Marketing Research

Marketing ResearchAaker, Kumar, DayNinth EditionInstructor’s Presentation Slides

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Marketing Research 10th Edition http://www.drvkumar.com/mr10/

Chapter Seventeen

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Hypothesis Testing:

Basic Concepts and Tests of Association


Hypothesis Testing: Basic Concepts

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

• Purpose of Hypothesis Testing ▫ To make a judgment about the difference between two

sample statistics or between sample statistic and a hypothesized population parameter

• 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

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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.

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Hypothesis Testing ProcessProblem Definition

Clearly state the null and alternative hypotheses

Choose the relevant test and the appropriate probability distribution

Choose the critical value

Compare test statistic & critical value

Reject null

Determine the significance level

Compute relevant test statistic

Determine the degrees of freedom

Decide if one-or two-tailed test

Do not reject null

Does the test statistic fall in the critical

region?


Basic Concepts of Hypothesis TestingThree Criteria Used To Decide Critical Value

(Whether To Accept or Reject Null Hypothesis):

•Significance Level

•Degrees of Freedom

•One or Two Tailed Test

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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 ()

• When choosing a level of significance, there is an inherent tradeoff between these two types of errors

• A good test of hypothesis should reject a null hypothesis when it is false

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Relationship between Type I & Type II Errors

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Relationship between Type I & Type II Errors (Contd.)

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Relationship between Type I & Type II Errors (Contd.)

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Choosing The Critical Value• Power of hypothesis test

▫ (1 - ) should be as high as possible

• Degrees 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

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Hypothesis Testing & Associated Statistical Tests

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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

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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

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Hypothesis Testing

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Data Analysis Outcome

Accept Null Hypothesis

Reject Null Hypothesis

Null Hypothesis is True Correct Decision Type I Error

Null Hypothesis is False Type II Error Correct Decision


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

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The Concept of Statistical Independence

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If n is equal to 200 and Ei is the number of outcomes expected in cell i,


Chi-Square As a Test of Independence

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Chi-Square As a Test of Independence (Contd.)

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.

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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

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Cutoff points of the chi-square distribution function


Degrees of Freedom• Number of degrees 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

Chi-square Distribution (Contd.)

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Chi-square Statistic (2)• Measures of the difference between the actual numbers observed in cell i

(Oi), and number expected (Ei) under assumption of statistical independence

if the null hypothesis were true

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

Oi = observed number in cell i

Ei = number in cell i expected under independence

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

n is the total number of observations

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i

iin

i E

EO 2

1

2 )(


Chi-square Step-by-Step

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Strength of Association• Measured by contingency coefficient

• 0 = no association (i.e., Variables are statistically independent)

• Maximum value depends on the size of table

• Compare only tables of same size

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Limitations of Chi-square 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

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Measures of Association for Nominal Variables

•Measures based on Chi-Square

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Phi-squared

Cramer’s V


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

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To determine 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

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