chapters 16-17 hypothesis testing, and determining and interpreting between two variables
TRANSCRIPT
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