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Bluman, Chapter 11 1

http://www.youtube.com/watch?v=0Lw0NLw75Lo&safe=active

http://www.youtube.com/watch?v=0Lw0NLw75Lo&safe=active

While you wait: Copy the table below.

Allow room for data to be added in each cell.

Fill in the last row and column.


Location No College

Four-Year

Degree

Advanced

Degree Total

Urban 15 12 8

Suburban 8 15 9

Rural 6 8 7

Total `

Tests Using

Contingency

Tables Sec: 11.2


11.2 Tests Using Contingency

Tables When data can be tabulated in table form in terms of

frequencies, several types of hypotheses can be tested

by using the chi-square test.

The test of independence of variables is used to

determine whether two variables are independent of or

related to each other when a single sample is selected.

The test of homogeneity of proportions is used to

determine whether the proportions for a variable are

equal when several samples are selected from

different populations.


Test for Independence

The chi-square goodness-of-fit test can be used

to test the independence of two variables.

The hypotheses are:

H0: There is no relationship between two

variables.

H1: There is a relationship between two

variables.

If the null hypothesis is rejected, there is some

relationship between the variables.



In order to test the null hypothesis, one must

compute the expected frequencies, assuming

the null hypothesis is true.

When data are arranged in table form for the

independence test, the table is called a

contingency table.


Contingency Tables


The dimension of the contingency table

is given: ROW by COLUMN

Contingency Tables

The degrees of freedom for any contingency

table are d.f. = (rows – 1) (columns – 1) =

(R – 1)(C – 1).


The table above is 2 by 3. Only count the

numerical cells.


The formula for the test for independence:

where

d.f. = (R – 1)(C – 1)

O = observed frequency

E = expected frequency =


2

2

O E

E

row sum column sum

grand total

Example (see page 618): College

Education and Place of Residence A sociologist wishes to see whether the number of years

of college a person has completed is related to her or his

place of residence. A sample of 88 people is selected

and classified as shown. At α = 0.05, can the sociologist

conclude that a person’s location is dependent on the

number of years of college?


Location

No

College

Four-Year

Degree

Advanced

Degree Total

Urban 15 12 8 35

Suburban 8 15 9 32

Rural 6 8 7 21

Total 29 35 24 88

Example 11-5: College Education and

Place of Residence


Step 1: State the hypotheses and identify the claim.

H0: A person’s place of residence is independent

of the number of years of college completed.

H1: A person’s place of residence is dependent on

the number of years of college completed

(claim).

Step 2: Find the critical value.

The critical value is 9.488, since the degrees of

freedom are (3 – 1)(3 – 1) = 4.


Location

No

College

Four-Year

Degree

Advanced

Degree Total

Urban 15

12 8 35

Suburban 8

15 9 32

Rural 6

8 7 21

Total 29 35 24 88

Compute the expected values.

row sum column sum

grand totalE

1,1

35 2911.53

88 E

(11.53)

(10.55)

(6.92)

(13.92)

(12.73)

(8.35)

(9.55)

(8.73)

(5.73)

See next two slides for

calculator use.


[A]

Enter Observed values

as Matrix 1: 2nd x-1

[B]

Enter Expected values as Matrix


Test statistics :

2=3.006

College Education and Place of

Residence


Step 3: Compute the test value.

2

2

O E

E

2 2 2

2 2 2

2 2 2

15 11.53 12 13.92 8 9.55

11.53 13.92 9.55

8 10.55 15 12.73 9 8.73

10.55 12.73 8.73

6 6.92 8 8.35 7 5.73

6.92 8.35 5.73

3.01

College Education and Place of

Residence


Step 4: Make the decision.

Do not reject the null hypothesis, since 3.01<9.488.

Step 5: Summarize the results.

There is not enough evidence to support the claim

that a person’s place of residence is dependent on

the number of years of college completed.

Chapter 11 Other Chi-Square Tests

Section 11-2

Example 11-6

Page # 610


Example 11-6: Alcohol and Gender

A researcher wishes to determine whether there is a

relationship between the gender of an individual and the

amount of alcohol consumed. A sample of 68 people is

selected, and the following data are obtained. At α = 0.10,

can the researcher conclude that alcohol consumption is

related to gender?


Gender

Alcohol Consumption

Total Low Moderate High

Male 10 9 8 27

Female 13 16 12 41

Total 23 25 20 68


Step 1: State the hypotheses and identify the claim.

H0: The amount of alcohol that a person

consumes is independent of the individual’s

gender.

H1: The amount of alcohol that a person

consumes is dependent on the individual’s

gender (claim).



freedom are (2 – 1 )(3 – 1) = (1)(2) = 2.




Gender

Alcohol Consumption

Total Low Moderate High

Male 10

9 8 27

Female 13

16 12 41

Total 23 25 20 68


row sum column sum

grand totalE

1,1

27 239.13

68 E

(9.13) (9.93) (7.94)

(13.87) (15.07) (12.06)




2

2

O E

E

2 2 2

2 2 2

10 9.13 9 9.93 8 7.94

9.13 9.93 7.94

13 13.87 16 15.07 12 12.06

13.87 15.07 12.06

χ2 = 0.281



Do not reject the null hypothesis, since

0.281 < 4.605.

.


There is not enough evidence to support the claim

that the amount of alcohol a person consumes is

dependent on the individual’s gender.



Please record the table below in your

notes. Allow for additional col and row Household

income

Less than

$30,000 ( 24%)

$30,000-

$74,999

(33%)

$75,000-

$99,999

(38%)

$100,000 or

more (49%)

YES 24 33 38 49


Please record the table below in your

notes. Allow for additional col and row

Viewing Preferences

Lone Ranger Sesame Street The Simpsons

Boys 50 30 20

Girls 50 80 70

Test for Homogeneity of

Proportions Homogeneity of proportions test is used

when samples are selected from several

different populations and the researcher is

interested in determining whether the

proportions of elements that have a common

characteristic are the same for each population.


Test for Homogeneity of

Proportions The hypotheses are:

H0: p1 = p2 = p3 = … = pn

H1: At least one proportion is different from

the others.

When the null hypothesis is rejected, it can be

assumed that the proportions are not all equal.


Assumptions for Homogeneity of

Proportions

1. The data are obtained from a random sample.

2. The expected frequency for each category

must be 5 or more.



Section 11-2

Example 11-7

Page #611


Money and Happiness A psychologist selected 100 people from

each of 4 income groups and asked them if

they were “VERY HAPPY”. The percent of

each group who said yes are shown. At

a=0.05 test the claim that there is no

difference in proportions.


Household

income

Less than

$30,000 ( 24%)

$30,000-

$74,999

(33%)

$75,000-

$99,999

(38%)

$100,000 or

more (49%)

YES 24 33 38 49

Complete the table:


Household

income

Less than

$30,000 (

24%)

$30,000-

$74,999

(33%)

$75,000-

$99,999

(38%)

$100,000 or

more (49%) Total

YES 24 33 38 49

NO

Total

Household

income

Less than

$30,000 (

24%)

$30,000-

$74,999

(33%)

$75,000-

$99,999

(38%)

$100,000 or

more (49%) Total

YES 24 33 38 49 144

NO 76 67 62 51 256

Total 100 100 100 100 400

Example 11-7: Money and Happiness

Step 1: State the hypotheses.

H0: p1 = p2 = p3 = p4

H1: At least one proportion differs from the other.



freedom are (2 – 1 )(4 – 1) = 3.


Example 11-7: Money and Happiness


Reject the null hypothesis, since 14.149 > 7.815

.


There is enough evidence to reject the claim that

there is a difference proportions. Hence it seems

that there is a difference in the proportions of the

income.


Example 11-7: Lost Luggage

A researcher selected 100 passengers from each of 3

airlines and asked them if the airline had lost their

luggage on their last flight. The data are shown in the

table. At α = 0.05, test the claim that the proportion of

passengers from each airline who lost luggage on the

flight is the same for each airline.


Airline 1 Airline 2 Airline 3 Total

Yes 10 7 4 21

No 90 93 96 279

Total 100 100 100 300


Step 1: State the hypotheses.

H0: p1 = p2 = p3

H1: At least one proportion differs from the other.



freedom are (2 – 1 )(3 – 1) = (1)(2) = 2.





row sum column sum

grand totalE

1,1

21 1007

300 E

Airline 1 Airline 2 Airline 3 Total

Yes 10

7 4 21

No 90

93 96 279

Total 100 100 100 300

(7) (7) (7)

(93) (93) (93)

Example 11-7: Luggage



2

2

O E

E

2 2 2

2 2 2

10 7 7 7 4 7

7 7 7

90 93 93 93 96 93

93 93 93

2.765



Do not reject the null hypothesis, since

2.765 < 5.991.

.


There is not enough evidence to reject the claim

that the proportions are equal. Hence it seems that

there is no difference in the proportions of the

luggage lost by each airline.


In a study of the television viewing habits of

children, a developmental psychologist selects a

random sample of 300 first graders: 100 boys

and 200 girls. Each child is asked which of the

following TV programs they like best: The Lone

Ranger, Sesame Street, or The Simpsons.

Results are shown in the contingency table to

follow.



Viewing Preferences


Boys 50 30 20

Girls 50 80 70

Do the boys' preferences for these TV

programs differ significantly from the

girls' preferences? Use a 0.05 level of

significance.

Step 1:


H0: Pboys who prefer Lone Ranger = Pgirls who prefer Lone Ranger

H0: Pboys who prefer Sesame Street = Pgirls who prefer Sesame Street

H0: Pboys who prefer The Simpsons = Pgirls who prefer The Simpsons

Null hypothesis: The null hypothesis

states that the proportion of boys who

prefer the Lone Ranger is identical to

the proportion of girls. Similarly, for the

other programs.

Alternative hypothesis: At least

one of the null hypothesis

statements is false.

Calculations:

Determine the expected values:


Viewing Preferences Row total


Boys 50 30 20 100

Girls 50 80 70 200

Column total 100 110 90 300

On your own

Read section 11.2

and all its examples.

Sec 11.2 page 614

#1-7 all;

And #9, 17, 23, 27, 31


Example

Suppose a new postoperative procedure is

administered to a number of patients in a large

hospital. Do the doctors feel differently about this

procedure from the nurses, or do they feel basically

the same?

The question is not whether the

doctors prefer the procedure but

whether there is a difference of

opinion between doctors and

nurses.

Hypothesis:

H0: the opinion about the procedure is

independent of the profession.

H1: the opinion about the procedure is

dependent of the profession.

Group Prefers

new

Prefers

old

No pref

nurses 100 80 20

doctors 50 120 30

Consider the result listed

below:


Section 11-2

Example 11-5

Page #606


Hospitals and Infections



Section 11-2

618

Page #606


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