M303 Hypothesis Testing III

FALL 2026 · LECTURE 16
M303
Hypothesis Testing III
Compare several means with ANOVA and test categorical associations with chi-square.

Where we are

Problem formulationResearch designData collectionData analysis & recommendation
COMPLETEDt-tests

One benchmark · two linked means · two independent means

TODAYMore groups and categorical outcomes

ANOVA F-test · chi-square test

Learning Objectives

01Choose

ANOVA for mean comparisons across two or more independent groups.

02Interpret

The global F-test, group means, and source output.

03Choose

Chi-square for association between two categorical variables.

04Compute

Expected counts and translate the test into a marketing conclusion.

Previous class review

01

A paired t-test keeps two measurements from the same respondent linked.

TRUE
02

An independent-samples t-test requires a continuous dependent variable and a two-level categorical grouping variable.

TRUE
03

If the first sample mean is larger, the alternative must always be right-tailed.

FALSE
04

With three independent groups, repeated two-sample tests are the preferred global comparison.

FALSE

Choose the analysis from the variables

One continuous outcome is measured across three independent prototype groups. Which test?
ONE BENCHMARKOne-sample t-test
TWO INDEPENDENT GROUPSIndependent t-test
TWO OR MORE GROUPSANOVA F-test
TWO CATEGORICAL VARIABLESChi-square

Use ANOVA to compare means across groups

DEPENDENT VARIABLE
Continuous

Liking · purchase amount · time on site.

INDEPENDENT VARIABLE
Categorical · two or more levels

Prototype 1, 2, and 3 · age groups · income groups.

With two groups, ANOVA and the independent t-test address the same mean-comparison question.

ANOVA is the natural extension when the categorical variable has three or more levels.

ANOVA example: three tablet prototypes

ASUSTABLETPROTOTYPES DATA

Three independent samples of 10 participants each try different ASUS tablet prototypes and rate liking on a 1–11 scale.

Are all three population mean evaluations equal?

Hypotheses

H₀: μ₁ = μ₂ = μ₃

Hₐ: At least one population mean differs.

ANOVA data: three independent samples

Observation
Prototype 1
Prototype 2
Prototype 3
111108267103169471075959 610411744886589447101056

Run ANOVA with the M303 Web Tools

F
One-way ANOVAkoraycosguner.github.io/tools.html

Instructions and required data files will be posted on Canvas.

01

Load one continuous outcome and one categorical grouping variable.

02

Select the outcome and the group variable with two or more levels.

03

Run the one-way ANOVA and inspect the group means.

04

Record the F statistic and p-value; follow up on specific groups only when the design supports it.

ANOVA output: exact source values

Group
n
Mean
Variance
Prototype 1106.810.4 Prototype 2106.05.333333333 Prototype 3108.32.233333333
Source
SS
df
MS
F / p / F crit
Between groups27.26666667213.633333332.276437848 / 0.122007081 / 3.354130829 Within groups161.7275.988888889— Total188.966666729——

A significant F-test says the means are not all equal

If p < .05, reject H₀.

Conclude that at least one group’s population mean differs from at least one other group’s mean.

THE F-TEST ANSWERS
Is there evidence of any mean difference?

It tests the global null that all group means are equal.

THE F-TEST ALONE DOES NOT ANSWER
Exactly which groups differ?

That requires planned comparisons or post-hoc analysis.

In-class exercise: purchase amount across groups

01

Does mean purchase amount differ across age groups?

Yes / No
02

Does mean purchase amount differ across income groups?

Yes / No
03

Does mean purchase amount differ across race groups?

Yes / No
For each question: identify the outcome, identify the groups, run ANOVA, and state the marketing conclusion.

Use the PurchaseBehavior data posted on Canvas.

Choose the next test from the variables

Advertising medium and campaign success are both categorical. Which test?
CONTINUOUS OUTCOMEMean-comparison tests
TWO CATEGORICAL VARIABLESChi-square test

Compare observed counts with expected counts under independence.

Use chi-square to test categorical association

H₀ · INDEPENDENT
One categorical variable does not depend on the other

Advertising success is independent of advertising medium.

Hₐ · ASSOCIATED
The distribution of one variable changes across the other

Advertising success depends on advertising medium.

Chi-square analyzes counts—not means.

Chi-square example: does medium matter?

PANERA BREAD

Campaigns use local radio or local television and are coded as successful or unsuccessful.

Is campaign success associated with advertising medium?

Hypotheses

H₀: Success is independent of medium.

Hₐ: Success is associated with medium.

Advertising-medium data: source records

Campaign
Medium
Outcome
1Local TVUnsuccessful2Local radioSuccessful3Local TVSuccessful4Local TVSuccessful5Local radioSuccessful 6Local TVSuccessful7Local radioSuccessful8Local TVSuccessful9Local TVSuccessful10Local radioSuccessful

Expected counts describe independence

OBSERVED COUNTS
Local radio
Local TV
Row total
Successful
8
12
20
Unsuccessful
12
8
20
Column total
20
20
40
If medium has no association with success, what count belongs in each cell?

Complete the expected-count table

EXPECTED COUNTS
Local radio
Local TV
Row total
Successful
10
10
20
Unsuccessful
10
10
20
Column total
20
20
40
Compute each cell from its margins.

Expected = row total × column total ÷ grand total = 20 × 20 ÷ 40

Run chi-square with the M303 Web Tools

χ²
Chi-square testkoraycosguner.github.io/tools.html

Instructions and required data files will be posted on Canvas.

01

Load two categorical variables or an observed frequency table.

02

Build the row-by-column cross-tabulation of observed counts.

03

Compare observed counts with the expected counts under independence.

04

Run the chi-square test; reject independence when p < .05.

In-class exercise: purchase decisions by segment

01

Do purchase frequencies differ across age groups?

Yes / No
02

Do purchase frequencies differ across education levels?

Yes / No
03

Do purchase frequencies differ across income levels?

Yes / No
Each question pairs purchase decision with one categorical customer characteristic.

Use the PurchaseBehavior data posted on Canvas and report association—not causation.

ANOVA or chi-square?

ANOVA F-TEST
Compare means

Continuous outcome · categorical groups.
Example: mean purchase amount across age groups.

CHI-SQUARE TEST
Compare count distributions

Categorical outcome · categorical groups.
Example: purchase yes/no across age groups.

The outcome variable decides which question you are asking.

To sum up

01

ANOVA: compare a continuous outcome across two or more independent groups.

02

A significant global F-test says the population means are not all equal; it does not identify every differing pair.

03

Chi-square: compare observed categorical counts with expected counts under independence.

04

Report association—not causation—unless the research design warrants a causal claim.

Next class

NEXT TOPIC
Integrated hypothesis-testing applications
BE READY TO
  1. Select among all five tests from the variable structure.
  2. Explain a p-value decision in careful language.
  3. Apply the methods to the historical “I’m a PC” campaign data.