1

Chapter 10

Two-Sample Tests

David Chow

Nov 2014

2

Learning Objectives

In this chapter, you learn how to use hypothesis testing to compare the:

a. 1 = 2? Means of two independent populations

b. 1 = 2? Proportions of two independent populations,

c. Variances of two independent populations (F-test)

Only part (c), F-test of two variances, is covered in the exam

3

Any Difference Between 1 & 2?

Comparing Means: Want to know if 1 = 2

The question can also be posed in the form of a one-

tailed test such as 1 > 2

Eg: TOFEL scores from HK and Singapore students

Comparing Proportions: Test if 1 = 2

Eg: Proportion of students in a relationship (yr1 vs yr4)

Eg: Passing proportion in math (local vs int schools )

Eg: Proportion of contact-lens users (male vs female)

4

Some Funny Findings DSME 2010, Fall 2011

Sleeping duration

related to societal activities, living in campus

not related to gender, part-time, dating

No. of FB friends

not related to gender, majors

related to no. of siblings

More phone conversations

related to smartphone users, gender

Skipping classes 1:58 PM

5

Two-Sample Tests

Independent Populations

Independent

Population Means

1 and 2 known

1 and 2 unknown

Use a Z test statistic

Use S to estimate unknown ,

use a t test statistic

Assumptions:

population distributions are normal Or sample size 30 for each sample

Samples are randomly and independently drawn Independence: sample selected from one

population has no effect on the sample

selected from the other population

6

F-Test of Population

Variances

7

Testing Population Variances

Purpose: To determine if two independent

populations have the same variability.

H0: 12 = 2

2

H1: 12 2

2

H0: 12 2

2

H1: 12 < 2

2

H0: 12 2

2

H1: 12 > 2

2

Two-tail test Lower-tail test Upper-tail test

8

Testing Population Variances

2

2

2

1

S

SF

The test statistic is:

= Variance of Sample 1 (usu the larger one)

n1 - 1 = numerator degrees of freedom

n2 - 1 = denominator degrees of freedom

= Variance of Sample 2

2

1S

2

2S

Assume each population distributions is normally distributed, then the ratio

of the two sample variances follows

the F distribution.

9

Testing Population Variances

The critical F-value is found from the F table.

There are two appropriate degrees of freedom:

numerator degrees of freedom (column),

denominator degrees of freedom (row).

10

Testing Population Variances

0

FL Reject H0 Do not reject H0

H0: 12 2

2

H1: 12 < 2

2

Reject H0 if F < FL

0

FU Reject H0 Do not

reject H0

H0: 12 2

2

H1: 12 > 2

2

Reject H0 if F > FU

Lower-tail test Upper-tail test

11

Testing Population Variances

L2

2

2

1

U2

2

2

1

FS

SF

FS

SF

Rejection region for a two-tail test is:

F 0 /2

Reject H0 Do not reject H0 FU

H0: 12 = 2

2

H1: 12 2

2

FL

/2

Two-tail test

12

Testing Population Variances

To find the critical F values, FU and FL:

1. Find FU from the F table for n1 1 numerator

and n2 1 denominator degrees of freedom

*U

LF

1F 2. Find FL using the formula:

Where FU* is from the F table with n2 1 numerator and n1 1

denominator degrees of freedom (i.e., switch the d.f. from FU)

Check FU

only if

13

Eg: Dividend Yield

Is there a difference in the variances between the

NYSE & NASDAQ return rates at the = 0.05 level?

NYSE NASDAQ

Number 21 25

Mean 3.27 2.53

Std dev 1.30 1.16

14

Example

Form the hypothesis test:

H0: 21

22 = 0 (there is no difference between variances)

H1: 21

22 0 (there is a difference between variances)

Numerator:

n1 1 = 21 1 = 20 d.f.

Denominator:

n2 1 = 25 1 = 24 d.f.

FU = F.025, 20, 24 = 2.33

Numerator:

n2 1 = 25 1 = 24 d.f.

Denominator:

n1 1 = 21 1 = 20 d.f.

FL = 1/F.025, 24, 20 = 0.41

FU: FL:

15

16

Testing Population Variances

The test statistic is:

256.116.1

30.12

2

2

2

2

1 S

SF

0

/2 = .025

FU=2.33 Reject H0 Do not

reject H0

FL=0.41

/2 = .025

Reject H0 F

F = 1.256 is not in the

rejection region, so we do

not reject H0

Conclusion: There is insufficient

evidence of a difference in

variances at = .05

Review Questions

10.36 Find the upper critical F-value for a two-tailed test if

a. = 0.01, n1=16, n2=21

b. = 0.05, n1=16, n2=21

c. = 0.10, n1=16, n2=21

17

Review Questions

10.46 At the = 0.05 level, is there a difference in the variances between the male & female anxiety level? The scale runs from 20 (no anxiety) to 100 (highest level of anxiety).

Male FEMALE Number 100 72

Mean 40.26 36.85

Std dev 13.35 9.42

18