Due in 8hrs. Very Urgent Statistics Calculations Homework Assignment . No Late Work

OPRE_Chapter102.zip

Home >Business & Finance homework help >Operations Management homework help >Due in 8hrs. Very Urgent Statistics Calculations Homework Assignment . No Late Work

Airlines.xlsx

DATA

Destination	Southwest Fare ($)	US Airways Fare ($)
Chicago	706	706
Nashville	718	674
Denver	980	980
Dallas	1078	1284
Atlanta	654	654
Orlando	820	748
Tampa	820	728
Fort Lauderdale	832	822
Phoenix	1114	1918
Las Vegas	1138	1918

BrandZTechFin.xlsx

DATA

Brand	Brand Value 2014 ($mil)	Brand Value Change (%)	Region	Product Sector
Apple	246992	67	North America	Technology
Google	173652	9	North America	Technology
Microsoft	115500	28	North America	Technology
IBM	93987	-13	North America	Technology
Visa	91962	16	North America	Financial Institutions
Tencent/QQ	76572	43	Asia	Technology
Facebook	71121	99	North America	Technology
Wells Fargo	59310	9	North America	Financial Institutions
Mastercard	40188	2	North America	Financial Institutions
Baidu	40041	35	Asia	Technology
ICBC Asia	38808	-8	Asia	Financial Institutions
SAP	38225	5	Continental Europe	Technology
American Express	38093	11	North America	Financial Institutions
HSBC	24029	-11	United Kingdom	Financial Institutions
RBC	23989	6	North America	Financial Institutions
hp	23039	18	North America	Technology
China Construction Bank	22065	-12	Asia	Financial Institutions
Oracle	21680	4	North America	Technology
Samsung	21602	-17	Asia	Technology
TD	20638	3	North America	Financial Institutions
Commonwealth Bank	20599	-2	Asia	Financial Institutions
Agricultural Bank of China	20189	11	Asia	Financial Institutions
Accenture	20183	11	North America	Technology
Intel	18385	58	North America	Technology
ANZ	17702	-7	Asia	Financial Institutions
Citi	17486	1	North America	Financial Institutions
Bank of China	16438	16	Asia	Financial Institutions
Cisco	16060	17	North America	Technology
Siemens	15496	-8	Continental Europe	Technology
Huawei	15335	ERROR:#N/A	Asia	Technology
US Bank	14786	-1	North America	Financial Institutions
JP Morgan	13522	9	North America	Financial Institutions
Westpac	12420	6	Asia	Financial Institutions
LinkedIn	12200	-2	North America	Technology
Santander	12181	10	Continental Europe	Financial Institutions
Chase	11661	0	North America	Financial Institutions
ING	11560	18	Continental Europe	Financial Institutions
Twitter	11447	-17	North America	Technology
Bank of America	11335	12	North America	Financial Institutions
Scotiabank	11044	-3	North America	Financial Institutions

lsxl8e_ppt_ch10.ppt

Chapter 10 - *

Two-Sample Tests

Chapter 10

Chapter 10 - *

Objectives

In this chapter, you learn:

How to compare the means of two independent populations.

How to compare the means of two related populations.

How to compare the proportions of two independent populations.

How to compare the variances of two independent populations.

Chapter 10 - *

Online Topic

Effect Size: Section 10.5.

Chapter 10 - *

Two-Sample Tests

Population Means, Independent Samples

Population Means, Related Samples

Population Variances

Group 1 vs. Group 2

Same group before vs. after treatment

Variance 1 vs.

Variance 2

Examples:

Population Proportions

Proportion 1 vs. Proportion 2

DCOVA

Chapter 10 - *

Difference Between Two Means

Population means, independent samples

Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ1 – μ2.

The point estimate for the difference is

X1 – X2

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

DCOVA

Chapter 10 - *

Difference Between Two Means: Independent Samples

Population means, independent samples

Use Sp to estimate unknown σ. Use a Pooled-Variance t test.

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

Use S1 and S2 to estimate unknown σ1 and σ2. Use a Separate-variance t test

Different data sources
Unrelated.
Independent.
Sample selected from one population has no effect on the sample selected from the other population.

DCOVA

Chapter 10 - *

Hypothesis Tests for
Two Population Means

Lower-tail test:

H0: μ1  μ2

H1: μ1 < μ2

i.e.,

H0: μ1 – μ2  0

H1: μ1 – μ2 < 0

Upper-tail test:

H0: μ1 ≤ μ2

H1: μ1 > μ2

i.e.,

H0: μ1 – μ2 ≤ 0

H1: μ1 – μ2 > 0

Two-tail test:

H0: μ1 = μ2

H1: μ1 ≠ μ2

i.e.,

H0: μ1 – μ2 = 0

H1: μ1 – μ2 ≠ 0

Two Population Means, Independent Samples

DCOVA

Chapter 10 - *

Two Population Means, Independent Samples

Lower-tail test:

H0: μ1 – μ2  0

H1: μ1 – μ2 < 0

Upper-tail test:

H0: μ1 – μ2 ≤ 0

H1: μ1 – μ2 > 0

Two-tail test:

H0: μ1 – μ2 = 0

H1: μ1 – μ2 ≠ 0

a/2

-ta

-ta/2

ta/2

Reject H0 if tSTAT < -ta

Reject H0 if tSTAT > ta

Reject H0 if tSTAT < -ta/2

or tSTAT > ta/2

Hypothesis tests for μ1 – μ2

DCOVA

Chapter 10 - *

Population means, independent samples

Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal

Assumptions:

Samples are randomly and

independently drawn.

Populations are normally

distributed or both sample

sizes are at least 30.

Population variances are

unknown but assumed equal.

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

DCOVA

Chapter 10 - *

Population means, independent samples

The pooled variance is:

The test statistic is:

Where tSTAT has d.f. = (n1 + n2 – 2).

(continued)

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal

DCOVA

Chapter 10 - *

Population means, independent samples

The confidence interval for

μ1 – μ2 is:

Where tα/2 has d.f. = n1 + n2 – 2.

Confidence interval for µ1 - µ2 with σ1 and σ2 unknown and assumed equal

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

DCOVA

Chapter 10 - *

Pooled-Variance t Test Example

You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data:

NYSE NASDAQ
Number 21 25

Sample mean 3.27 2.53

Sample std dev 1.30 1.16

Assuming both populations are

approximately normal with equal variances, is
there a difference in mean dividend yield ( = 0.05)?

DCOVA

Chapter 10 - *

Pooled-Variance t Test Example: Calculating the Test Statistic

The test statistic is:

(continued)

H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)

H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)

DCOVA

Chapter 10 - *

H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)

H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)

 = 0.05

df = 21 + 25 - 2 = 44

Critical Values: t = ± 2.0154

Test Statistic:

Pooled-Variance t Test Example: Hypothesis Test Solution

Decision:

Conclusion:

Reject H0 at a = 0.05.

There is evidence of a difference in means.

2.0154

-2.0154

.025

Reject H0

.025

2.040

DCOVA

Chapter 10 - *

Pooled Variance t Test In Excel

DCOVA

Chapter 10 - *

Since we rejected H0 can we be 95% confident that µNYSE > µNASDAQ?

95% Confidence Interval for µNYSE - µNASDAQ:

Since 0 is less than the entire interval, we can be 95% confident that µNYSE > µNASDAQ.

Pooled-Variance t Test Example: Confidence Interval for µ1 - µ2

DCOVA

Chapter 10 - *

Pooled Variance t Confidence Interval In Excel

DCOVA

Chapter 10 - *

Population means, independent samples

Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown, not assumed equal

Assumptions:

Samples are randomly and

independently drawn.

Populations are normally

distributed or both sample

sizes are at least 30.

Population variances are

unknown and cannot be

assumed to be equal.

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

DCOVA

Chapter 10 - *

Population means, independent samples

(continued)

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and not assumed equal

DCOVA

This test is known at the separate-variance t test.

The formulae for this test are not covered in this chapter. See reference 3 for more details.

This test done in Excel is shown on the next slide.

Chapter 10 - *

Separate-Variance t Test In Excel

(continued)

DCOVA

NYSE NASDAQ
Number 21 25

Sample mean 3.27 2.53

Sample std dev 1.30 1.16

Using α=0.05 this test fails to reject the null.

For this data whether we can assume equal

variances or not is important to determine

because when we assumed equal variances

the null was rejected.

In Section 10.4 a test to help determine whether

this is a reasonable assumption or not is discussed.

Chapter 10 - *

Related Populations
The Paired Difference Test

Tests Means of 2 Related Populations

Paired or matched samples.

Repeated measures (before/after).

Use difference between paired values:

Eliminates Variation Among Subjects.
Assumptions:
Differences are normally distributed.
Or, if not Normal, use large samples.

Related samples

Di = X1i - X2i

DCOVA

Chapter 10 - *

Related Populations
The Paired Difference Test

The ith paired difference is Di , where

Related samples

Di = X1i - X2i

The point estimate for the paired difference population mean μD is D :

n is the number of pairs in the paired sample

The sample standard deviation is SD.

(continued)

DCOVA

Chapter 10 - *

The test statistic for μD is:

Paired samples

Where tSTAT has n - 1 d.f.

The Paired Difference Test:
Finding tSTAT

DCOVA

Chapter 10 - *

Lower-tail test:

H0: μD  0

H1: μD < 0

Upper-tail test:

H0: μD ≤ 0

H1: μD > 0

Two-tail test:

H0: μD = 0

H1: μD ≠ 0

Paired Samples

The Paired Difference Test: Possible Hypotheses

a/2

-ta

-ta/2

ta/2

Reject H0 if tSTAT < -ta

Reject H0 if tSTAT > ta

Reject H0 if tSTAT < -ta/2

or tSTAT > ta/2

Where tSTAT has n - 1 d.f.

DCOVA

Chapter 10 - *

The confidence interval for μD is:

Paired samples

where

The Paired Difference Confidence Interval

DCOVA

Chapter 10 - *

Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:

Paired Difference Test: Example

Number of Complaints: (2) - (1)

Salesperson Before (1) After (2) Difference, Di

C.B. 6 4 - 2

T.F. 20 6 -14

M.H. 3 2 - 1

R.K. 0 0 0

M.O. 4 0 - 4

-21

DCOVA



D =

= -4.2

Chapter 10 - *

Has the training made a difference in the number of complaints (at the 0.01 level)?

- 4.2

D =

H0: μD = 0

H1: μD  0

Test Statistic:

t0.005 = ± 4.604 d.f. = n - 1 = 4

Reject

/2

- 4.604 4.604

Decision: Do not reject H0

(tstat is not in the rejection region).

Conclusion: There is insufficient evidence of a change in the number of complaints.

Paired Difference Test: Solution

Reject

/2

- 1.66

 = .01

DCOVA

Chapter 10 - *

Paired Difference t Test In Excel

DCOVA

Chapter 10 - *

The confidence interval for μD is:

The probability this interval contains the true value of μD is 99%.

The Paired Difference Confidence Interval -- Example

DCOVA

D = -4.2, SD = 5.67

Chapter 10 - *

Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π1 – π2

Two Population Proportions

The point estimate for the difference is

Population proportions

Assumptions:

n1 π1  5 , n1(1- π1)  5

n2 π2  5 , n2(1- π2)  5

DCOVA

Chapter 10 - *

Two Population Proportions

Population proportions

The pooled estimate for the overall proportion is:

where X1 and X2 are the number of items of interest in samples 1 and 2.

In the null hypothesis we assume the null hypothesis is true, so we assume π1 = π2 and pool the two sample estimates.

DCOVA

Chapter 10 - *

Two Population Proportions

Population proportions

The test statistic for

π1 – π2 is a Z statistic:

(continued)

where

DCOVA

Chapter 10 - *

Hypothesis Tests for
Two Population Proportions

Population proportions

Lower-tail test:

H0: π1  π2

H1: π1 < π2

i.e.,

H0: π1 – π2  0

H1: π1 – π2 < 0

Upper-tail test:

H0: π1 ≤ π2

H1: π1 > π2

i.e.,

H0: π1 – π2 ≤ 0

H1: π1 – π2 > 0

Two-tail test:

H0: π1 = π2

H1: π1 ≠ π2

i.e.,

H0: π1 – π2 = 0

H1: π1 – π2 ≠ 0

DCOVA

Chapter 10 - *

Two-tail test:

H0: π1 – π2 = 0

H1: π1 – π2 ≠ 0

Lower-tail test:

H0: π1 – π2  0

H1: π1 – π2 < 0

Hypothesis Tests for
Two Population Proportions

Population proportions

Upper-tail test:

H0: π1 – π2 ≤ 0

H1: π1 – π2 > 0

a/2

-za

-za/2

za/2

Reject H0 if ZSTAT < -Za

Reject H0 if ZSTAT > Za

Reject H0 if ZSTAT < -Za/2

or ZSTAT > Za/2

(continued)

DCOVA

Chapter 10 - *

Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?

In a random sample, 36 of 72 men and 35 of 50 women indicated they would vote Yes.

Test at the .05 level of significance.

Hypothesis Test Example:
Two Population Proportions

DCOVA

Chapter 10 - *

The hypothesis test is:

H0: π1 – π2 = 0 (the two proportions are equal).

H1: π1 – π2 ≠ 0 (there is a significant difference between proportions).

The sample proportions are:
Men: p1 = 36/72 = 0.50
Women: p2 = 35/50 = 0.70

The pooled estimate for the overall proportion is:

Hypothesis Test Example:
Two Population Proportions

(continued)

DCOVA

Chapter 10 - *

The test statistic for π1 – π2 is:

Hypothesis Test Example:
Two Population Proportions

(continued)

.025

-1.96

1.96

.025

-2.20

Decision: Reject H0.

Conclusion: There is evidence of a significant difference in the proportion of men and women who will vote yes.

Reject H0

Critical Values = ±1.96

For  = .05

DCOVA

Chapter 10 - *

Comparing Two Population Proportions In Excel

DCOVA

Decision: Reject H0.

Conclusion: There is evidence of a significant difference in the proportion of men and women who will vote yes.

Chapter 10 - *

Confidence Interval for
Two Population Proportions

Population proportions

The confidence interval for

π1 – π2 is:

DCOVA

Chapter 10 - *

Confidence Interval for Two Population Proportions -- Example

The 95% confidence interval for π1 – π2 is:

Since this interval does not contain 0 can be 95% confident the two proportions are different.

DCOVA

Chapter 10 - *

H0: σ12 = σ22

H1: σ12 ≠ σ22

Testing for the Ratio Of Two Population Variances

Tests for Two

Population

Variances

F test statistic

H0: σ12 ≤ σ22

H1: σ12 > σ22

Hypotheses FSTAT

S12 / S22

S12 = Variance of sample 1 (the larger sample variance)

n1 = sample size of sample 1

S22 = Variance of sample 2 (the smaller sample variance)

n2 = sample size of sample 2

n1 –1 = numerator degrees of freedom

n2 – 1 = denominator degrees of freedom

Where:

DCOVA

Chapter 10 - *

The F critical value is found from the F table.
There are two degrees of freedom required: numerator and denominator.
The larger sample variance is always the numerator.

When

In the F table;
numerator degrees of freedom determine the column.
denominator degrees of freedom determine the row.

The F Distribution

df1 = n1 – 1 ; df2 = n2 – 1.

DCOVA

Chapter 10 - *

Finding the Rejection Region

H0: σ12 = σ22

H1: σ12 ≠ σ22

H0: σ12 ≤ σ22

H1: σ12 > σ22

Reject H0 if FSTAT > Fα

Reject H0 if FSTAT > Fα/2

DCOVA



Fα

Reject H0

Do not

reject H0

/2

Reject H0

Do not

reject H0

Fα/2

Chapter 10 - *

You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data:

NYSE NASDAQ
Number 21 25

Mean 3.27 2.53

Std dev 1.30 1.16

Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.05 level?

F Test: An Example

DCOVA

Chapter 10 - *

Find the F critical value for  = 0.05.

Numerator d.f. = n1 – 1 = 21 –1 =20.

Denominator d.f. = n2 – 1 = 25 –1 = 24.

Fα/2 = F.025, 20, 24 = 2.33.

Form the hypothesis test:

H0: σ21 = σ22 (there is no difference between variances.)

H1: σ21 ≠ σ22 (there is a difference between variances.)

F Test: Example Solution

DCOVA

Chapter 10 - *

H0: σ12 = σ22

H1: σ12 ≠ σ22

The test statistic is:

/2 = .025

F0.025=2.33

Reject H0

Do not

reject H0

F Test: Example Solution

FSTAT = 1.256 is not in the rejection region, so we do not reject H0.

(continued)

Conclusion: There is not sufficient evidence of a difference in variances at  = .05.

DCOVA

Chapter 10 - *

F Test in Excel

DCOVA

Conclusion: There is not sufficient evidence of a difference in variances at  = .05.

Chapter 10 - *

Chapter Summary

In this chapter we discussed:

Comparing the means of two independent populations.

Comparing the means of two related populations.

Comparing the proportions of two independent populations.

Comparing the variances of two independent populations.

(

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(

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(

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(

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(

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1.16

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2.040

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2.53

3.27

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STAT

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5.67

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55.67/

04.2

n/S

STAT

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

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(-15.87,

604

for

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(

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(

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STAT

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.582

122

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.70

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STAT

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(-0.37,

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1.96

0.70

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STAT

256

STAT

Luggage.xlsx

DATA

Wing A	Wing B
10.70	7.20
9.89	6.68
11.83	9.29
9.04	8.95
9.37	6.61
11.68	8.53
8.36	8.92
9.76	7.95
13.67	7.57
8.96	6.38
9.51	8.89
10.85	10.03
10.57	9.30
11.06	5.28
8.91	9.23
11.79	9.25
10.59	8.44
9.13	6.57
12.37	10.61
9.91	6.77

Phone.xlsx

DATA

Time	Location
1.48	1
1.75	1
0.78	1
2.85	1
0.52	1
1.60	1
4.15	1
3.97	1
1.48	1
3.10	1
1.02	1
0.53	1
0.93	1
1.60	1
0.80	1
1.05	1
6.32	1
3.93	1
5.45	1
0.97	1
7.55	2
3.75	2
0.10	2
1.10	2
0.60	2
0.52	2
3.30	2
2.10	2
0.58	2
4.02	2
3.75	2
0.65	2
1.92	2
0.60	2
1.53	2
4.23	2
0.08	2
1.48	2
1.65	2
0.72	2

Tablets.xlsx

DATA

WiFi Only	3G/4G+WiFi
12.40	11.70
10.60	11.40
11.50	10.70
12.60	11.50
8.00	11.30
10.90	11.60
11.60	8.90
10.00
10.90
8.90
9.50
10.50

Telecom.xlsx

DATA

Provider	TV	Internet
Verizon FIOS	73	74
WOW	74	76
Bright House Networks	68	70
A T & T U-verse	68	68
Cox	64	68
SuddenLink	65	70
Cablevision/Optimum	63	67
RCN	65	71
Comcast/Xfinity	59	62
TimeWarner	58	63
Charter	59	64
Mediacom	54	58
Wave/Astound	72	74
Cable One	63	68

CollegeDebt.xlsx

DATA

Option	Average Debt at Graduation ($)
Private	5552
Private	13009
Private	12560
Private	17856
Private	18456
Private	20303
Private	15010
Private	17891
Private	16640
Private	15660
Private	19798
Private	20577
Private	24382
Private	26055
Private	23978
Private	24064
Private	22808
Private	27827
Private	23082
Private	24334
Private	24266
Private	21139
Private	15769
Private	24990
Private	33309
Private	35902
Private	28474
Private	29914
Private	22718
Private	14315
Private	35853
Private	31208
Private	28647
Private	32106
Private	28327
Private	32347
Private	38540
Private	29026
Private	35510
Private	31337
Private	23750
Private	27827
Private	30688
Private	30798
Private	24465
Private	34998
Private	34610
Private	26656
Private	37694
Private	32307
Private	30311
Private	16817
Private	31073
Private	24451
Private	33775
Private	26989
Private	29122
Private	33877
Private	29829
Private	34922
Private	32000
Private	44584
Private	32914
Private	31501
Private	27164
Private	31070
Private	26806
Private	36808
Private	29702
Private	29574
Private	29296
Private	26119
Private	26088
Private	29101
Private	32746
Private	37607
Private	37551
Private	31214
Private	29115
Private	28518
Private	31804
Private	31891
Private	29699
Private	36523
Private	29439
Private	27835
Private	36955
Private	33455
Private	23760
Private	30540
Private	28356
Private	34884
Private	34012
Private	33743
Private	30634
Private	28918
Private	26000
Private	33657
Private	31267
Private	26848
Public	17602
Public	21815
Public	20708
Public	17468
Public	20229
Public	27163
Public	24400
Public	25664
Public	25254
Public	20254
Public	21263
Public	19530
Public	17927
Public	25300
Public	26472
Public	20452
Public	25027
Public	23912
Public	23761
Public	19880
Public	23562
Public	23782
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Public	29121
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Public	20728
Public	21653
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Public	23089
Public	20504
Public	23575
Public	24657
Public	19970
Public	9949
Public	20467
Public	27619
Public	32571
Public	23186
Public	24284
Public	28508
Public	16105
Public	13000
Public	34623
Public	25711
Public	24624
Public	25821
Public	17696
Public	22719
Public	14000
Public	24313
Public	27181
Public	35430
Public	24875
Public	24111
Public	25613
Public	25741
Public	17893
Public	22140
Public	22523
Public	21613
Public	26072
Public	17617
Public	20273
Public	23951
Public	23964
Public	29898
Public	28999
Public	15373
Public	22541
Public	30396
Public	13386
Public	22603
Public	21742
Public	23729
Public	21659
Public	26894
Public	18100
Public	30755
Public	25159
Public	23545
Public	31526
Public	23151
Public	21137
Public	27146
Public	27324
Public	23838
Public	21173
Public	15626
Public	23726
Public	20636
Public	34382
Public	25729
Public	28109
Public	26946
Public	29121
Public	33944

InternetMobileTime2.xlsx

DATA

Gender	Minutes
F	72
F	144
F	48
F	72
F	36
F	360
F	44
F	30
F	432
F	24
F	288
F	144
F	144
F	240
F	432
F	144
F	144
F	144
F	576
F	216
F	72
F	72
F	144
F	288
F	144
F	36
F	288
F	48
F	288
F	144
M	432
M	2304
M	108
M	72
M	24
M	12
M	1728
M	720
M	144
M	432
M	432
M	432
M	576
M	576
M	432
M	144
M	216
M	576
M	96
M	288
M	48
M	144
M	144
M	432
M	144
M	36
M	72
M	24
M	288
M	240