CHAPTERS 16-17

HYPOTHESIS TESTING, AND DETERMINING AND

INTERPRETING BETWEEN TWO VARIABLES

Important Topics of These Chapters

Hypothesis and testing.Steps involved in hypotheses testing.Type I and Type II errors.Independent and related samples.Degrees of freedom.Hypotheses about single mean.Cross-tabulations.Goodness of fit and chi-square tests.How to interpret a chi-square result.

Hypotheses and Hypothesis Testing

Hypotheses:Assumptions, intuition, prior knowledge or theories that a researcher or manager makes statements about population parameter under study. Most commonly takes the form of exact specification as what the population parameter value is.

Hypothesis Testing:Statistical procedure used to ‘accept’ or ‘reject’ the hypothesis based on sample information. For hypothesis testing, sample is the most current information about population.

Hypothesis Testing (cont.)

• Statement of hypotheses:• Null Hypothesis(Ho:):

• Stated hypothesis:• Mean Income of population is equal to $36,000.

• Alternative Hypothesis (Ha:):• Alternative that tested against the ‘Null hypothesis’.

• Mean income of population is not equal to $36,000. (Two tails test).,or• Directional Hypothesis:

• Mean income of population is less than $36,000.(one tail test), or• Mean income of population is higher than $36,000. (one tail test).

Hypothesis testing (cont.)• Statistical technique to test the Null and Alternative Hypotheses:

• Cross tabulation, or other available statistical techniques.• Decision rule as the basis for determining whether to reject or fail to reject the null hypothesis:

• Computed and table value of test statistic (for cross-tabulation, it is the table value of Chi-Square statistics at certain degree of freedom (d.f .= c-1 X r-1), against to computed value of Chi-Square statistic.

• Significance level:• At .01 level (99% confidence), or at .05 level (95% confidence), or at .10 level (90% confidence).

• Reject, or fail to reject the Null Hypothesis by basing upon decision rule.• State the conclusion from the perspective of the original research problem or question.

Formulate H0 and H1

Steps Involved in Hypothesis TestingSteps Involved in Hypothesis Testing

Select Appropriate Test

Collect Data and Calculate Test Statistic

Determine Probability Associated with Test

Statistic

Choose Level of Significance,

Draw Marketing Research Conclusion

Reject, or Fail to Reject H0

Determine Critical Value of Test Statistic

TSCR

Determine if TSCR falls into (Non)

Rejection Region

Compare with Level of Significance,

Other Issues in Hypothesis Testing

Types of errors in hypothesis testing.

Type II error

Fail to reject the null hypothesis

when, in fact, it is false

Type I error

Rejection of a null hypothesis when, in

fact, it is true

Other Issues in Hypothesis Testing (cont.)

Type I and Type II Errors

Actual State ofthe Null Hypothesis

Fail to Reject Ho Reject Ho

Ho is true Correct (1 - ) no error Type I Error ()

Ho is false Type II error () Correct (1 - )no error

Probabilities of Type I & Type II Probabilities of Type I & Type II ErrorError

99% of Total Area

Critical Value of Z

= 15

= 17

= 0.01

= 1.645Z

= -2.33Z

Z

Z

95% of Total Area

= 0.05

Other Issues in Hypothesis Testing (cont.)

Accepting Ho or Failing to Reject (FTR) Ho:Researchers often fail to make a distinction between accepting and failing to reject (FTR) Ho.

One-Tailed Test or Two-Tailed Test:The decision of whether to use a one-tailed test or a two-tailed test depends on the nature of the situation and what you are trying to demonstrate when you stated the Null Hypothesis.

Hypothesis Tests

Independent versus Related SamplesIndependent Samples:

Samples in which measurement of a variable in one population has no effect on the measurement of the variable in another.

Related Samples:Samples in which the measurement of a variable in one population may influence the measurement of the variable in another.

Hypothesis Tests (cont.)

Degrees of Freedom:Degrees of freedom are the number of observations in a statistical problem that are not restricted or are free to vary.The number of degrees of freedom (d.f.) is equal to the number of observations minus the number of assumptions or constraints necessary to calculate a statistic.

Hypotheses About One Mean

Z-test:Hypothesis test about a single mean if the sample is large enough (n > 30) and drawn from a normal population.

Calculation of the Test Statistic:

Z =estimated standard error of the mean

(sample mean) - population mean under the

null hypothesis(

)

Hypotheses About One Mean(cont.)

T-test:Hypothesis test about a single mean if the sample is too small (n < 30) to use the Z-test.

Calculation of the Test Statistic:

t =estimated standard error of the mean

(sample mean) - population mean specified under the null hypothesis( )

Unshaped Area

= 0.0336

Probability of z with a One-Tailed TestProbability of z with a One-Tailed Test

Shaded Area

= 0.9664

z = 1.830

Hypothesis Tests

Distributions

A Broad Classification of Hypothesis TestsA Broad Classification of Hypothesis Tests

Tests of Association

Tests of Differences

Median/ Rankings

Means Proportions

Cross-TabulationsMonotonic relationships:

Researcher can assign only a general direction (increase or decrease) between two variables.

Non-monotonic relationships:The presence(or absence) of one variable is systematically associated with the presence(or absence) of another variable.

Cross tabulation and associated chi-square:It used to assess whether a non-monotonic relationship exits between two nominal-scaled variables.

Gender and Internet UsageGender and Internet Usage

SexRow

Internet Usage Male Female Total

Light (1) 5 10 15

Heavy (2) 10 5 15

Column Total 15 15

Internet Usage by SexInternet Usage by Sex

Sex

Internet Usage Male Female

Light 33.3% 66.7%

Heavy 66.7% 33.3%

Column total 100% 100%

Sex by Internet UsageSex by Internet Usage

Internet Usage

Sex Light Heavy Total

Male 33.3% 66.7% 100.0%

Female 66.7% 33.3% 100.0%

Purchase of Fashion Clothing by Purchase of Fashion Clothing by Marital StatusMarital Status

Purchase ofFashion

Current Marital Status

Clothing Married Unmarried

High 31% 52%

Low 69% 48%

Column 100% 100%

Number ofrespondents

700 300

Purchase of Fashion Clothing by Purchase of Fashion Clothing by Marital Status and GenderMarital Status and Gender

Purchase ofFashion

SexMale Female

Clothing Marr ied NotMarr ied

Marr ied NotMarr ied

High 35% 40% 25% 60%

Low 65% 60% 75% 40%

Columntotals

100% 100% 100% 100%

Number ofcases

400 120 300 180

Unmarried Unmarried

Ownership of Expensive Ownership of Expensive Automobiles by Education LevelAutomobiles by Education Level

Own ExpensiveAutomobile

Education

College Degree No College Degree

Yes 32% 21%

No 68% 79%

Column totals 100% 100%

Number of cases 250 750

Desire to Travel Abroad by AgeDesire to Travel Abroad by Age

Desire to Travel Abroad Age

Less than 45 45 or More

Yes 50% 50%

No 50% 50%

Column totals 100% 100%

Number of respondents 500 500

Eating Frequently in Fast Food Eating Frequently in Fast Food Restaurants by Family SizeRestaurants by Family Size

Eat Frequently in FastFood Restaurants

Family Size

Small Large

Yes 65% 65%

No 35% 35%

Column totals 100% 100%

Number of cases 500 500

Ownership of Expensive Automobiles Ownership of Expensive Automobiles by Education Level and Income Levelsby Education Level and Income Levels

OwnExpensive

IncomeLow Income High Income

Automobile CollegeDegree

NoCollegeDegree

CollegeDegree

NoCollegeDegree

Yes 20% 20% 40% 40%

No 80% 80% 60% 60%

Columntotals

100% 100% 100% 100%

Number ofrespondents

100 700 150 50

Eating Frequently in Fast Food Eating Frequently in Fast Food Restaurants by Family Size & IncomeRestaurants by Family Size & Income

EatFrequentlyin Fast FoodRestaurants

Income Low Family size

High Family size

Small Large Small Large

Yes 65% 65% 65% 65%

No 35% 35% 35% 35%

Columntotals

100% 100% 100% 100%

Number ofRespondents

250 250 250 250

Desire to Travel AbroadDesire to Travel Abroadby Age and Genderby Age and Gender

Desire toTravelAbroad

Sex Male Age

Female Age

< 45 >=45 <45 >=45

Yes 60% 40% 35% 65%

No 40% 60% 65% 35%

Columntotals

100% 100% 100% 100%

Number ofCases

300 300 200 200

Goodness of Fit

Chi-Square Test:Test of the goodness of fit between the observed distribution and the expected distribution of a variable.Statement of Hypotheses:

Ho: There is not an association (relationship) between variable ‘X’ and variable ‘Y’.Ha: There is an association (relationship) between variable ‘X’ and variable ‘Y’.

Chi-Square AnalysisThe computed Chi-Square value: n (observed - Expected)X2 = -------------------------------- i - 1 ExpectedWhere: Observed: Observed frequency of cell i. Expected: Expected frequency of cell I. n: number of cells.

Chi-Square Analysis (cont.) The Chi-Square Distribution:

Table value or critical value of Chi-Square at certain degree of freedom (d.f).Degrees of freedom (d.f.) for Chi-Square statistics:

(r-1) (c-1)

Chi-Square Distribution: One Tail Chi-Square Distribution: One Tail TestTest

Reject H0

Fail to Reject H0

CriticalValue, or table value of X2

2

How to Interpret a Chi-Square Result

It yields the probability that researcher find evidence in support of the null hypothesis.It should be pointed out that whether or not a non-monotonic relationship exits between variable ‘X’ and variable “Y’.The chi-square test does not indicate the nature or direction of association between the two variables.The chi-square test indicated the strength of association that exits between two variables.

Independent Samples

One Sample Two or More Samples

One Sample Two or More Samples

Paired Samples Independent

SamplesPaired

Samples

* t test * Z test

* Chi-Square * K-S * Runs* Binomial

* Two-Group t test

* Z test

* Pairedt test * Chi-Square

* Mann-Whitney* Median* K-S

* Sign* Wilcoxon* McNemar* Chi-Square

Hypothesis Tests

Parametric Tests (Metric Tests)

Non-parametric Tests (Nonmetric Tests)

A Classification of Hypothesis Testing A Classification of Hypothesis Testing Procedures for Examining DifferencesProcedures for Examining Differences