Leson 6-Leson 6-11

Statistics for Management

Lesson 6

Hypothesis testing:

The comparison of two populations

Lesson. 6 - 2

Lesson Topics

1. Comparing two independent samples

- Comparing Two Means

- Comparing two proportions

2. Comparing two dependent samples

Lesson. 6 - 3

Comparing two independent samples

•Comparing Two Means:• Z Test for the Difference in Two Means

(Variances Known)

• t Test for Difference in Two Means (Variances Unknown)

•Comparing two proportions Z Test for Differences in Two Proportions

Lesson. 6 - 4

• Different Data Sources: Unrelated Independent

Sample selected from one population has no effect or bearing on the sample selected from the other population.

• Use Difference Between the 2 Sample Means

• Use Pooled Variance t Test

Independent Samples

Lesson. 6 - 5

• Assumptions: Samples are Randomly and Independently drawn Data Collected are Numerical Population Variances Are Known Samples drawn are Large

• Test Statistic:

Z Test for Differences in Two Means (Variances Known)

2

22

1

12

2121

nn

)()XX(Z

Lesson. 6 - 6

• Assumptions: Both Populations Are Normally Distributed Or, If Not Normal, Can Be Approximated by Normal Distribution Samples are Randomly and Independently drawn Population Variances Are Unknown But Assumed Equal

t Test for Differences in Two Means (Variances Unknown)

Lesson. 6 - 7

Developing the Pooled-Variance t Test (Part 1)

•Setting Up the Hypothesis:

H0: 1 2

H1: 1 > 2

H0: 1 -2 = 0

H1: 1 - 2

0

H0: 1 = 2

H1: 1 2

H0: 1

2

H0: 1 - 2 0

H1: 1 - 2 > 0

H0: 1 - 2

H1: 1 -

2 < 0

OR

OR

OR Left Tail

Right Tail

Two Tail

H1: 1 < 2

Lesson. 6 - 8

Developing the Pooled-Variance t Test (Part 2)

•Calculate the Pooled Sample Variances as an Estimate of the Common Populations Variance:

)n()n(

S)n(S)n(Sp 11

11

21

222

2112

2pS

21S

22S

1n

2n

= Pooled-Variance

= Variance of Sample 1

= Variance of sample 2

= Size of Sample 1

= Size of Sample 2

Lesson. 6 - 9

tX X

Sn S n S

n n

df n n

P

1 2 1 2

2 1 12

2 22

1 2

1 2

1 1

1 1

2

Hypothesized Difference

Developing the Pooled-Variance t Test (Part 3)

•Compute the Test Statistic:

( ))(

( ) ( )( ) ( )

112pS n1 n2

_ _

Lesson. 6 - 10

You’re a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQNumber 21 25Mean 3.27 2.53Std Dev 1.30 1.16Assuming equal variances, isthere a difference in average yield (= 0.05)?

© 1984-1994 T/Maker Co.

Pooled-Variance t Test: Example

Lesson. 6 - 11

tX X

Sn n

Sn S n S

n n

P

P

1 2 1 2

2

1 2

2 1 12

2 22

1 2

2 2

3 27 2 53 0

151021 25

2 03

1 1

1 1

21 1 1 30 25 1 116

21 1 25 11 510

. .

.

.

. ..

Calculating the Test Statistic:

(

((((

(

((

(

(

(

)

)

))

))

))

)))

11

11

Lesson. 6 - 12

H0: 1 - 2 = 0 (1 = 2)

H1: 1 - 2 0 (12)= 0.05df = 21 + 25 - 2 = 44Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Reject at = 0.05

There is evidence of a difference in means.t0 2.0154-2.0154

.025

Reject H0 Reject H0

.025

t

3 27 2 53

151021 25

2 03. .

.

.

Solution

11

Lesson. 6 - 13

Z Test for Differences in Two Proportions

Assumption: Sample is large enough

21

ss

n1

n1

)p1(p

PPZ 21

21

A2A1

21

s2s1

nn

nn

nn

pnpnp 21

5)p1(n);p1(n & 5pn;pn 22112211

Lesson. 6 - 14

Lesson 5 + 6 Summary•Addressed Hypothesis Testing Methodology

•Performed Z Test for the Mean (Known)

• Discussed p-Value Approach to Hypothesis Testing

•Made Connection to Confidence Interval Estimation

•Performed One Tail and Two Tail Tests

• Performed t Test of Hypothesis for the Mean

•Performed Z Test of Hypothesis for the Proportion

•Comparing two independent samples