Chapter10 - Hypothesis Testing Two-Sample Tests

Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 10

Learning Objectives

In this chapter, you learn how to use hypothesis

testing for comparing the difference between:



_{The means of two independent populations}



_{The means of two related populations}



_{The proportions of two independent}

populations



_{The variances of two independent}

Two-Sample Tests Overview

Two Sample Tests

Independent

Population

Means

Means,

Related

Populations

Independent

Population

Variances

Group 1 vs.

Group 2

Same group

before vs. after

treatment

Variance 1 vs.

Variance 2

Examples

Independent

Population

Proportions

Two-Sample Tests

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

Goal: Test hypothesis or form

a confidence interval for the

difference between two

population means, μ

₁

– μ

₂

The point estimate for the

difference between sample

means:

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown



_{Different data sources}



_{Independent: Sample selected}

from one population has no

effect on the sample selected

from the other population



_{Use the difference between 2}

sample means



_{Use Z test, pooled variance t}

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

Use a

Z

test statistic

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

Assumptions:



Samples are randomly and

independently drawn

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

When σ

₁

and σ

₂

are known and both

populations are normal, the test

statistic is a Z-value and the

standard error of X

₁

– X

₂

is

2

2

2

1

2

1

X

X

_n

σ

n

σ

σ

2

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

Two-Sample Tests

Independent Populations

Lower-tail test:

H

₀

: μ

₁

_

μ

₂

H

₁

: μ

₁

< μ

₂

i.e.,

H

₀

: μ

₁

– μ

₂

_

0

H

: μ

– μ

< 0

Upper-tail test:

H

₀

: μ

₁

≤ μ

₂

H

₁

: μ

₁

> μ

₂

i.e.,

H

₀

: μ

₁

– μ

₂

≤ 0

H

: μ

– μ

> 0

Two-tail test:

H

₀

: μ

₁

= μ

₂

H

₁

: μ

₁

≠ μ

₂

i.e.,

Two-Sample Tests

Independent Populations

Two Independent Populations, Comparing Means

Lower-tail test:

H

₀

: μ

₁

– μ

₂

_

0

H

₁

: μ

₁

– μ

₂

< 0

Upper-tail test:

H

₀

: μ

₁

– μ

₂

≤ 0

H

₁

: μ

₁

– μ

₂

> 0

Two-tail test:

H

₀

: μ

₁

– μ

₂

= 0

H

₁

: μ

₁

– μ

₂

≠ 0





/2



/2



-z

_

z

_

-z

_

_/2

z

_

_/2

Reject H

₀

if Z < -Z

_a

Reject H

₀

if Z > Z

_a

Reject H

₀

if Z < -Z

_a/2

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

Assumptions:



_{Samples are randomly and}

independently drawn



Populations are normally

distributed



_{Population variances are}

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

Forming interval estimates:



The population variances

are assumed equal, so use

the two sample standard

deviations and pool them to

estimate σ

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown

The pooled standard

deviation is:

Two-Sample Tests

Independent Populations

Where t has (n

₁

+ n

₂

– 2) d.f., and

The test statistic is:









Independent

Population Means

σ

₁

and σ

₂

known

Two-Sample Tests

Independent Populations



_{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

Two-Sample Tests

Independent Populations

















_1.5021

Two-Sample Tests

Independent Populations



_H

₀

_{: μ}

₁

_{- μ}

₂

_{= 0 i.e. (μ}

₁

_{= μ}

₂

₎



_H

₁

_{: μ}

₁

_{- μ}

₂

_{≠ 0 i.e. (μ}

₁

_{≠ μ}

₂

₎



_

_{= 0.05}



_{df = 21 + 25 - 2 = 44}



_{Critical Values: t = ± 2.0154}



_{Test Statistic: 2.040}

t

0

2.0154

-2.0154

.025

Reject H

₀

Reject H

₀

.025

Decision: Reject H

₀

at

α

= 0.05

2.040

Independent Populations

Unequal Variance



_{If you cannot assume population variances}

are equal, the pooled-variance t test is

inappropriate



_{Instead, use a separate-variance t test, which}

includes the two separate sample variances in

the computation of the test statistic



_{The computations are complicated and are}

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

σ

₁

and σ

₂

unknown





2

2

2

1

2

1

2

1

n

σ

n

σ

X

X

_

_

Z

_

The confidence interval for

Two-Sample Tests

Independent Populations

Independent

Population Means

σ

₁

and σ

₂

known

Two-Sample Tests

Related Populations

D = X

₁

- X

₂

Tests Means of 2 Related Populations



_{Paired or matched samples}



_{Repeated measures (before/after)}



_{Use difference between paired values:}



_{Eliminates Variation Among Subjects}



_Assumptions:

Two-Sample Tests

Related Populations

The ith paired difference is D

_i

, where

n

D

D

n

1

i

i







D

_i

= X

_1i

- X

_2i

The point estimate for the population mean

paired difference is D :

Two-Sample Tests

Related Populations

The test statistic for the mean difference is a Z

value:

n

σ

μ

D

Z

D

D





Where

μ

D

= hypothesized mean difference

σ

D

= population standard deviation of differences

Two-Sample Tests

Related Populations

If σ

_D

is unknown, you can estimate the

unknown population standard deviation with a

sample standard deviation:

1

n

)

D

(D

S

n

1

i

2

i

D

Two-Sample Tests

Related Populations

1

The test statistic for D is now a t statistic:

Two-Sample Tests

Related Populations

Lower-tail test:

H

₀

: μ

_D

_

0

H

₁

: μ

_D

< 0

Upper-tail test:

H

₀

: μ

_D

≤ 0

H

₁

: μ

_D

> 0

Two-tail test:

H

₀

: μ

_D

= 0

H

₁

: μ

_D

≠ 0





/2



/2



-t

_

t

_

-t

_

_/2

t

_

_/2

Two-Sample Tests

Related Populations Example

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:

Salesperson

Number of Complaints

Difference, D

_i

(2-1)

Before (1)

After (2)

C.B.

6

4

-2

T.F.

20

6

-14

M.H.

3

2

-1

Two-Sample Tests

Related Populations Example

2

Salesperson

Number of Complaints

Difference, D

_i

(2-1)

Before (1)

After (2)

C.B.

6

4

-2

T.F.

20

6

-14

M.H.

3

2

-1

R.K.

0

0

0

Two-Sample Tests

Related Populations Example

Has the training made a difference in the number of

complaints (at the α = 0.01 level)?

H

₀

: μ

_D

= 0

H

₁

:



μ

_D

_

0

Critical Value = ± 4.604

d.f. = n - 1 = 4

Test Statistic:

Two-Sample Tests

Related Populations Example

Reject

- 4.604

4.604

Reject



/2

- 1.66

Decision: Do not reject

H

₀

(t statistic is not in the reject

region)

Conclusion: There is no

evidence of a significant change

in the number of complaints

Two-Sample Tests

Related Populations



_{The confidence interval for μ}

_D

_{(σ known) is:}

n

σ

_D

Z

D

_

Where

Two-Sample Tests

Related Populations



_{The confidence interval for μ}

_D

_{(σ unknown) is:}

1

n

)

D

(D

S

n

1

i

2

i

D











n

S

t

D

D

1

n

_



Two Population Proportions

Goal: Test a hypothesis or form a confidence

interval for the difference between two

independent population proportions, π

₁

– π

₂

Assumptions:

n

₁

π

₁

_

5 , n

₁

(1-π

₁

)

_

5

n

₂

π

₂

_

5 , n

₂

(1-π

₂

)

_

5

Two Population Proportions

Since you begin by assuming the null

hypothesis is true, you assume π

₁

= π

₂

and pool

the two sample (p) estimates.

2

1

2

1

n

n

X

X

p







The pooled estimate for

the overall proportion is:

Two Population Proportions

Hypothesis for Population Proportions

Lower-tail test:

H

₀

: π

₁

_

π

₂

H

₁

: π

₁

< π

₂

i.e.,

H

₀

: π

₁

– π

₂

_

0

H

₁

: π

₁

– π

₂

< 0

Upper-tail test:

H

₀

: π

₁

≤ π

₂

H

₁

: π

₁

> π

₂

i.e.,

H

₀

: π

₁

– π

₂

≤ 0

H

₁

: π

₁

– π

₂

> 0

Two-tail test:

H

₀

: π

₁

= π

₂

H

₁

: π

₁

≠ π

₂

i.e.,

Two Population Proportions

Hypothesis for Population Proportions

Lower-tail test:

H

₀

: π

₁

– π

₂

_

0

H

₁

: π

₁

– π

₂

< 0

Upper-tail test:

H

₀

: π

₁

– π

₂

≤ 0

H

₁

: π

₁

– π

₂

> 0

Two-tail test:

H

₀

: π

₁

– π

₂

= 0

H

₁

: π

₁

– π

₂

≠ 0





/2



/2



-z

_

z

_

-z

_

_/2

z

_

_/2

Two Independent Population

Proportions: Example



_{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 of 72 men, 36 indicated}

they would vote Yes and, in a sample of 50

women, 31 indicated they would vote Yes

Two Independent Population

Proportions: Example



_H

₀

_{: π}

₁

_{– π}

₂

_{= 0 (the two proportions are equal)}



_H

₁

_{: π}

₁

_{– π}

₂

_{≠ 0 (there is a significant difference}

between proportions)



_{The sample proportions are:}



_Men:

_p

₁

_{= 36/72 = .50}



_Women:

_p

₂

_{= 31/50 = .62}



_{The pooled estimate for the overall proportion is:}

Two Independent Population

Proportions: Example

The test statistic for

π

₁

– π

₂

is:

Critical Values = ±1.96

For

_

= .05

.025

-1.96

1.96

.025

-1.31

Decision: Do not reject H

₀

Conclusion: There is no evidence of a

significant difference in proportions who

will vote yes between men and women.

Two Independent Population

Proportions





2

2

2

1

1

1

2

1

n

)

(1

n

)

(1

p

p

p

p

Z

p

p

_

_



_



Testing Population Variances



_{Purpose: To determine if two independent}

populations have the same variability.

H

₀

: σ

₁

2

= σ

2

2

H

₁

: σ

₁

2

≠ σ

2

2

H

₀

: σ

₁

2

_

σ

2

2

H

₁

: σ

₁

2

< σ

2

2

H

₀

: σ

₁

2

≤ σ

2

2

H

₁

: σ

₁

2

> σ

2

2

Testing Population Variances

2

2

2

1

S

S

F

_

The F test statistic is:

= Variance of Sample 1

n

₁

- 1 = numerator degrees of freedom

= Variance of Sample 2

2

1

S

2

2

Testing Population Variances



_{The F critical value}

_{is found from the F table}



_{There are two appropriate degrees of}

freedom: numerator and denominator.



_{In the F table,}



_{numerator degrees of freedom determine the}

column



_{denominator degrees of freedom determine the}

Testing Population Variances

0



F

_L

Reject

H

₀

Do not

reject H

₀

H

₀

: σ

₁

2

_

σ

2

2

H

₁

: σ

₁

2

< σ

2

2

Reject H

₀

if F < F

_L

0



F

_U

Reject H

0

Do not

reject H

₀

H

₀

: σ

₁

2

≤ σ

2

2

H

₁

: σ

₁

2

> σ

2

2

Reject H

if F > F

Testing Population Variances



rejection region

for a two-tail test is:

F

Testing Population Variances

To find the critical F values:

1. Find F

_U

from the F table for n

₁

– 1

numerator and n

₂

– 1 denominator degrees

of freedom.

*

U

L

F

1

F

_

2. Find F

_L

using the formula:

Where F

_U*

is from the F table with n

₂

– 1

Testing Population Variances



_{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}

Testing Population Variances



_{Form the hypothesis test:}



_H

₀

_{: σ}

₂₁

_{– σ}

₂₂

_{= 0 (there is no difference between variances)}



_H

₁

_{: σ}

2

₁

– σ

2

₂

≠ 0 (there is a difference between variances)



_Numerator:



_n

₁

_{– 1 = 21 – 1 = 20 d.f.}



_Denominator:



_n

₂

_{– 1 = 25 – 1 = 24 d.f.}



_Numerator:



_n

₂

_{– 1 = 25 – 1 = 24 d.f.}



_Denominator:



_n

₁

_{– 1 = 21 – 1 = 20 d.f.}

Testing Population Variances



_{The test statistic is:}

256

rejection region, so we do

not reject H

₀



_{Conclusion: There is insufficient}

Chapter Summary

In this chapter, we have



_{Compared two independent samples}



_{Performed Z test for the differences in two means}



_{Performed pooled variance t test for the differences}

in two means



_{Formed confidence intervals for the differences}

between two means



_{Compared two related samples (paired samples)}



_{Performed paired sample Z and t tests for the mean}

difference



_{Formed confidence intervals for the paired}

Chapter Summary



_{Compared two population proportions}



_{Formed confidence intervals for the difference between}

two population proportions



_{Performed Z-test for two population proportions}



_{Performed F tests for the difference between two}

population variances



_{Used the F table to find F critical values}