

3 3 Slide

Slide


Statistical Inferences About MeansStatistical Inferences About Meansand Proportions with Two Populationsand Proportions with Two Populations

Inferences About the Difference BetweenInferences About the Difference Between

Two Population Means: Two Population Means: 11 and and 22 Known Known Inferences About the Difference BetweenInferences About the Difference Between

Two Population Means: Two Population Means: 11 and and 22 Unknown Unknown

4 4 Slide

Slide


Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Two Population Means: 1 1 and and 2 2 Known Known

Interval Estimation of Interval Estimation of 11 – – 22

Hypothesis Tests About Hypothesis Tests About 11 – – 22

5 5 Slide

Slide


Estimating the Difference BetweenEstimating the Difference BetweenTwo Population MeansTwo Population Means

Let Let 11 equal the mean of population 1 and equal the mean of population 1 and 22 equalequal

the mean of population 2.the mean of population 2. The difference between the two population The difference between the two population means ismeans is 11 - - 22.. To estimate To estimate 11 - - 22, we will select a simple , we will select a simple randomrandom

sample of size sample of size nn11 from population 1 and a from population 1 and a simplesimple

random sample of size random sample of size nn22 from population 2. from population 2. Let equal the mean of sample 1 and Let equal the mean of sample 1 and

equal theequal the

mean of sample 2.mean of sample 2.

x1x1 x2x2

The point estimator of the difference between The point estimator of the difference between thethe

means of the populations 1 and 2 is .means of the populations 1 and 2 is .x x1 2x x1 2

6 6 Slide

Slide


Expected ValueExpected Value

Sampling Distribution of Sampling Distribution of x x1 2x x1 2

E x x( )1 2 1 2 E x x( )1 2 1 2

Standard Deviation (Standard Error)Standard Deviation (Standard Error)

x x n n1 2

12

1

22

2

x x n n1 2

12

1

22

2

where: where: 1 1 = standard deviation of population 1 = standard deviation of population 1

2 2 = standard deviation of population 2 = standard deviation of population 2

nn1 1 = sample size from population 1= sample size from population 1

nn22 = sample size from population 2 = sample size from population 2

7 7 Slide

Slide


Interval EstimateInterval Estimate

Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Known Known

2 21 2

1 2 / 21 2

x x zn n

2 21 2

1 2 / 21 2

x x zn n

where:where:

1 - 1 - is the confidence coefficient is the confidence coefficient

8 8 Slide

Slide



In a test of driving distance using a In a test of driving distance using a mechanicalmechanical

driving device, a sample of Par golf balls wasdriving device, a sample of Par golf balls was

compared with a sample of golf balls made by compared with a sample of golf balls made by Rap,Rap,

Ltd., a competitor. The sample statistics appear Ltd., a competitor. The sample statistics appear on theon the

next slide.next slide.

Par, Inc. is a manufacturerPar, Inc. is a manufacturer

of golf equipment and hasof golf equipment and has

developed a new golf balldeveloped a new golf ball

that has been designed tothat has been designed to

provide “extra distance.”provide “extra distance.”

Example: Par, Inc.Example: Par, Inc.

9 9 Slide

Slide




Sample SizeSample Size

Sample MeanSample Mean

Sample #1Sample #1Par, Inc.Par, Inc.

Sample #2Sample #2Rap, Ltd.Rap, Ltd.

120 balls120 balls 80 balls80 balls

275 yards 258 yards275 yards 258 yards

Based on data from previous driving distanceBased on data from previous driving distancetests, the two population standard deviations aretests, the two population standard deviations areknown with known with 1 1 = 15 yards and = 15 yards and 2 2 = 20 yards. = 20 yards.

10 10 Slide

Slide




Let us develop a 95% confidence interval Let us develop a 95% confidence interval estimateestimate

of the difference between the mean driving of the difference between the mean driving distances ofdistances of

the two brands of golf ball.the two brands of golf ball.

11 11 Slide

Slide


Estimating the Difference BetweenEstimating the Difference BetweenTwo Population MeansTwo Population Means

mm11 – – 22 = difference between= difference between the mean distancesthe mean distances

xx11 - - xx22 = Point Estimate of = Point Estimate of mm11 – – 22

Population 1Population 1Par, Inc. Golf BallsPar, Inc. Golf Balls

11 = mean driving = mean driving distance of Pardistance of Par

golf ballsgolf balls

Population 1Population 1Par, Inc. Golf BallsPar, Inc. Golf Balls

11 = mean driving = mean driving distance of Pardistance of Par


Population 2Population 2Rap, Ltd. Golf BallsRap, Ltd. Golf Balls

22 = mean driving = mean driving distance of Rapdistance of Rap


Population 2Population 2Rap, Ltd. Golf BallsRap, Ltd. Golf Balls

22 = mean driving = mean driving distance of Rapdistance of Rap


Simple random sampleSimple random sample of of nn22 Rap golf balls Rap golf balls

xx22 = sample mean distance = sample mean distance for the Rap golf ballsfor the Rap golf balls

Simple random sampleSimple random sample of of nn22 Rap golf balls Rap golf balls

xx22 = sample mean distance = sample mean distance for the Rap golf ballsfor the Rap golf balls

Simple random sampleSimple random sample of of nn11 Par golf balls Par golf balls

xx11 = sample mean distance = sample mean distance for the Par golf ballsfor the Par golf balls

Simple random sampleSimple random sample of of nn11 Par golf balls Par golf balls

xx11 = sample mean distance = sample mean distance for the Par golf ballsfor the Par golf balls

12 12 Slide

Slide


Point Estimate of Point Estimate of 11 - - 22

Point estimate of Point estimate of 11 2 2 ==x x1 2x x1 2

where:where:

11 = mean distance for the population = mean distance for the population of Par, Inc. golf ballsof Par, Inc. golf balls

22 = mean distance for the population = mean distance for the population of Rap, Ltd. golf ballsof Rap, Ltd. golf balls

= 275 = 275 258 258

= 17 yards= 17 yards

13 13 Slide

Slide


x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

Interval Estimation of Interval Estimation of 11 - - 22::11 and and 22 Known Known

We are 95% confident that the difference betweenWe are 95% confident that the difference betweenthe mean driving distances of Par, Inc. balls and Rap,the mean driving distances of Par, Inc. balls and Rap,Ltd. balls is 11.86 to 22.14 yards.Ltd. balls is 11.86 to 22.14 yards.

17 17 ++ 5.14 or 11.86 yards to 22.14 yards 5.14 or 11.86 yards to 22.14 yards

14 14 Slide

Slide


Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

HypothesesHypotheses

1 2 0

2 21 2

1 2

( )x x Dz

n n

1 2 0

2 21 2

1 2

( )x x Dz

n n

1 2 0: aH D 1 2 0: aH D 0 1 2 0: H D 0 1 2 0: H D 0 1 2 0: H D 0 1 2 0: H D

1 2 0: aH D 1 2 0: aH D 0 1 2 0: H D 0 1 2 0: H D 1 2 0: aH D 1 2 0: aH D

Left-tailedLeft-tailed Right-tailedRight-tailed Two-tailedTwo-tailed

Test StatisticTest Statistic

15 15 Slide

Slide




Can we conclude, usingCan we conclude, using

= .01, that the mean driving= .01, that the mean driving

distance of Par, Inc. golf ballsdistance of Par, Inc. golf balls

is greater than the mean drivingis greater than the mean driving

distance of Rap, Ltd. golf balls?distance of Rap, Ltd. golf balls?

16 16 Slide

Slide


HH00: : 1 1 - - 22 << 0 0

HHaa: : 1 1 - - 22 > 0 > 0where: where: 11 = mean distance for the population = mean distance for the population of Par, Inc. golf ballsof Par, Inc. golf balls22 = mean distance for the population = mean distance for the population of Rap, Ltd. golf ballsof Rap, Ltd. golf balls

1. Develop the hypotheses.1. Develop the hypotheses.

pp –Value and Critical Value Approaches –Value and Critical Value Approaches


2. Specify the level of significance.2. Specify the level of significance. = .01= .01

17 17 Slide

Slide


3. Compute the value of the test statistic.3. Compute the value of the test statistic.



1 2 0

2 21 2

1 2

( )x x Dz

n n

1 2 0

2 21 2

1 2

( )x x Dz

n n

2 2

(235 218) 0 17 6.49

2.62(15) (20)120 80

z

2 2

(235 218) 0 17 6.49

2.62(15) (20)120 80

z

18 18 Slide

Slide


p p –Value Approach–Value Approach

4. Compute the 4. Compute the pp–value.–value.

For For zz = 6.49, the = 6.49, the pp –value < .0001. –value < .0001.


5. Determine whether to reject 5. Determine whether to reject HH00..

Because Because pp–value –value << = .01, we reject = .01, we reject HH00..

At the .01 level of significance, the sample At the .01 level of significance, the sample evidenceevidenceindicates the mean driving distance of Par, Inc. indicates the mean driving distance of Par, Inc. golfgolfballs is greater than the mean driving distance balls is greater than the mean driving distance of Rap,of Rap,Ltd. golf balls.Ltd. golf balls.

19 19 Slide

Slide



5. Determine whether to reject 5. Determine whether to reject HH00..

Because Because zz = 6.49 = 6.49 >> 2.33, we reject 2.33, we reject HH00..

Critical Value ApproachCritical Value Approach

For For = .01, = .01, zz.01.01 = 2.33 = 2.33

4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.

Reject Reject HH00 if if zz >> 2.33 2.33

The sample evidence indicates the mean The sample evidence indicates the mean drivingdrivingdistance of Par, Inc. golf balls is greater than distance of Par, Inc. golf balls is greater than the meanthe meandriving distance of Rap, Ltd. golf balls.driving distance of Rap, Ltd. golf balls.

20 20 Slide

Slide


Sample ProblemSample Problem

Problem # 7 (10-Page 401; 11-Page 414)Problem # 7 (10-Page 401; 11-Page 414)

a. H : µ = µ H : µ > µa. H : µ = µ H : µ > µ

0 1 2 a 1 20 1 2 a 1 2

b.b. Point reduction in the mean duration of games during 2003 = 172 – 166Point reduction in the mean duration of games during 2003 = 172 – 166

= 6 = 6 minutesminutes

_ _ 2 2_ _ 2 2

c. Test-statistic, z = [( x - x ) – 0] /√ [ (c. Test-statistic, z = [( x - x ) – 0] /√ [ (σσ / n ) + ( / n ) + (σσ / n )] / n )]

1 2 1 1 2 21 2 1 1 2 2

=(172 – 166)/√[ (144/60 + 144/50)]=(172 – 166)/√[ (144/60 + 144/50)]

= 6/2.3 = 2.61= 6/2.3 = 2.61

Critical z at = 1.645 Reject HCritical z at = 1.645 Reject H

0.05 00.05 0

Statistical test supports that the mean duration of games in 2003 is less Statistical test supports that the mean duration of games in 2003 is less than that in 2002.than that in 2002.

p-value = 1 – 0.9955 = 0.0045p-value = 1 – 0.9955 = 0.0045

21 21 Slide

Slide


Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Two Population Means: 1 1 and and 2 2

UnknownUnknown

Interval Estimation of Interval Estimation of 11 – – 22

Hypothesis Tests About Hypothesis Tests About 11 – – 22

22 22 Slide

Slide


Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Unknown Unknown

When When 1 1 and and 2 2 are unknown, we will: are unknown, we will:

• replace replace zz/2/2 with with tt/2/2. .

• use the sample standard deviations use the sample standard deviations ss11 and and ss22

as estimates of as estimates of 1 1 and and 2 2 , and , and

23 23 Slide

Slide


Interval Estimation of µ - µInterval Estimation of µ - µ 1 2 1 2

(Unknown and )(Unknown and ) 1 21 2

Interval estimateInterval estimate _ _ 2 2_ _ 2 2 (x - x ) ± t √ (s /n + s /n )(x - x ) ± t √ (s /n + s /n ) 1 2 1 2 /2 1 1 2 2/2 1 1 2 2

Degree of freedom = n + n - 2Degree of freedom = n + n - 2 1 21 2

24 24 Slide

Slide


Example: Specific MotorsExample: Specific Motors

Difference Between Two Population Difference Between Two Population Means:Means:

11 and and 2 2 Unknown Unknown

Specific Motors of DetroitSpecific Motors of Detroit

has developed a new automobilehas developed a new automobile

known as the M car. 24 M carsknown as the M car. 24 M cars

and 28 J cars (from Japan) were roadand 28 J cars (from Japan) were road

tested to compare miles-per-gallon (mpg) performance. tested to compare miles-per-gallon (mpg) performance.

The sample statistics are shown on the next slide.The sample statistics are shown on the next slide.

25 25 Slide

Slide



11 and and 2 2 Unknown Unknown Example: Specific MotorsExample: Specific Motors

Sample SizeSample Size

Sample MeanSample Mean

Sample Std. Dev.Sample Std. Dev.

Sample #1Sample #1M CarsM Cars

Sample #2Sample #2J CarsJ Cars

24 cars24 cars 2 28 cars8 cars

29.8 mpg 27.3 mpg29.8 mpg 27.3 mpg

2.56 mpg 1.81 mpg2.56 mpg 1.81 mpg

26 26 Slide

Slide



11 and and 2 2 Unknown Unknown

Let us develop a 90% confidenceLet us develop a 90% confidence

interval estimate of the differenceinterval estimate of the difference

between the mpg performances ofbetween the mpg performances of

the two models of automobile.the two models of automobile.


27 27 Slide

Slide


Point estimate of Point estimate of 11 2 2 ==x x1 2x x1 2

Point Estimate of Point Estimate of 1 1 2 2

where:where:

11 = mean miles-per-gallon for the = mean miles-per-gallon for the population of M carspopulation of M cars

22 = mean miles-per-gallon for the = mean miles-per-gallon for the population of J carspopulation of J cars

= 29.8 - 27.3= 29.8 - 27.3

= 2.5 mpg= 2.5 mpg

28 28 Slide

Slide


Interval Estimate of µ - µInterval Estimate of µ - µ 1 2 1 2

Interval estimateInterval estimate 2 22 2 29.8 – 27.3 ± t √ (2.56) /24 + (1.81) /28)29.8 – 27.3 ± t √ (2.56) /24 + (1.81) /28) 0.1/2 0.1/2 2.5 ± 1.676 (0.62)2.5 ± 1.676 (0.62) 2.5 ± 1.042.5 ± 1.04 1.46 and 3.541.46 and 3.54 We are 90% confident that the difference We are 90% confident that the difference

between the average miles per gallon between between the average miles per gallon between the J cars and M cars is between 1.46 and 3.54.the J cars and M cars is between 1.46 and 3.54.

29 29 Slide

Slide


Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

HypothesesHypotheses

1 2 0

2 21 2

1 2

( )x x Dt

s sn n

1 2 0

2 21 2

1 2

( )x x Dt

s sn n

1 2 0: aH D 1 2 0: aH D 0 1 2 0: H D 0 1 2 0: H D 0 1 2 0: H D 0 1 2 0: H D

1 2 0: aH D 1 2 0: aH D 0 1 2 0: H D 0 1 2 0: H D 1 2 0: aH D 1 2 0: aH D

Left-tailedLeft-tailed Right-tailedRight-tailed Two-tailedTwo-tailed Test StatisticTest Statistic

30 30 Slide

Slide




Can we conclude, using aCan we conclude, using a

.05 level of significance, that the.05 level of significance, that the

miles-per-gallon (miles-per-gallon (mpgmpg) performance) performance

of M cars is greater than the miles-per-of M cars is greater than the miles-per-

gallon performance of J cars?gallon performance of J cars?

31 31 Slide

Slide


2. Specify the level of significance.2. Specify the level of significance.

3. Compute the value of the test statistic.3. Compute the value of the test statistic.

= .05= .05



1 2 0

2 2 2 21 2

1 2

( ) (29.8 27.3) 0 4.003

(2.56) (1.81)24 28

x x Dt

s sn n

1 2 0

2 2 2 21 2

1 2

( ) (29.8 27.3) 0 4.003

(2.56) (1.81)24 28

x x Dt

s sn n

32 32 Slide

Slide


Hypothesis Tests of µ - µHypothesis Tests of µ - µ 1 2 1 2

H : µ = µ H : µ > µH : µ = µ H : µ > µ 0 1 2 a 1 20 1 2 a 1 2

Where µ average miles per gallon of M carsWhere µ average miles per gallon of M cars 11 µ average miles per gallon of J carsµ average miles per gallon of J cars 22

At = 0.05 with 50 degree of freedom, critical t = 1.676At = 0.05 with 50 degree of freedom, critical t = 1.676

Since t-statistic (4.003) is larger than critical t (1.676), Since t-statistic (4.003) is larger than critical t (1.676), we reject the null hypothesis. This means that the we reject the null hypothesis. This means that the average MPG of M cars is not equal to that of J carsaverage MPG of M cars is not equal to that of J cars

33 33 Slide

Slide


End of Chapter 10End of Chapter 10Part APart A

Chapter 10 Business Analytics

Documents

Transcript of Chapter 10 Business Analytics