marketing research - dadang iskandar · pdf filechapter seventeen hypothesis testing: ......

Marketing Research 8th Edition

Aaker, Kumar, Day

Marketing Research

Aaker, Kumar, Day

Eighth Edition

Instructor‟s Presentation Slides


Aaker, Kumar, Day

Chapter Seventeen

Hypothesis Testing:

Basic Concepts and Tests of

Association


Aaker, Kumar, Day

Hypothesis Testing:

Basic Concepts

Assumption (hypothesis) made about a population parameter (not sample parameter)

Purpose of Hypothesis Testing

To make a judgement about the difference between two sample statistics or the sample statistic and a hypothesized population parameter

Evidence has to be evaluated statistically before arriving at a conclusion regarding the hypothesis.


Aaker, Kumar, Day

Hypothesis Testing

The null hypothesis (Ho) is tested against the

alternative hypothesis (Ha).

At least the null hypothesis is stated.

Decide upon the criteria to be used in making

the decision whether to “reject” or "not reject"

the null hypothesis.


Aaker, Kumar, Day

The Logic of Hypothesis Testing

Evidence has to be evaluated statistically

before arriving at a conclusion regarding the

hypothesis

Depends on whether information generated

from the sample is with fewer or larger

observations


Aaker, Kumar, Day

Problem Definition

Clearly state the null and

alternative hypotheses.

Choose the relevant test and

the appropriate probability

distribution

Choose the critical value

Compare test statistic and

critical value

Reject null

Does the test statistic fall in

the critical region?

Determine the

significance level

Compute relevant

test statistic

Determine the

degrees of

freedom

Decide if one-or

two-tailed test

Do not reject null


Aaker, Kumar, Day

Basic Concepts of Hypothesis

Testing (Contd.)

The Three Criteria Used Are

Significance Level

Degrees of Freedom

One or Two Tailed Test


Aaker, Kumar, Day

Significance Level

Indicates the percentage of sample means that is outside the cut-off limits (critical value)

The higher the significance level () used for testing a hypothesis, the higher the probability of rejecting a null hypothesis when it is true (Type I error)

Accepting a null hypothesis when it is false is called a Type II error and its probability is ()


Aaker, Kumar, Day

Significance Level (Contd.)

When choosing a level of significance, there is

an inherent tradeoff between these two types

of errors

Power of hypothesis test (1 - )

A good test of hypothesis ought to reject a null

hypothesis when it is false

1 - should be as high a value as possible


Aaker, Kumar, Day

Degree of Freedom

The number or bits of "free" or unconstrained

data used in calculating a sample statistic or

test statistic

A sample mean (X) has `n' degree of freedom

A sample variance (s2) has (n-1) degrees of

freedom


Aaker, Kumar, Day

One or Two-tail Test

One-tailed Hypothesis Test

Determines whether a particular population parameter is

larger or smaller than some predefined value

Uses one critical value of test statistic

Two-tailed Hypothesis Test

Determines the likelihood that a population parameter is

within certain upper and lower bounds

May use one or two critical values


Aaker, Kumar, Day

Basic Concepts of Hypothesis

Testing (Contd.)

Select the appropriate probability distribution

based on two criteria

Size of the sample

Whether the population standard deviation is

known or not


Aaker, Kumar, Day

Hypothesis Testing

DATA ANALYSIS

OUTCOME

In Population Accept Null

Hypothesis

Reject Null

Hypothesis

Null Hypothesis

True

Correct Decision Type I Error

Null Hypothesis

False

Type II Error Correct

Decision


Aaker, Kumar, Day

Hypothesis Testing

Tests in this class Statistical Test

Frequency Distributions 2

Means (one) z (if is known)

t (if is unknown)

Means (two) t

Means (more than two) ANOVA


Aaker, Kumar, Day

Cross-tabulation and Chi Square

In Marketing Applications, Chi-square Statistic Is Used As

Test of Independence

Are there associations between two or more variables in a study?

Test of Goodness of Fit

Is there a significant difference between an observed frequency

distribution and a theoretical frequency distribution?

Statistical Independence

Two variables are statistically independent if a knowledge of one would

offer no information as to the identity of the other


Aaker, Kumar, Day

Chi-Square As a Test of

Independence

Null Hypothesis Ho

Two (nominally scaled) variables are statistically

independent

Alternative Hypothesis Ha

The two variables are not independent

Use Chi-square distribution to test.


Aaker, Kumar, Day

Chi-square As a Test of

Independence (Contd.)

Chi-square Distribution

A probability distribution

Total area under the curve is 1.0

A different chi-square distribution is

associated with different degrees of freedom


Aaker, Kumar, Day


Independence (Contd.) Degree of Freedom

v = (r - 1) * (c - 1)

r = number of rows in contingency table

c = number of columns

Mean of chi-squared distribution

= Degree of freedom (v)

Variance = 2v


Aaker, Kumar, Day

Chi-square Statistic (2)

Measures of the difference between the actual numbers observed in cell i (Oi), and number expected (Ei) under independence if the null hypothesis were true

With (r-1)*(c-1) degrees of freedom

r = number of rows c = number of columns

Expected frequency in each cell: Ei = pc * pr * n

Where pc and pr are proportions for independent variables and n is the total number of observations

i

iin

i E

EO 2

1

2 )(


Aaker, Kumar, Day

Chi-square Step-by-Step

1) Formulate Hypotheses

2) Calculate row and column totals

3) Calculate row and column proportions

4) Calculate expected frequencies (Ei)

5) Calculate 2 statistic

6) Calculate degrees of freedom

7) Obtain Critical Value from table

8) Make decision regarding the Null-hypothesis


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Example of Chi-square as a Test of

Independence

Class

1 2

A 10 8

Grade B 20 16

C 45 18

D 16 6

E 9 2

This is a „Cell‟


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Independence - Exercise

Own Income

Expensive Low Middle High

Automobile

Yes 45 34 55

No 52 53 27

Task: Make a decision whether the two variables are independent!


Aaker, Kumar, Day

The chi-square distribution

Probability distributions that are continuous, have one mode, and are skewed to the

right.

Exact shape varies according to the number of degrees of freedom.

The critical value of a test statistic in a chi-square distribution is determined by

specifying a significance level and the degrees of freedom.

Ex: Significance level = .05

Degrees of freedom = 4

CVx2 = 9.49

The decision rule when testing hypotheses by means of chi-square distribution is:

If x2 is <= CVx2, accept H0 Thus, for 4 df and = .05

If x2 is > CVx2, reject H0 If If x2 is <= 9.49, accept H0

df = 4

x2

F(x2) Critical value = 9.49

5% of area under curve

= .05


Aaker, Kumar, Day

Cross Tabulation Example

In a nationwide study of 1,402 adults a question was asked about institutions:

“I am going to name some institutions in this country. As far as the people running

these institutions are concerned, would you say have a great deal of confidence,

only some confidence, or hardly any confidence at all in them?”

One of the institutions was television.

Answers to the question about television are cross-tabulated with three levels of

income below.

Annual Family Income

Under

$10,000

$10,000 –

20,000

Over

$20,000

95 57 39 191

272 274 214 760

140 163 148 451

507 494 401 1,402

A great deal

Only some

Hardly any

Amount of confidence

in television


Aaker, Kumar, Day

Calculations for income-confidence data

Cell Observed Expected Contribution

(Ou – Eu)2/ Eu

Cell11 95 69.1 9.71

Cell12 57 67.3 1.58

Cell13 39 54.6 4.46

Cell21 272 274.8 .03

Cell22 274 267.8 .14

Cell23 214 217.4 .05

Cell31 140 163.1 3.27

Cell32 163 158.9 .11

Cell33 148 129.0 2.80

X2ts = 22.15


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= .05

df = 4 [(r-1) (c-1)]

n = 1402

X2cv = 9.5

X2ts = 22.15

df = 4 F(x2) X2

cv = 9.5

5% of area under curve

= .05

22.15


Aaker, Kumar, Day

Strength of Association

Measured by contingency coefficient

C = x2 o< c < 1

x2 + n

0 - no association (i.E. Variables are

statistically independent)

Maximum value depends on the size of

table-compare only tables of same size


Aaker, Kumar, Day

Limitations As an Association

Measure

It Is Basically Proportional to Sample Size

Difficult to interpret in absolute sense and compare

cross-tabs of unequal size

It Has No Upper Bound

Difficult to obtain a feel for its value

Does not indicate how two variables are related


Aaker, Kumar, Day

Chi-square Goodness of Fit

Used to investigate how well the observed

pattern fits the expected pattern

Researcher may determine whether population

distribution corresponds to either a normal,

poisson or binomial distribution


Aaker, Kumar, Day

Chi-square Degrees of Freedom

Employ (k-1) rule

Subtract an additional degree of freedom for

each population parameter that has to be

estimated from the sample data


Aaker, Kumar, Day

Goodness-of-Fit Test

Suppose a researcher is investigating preferences for four possible names of a

new lightweight brand of sandals: Camfo, Kenilay, Nemlads, and Dics. Since

the names are generated from random combinations of syllables, thre researcher

expects preferences will be equally distributed across the four names (that is,

each name will receive 25 percent of the available preferences). After sampling

300 people at reandom and asking them which one of the four names was most

preferred, the following distribution resulted (each expected value is 300 * .25 =

75).

Possible Name Observed

Preferences

Expected

Preferences

Camfo 30 75

Kenilay 80 75

Nemlads 120 75

Dics 70 75


Aaker, Kumar, Day

Goodness-of-Fit Test cont.

There are (d – 1) or three degrees of freedom in this instance. If is specified as 0.01, the critical value is 11.325 from Statistical Appendix Table 3.18 Given this information, the hypothesis to be tested can be stated as:

H0: preferences are equal for the names

Ha: preferences are not equal for the names

And the decision rule is

If x2 is <= 11.325, accept H0.

If x2 is > 11.325, reject H0.

The test statistic is calculated as

x2 = (30-75)2 / 75 + (80-75)2 / 75 + (120-75)2 / 75 + (70-75)2 / 75

= 27.00 + .33 + 27.00 + .33

= 54.66

marketing research - dadang iskandar · pdf filechapter seventeen hypothesis testing: ......

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