1. MKTG 368 All Statistics PowerPoints
Setting Up Null and Alternative Hypotheses
One-tailed vs. Two-Tailed Hypotheses
Single Sample T-Test
Paired Samples T-Test
Independent Samples T-Test
ANOVA
Correlation and Regression
One-Way and Two-Way Chi-Square

2. Translating a Problem Statement Into the Null and Alternative Hypotheses

3. Initial Problem Statement
Example:
Let’s say we are interested in whether a flyer increases contributions to National Public Radio. We know that last year the average contribution was $52. This year, we sent out a flyer to 30 people explaining the benefits of NPR and asked for donations. This year’s average contribution with the flyer ended up being $55, with a standard deviation of $12.
How do we translate this into the null and alternative hypotheses (in terms of both a sentence and a formula)?

4. Gleaning Information from the Statement
Example:
Let’s say we are interested in whether a flyer increases contributions to National Public Radio. We know that last year the average contribution was $52. This year, we sent out a flyer to 30 people explaining the benefits of NPR and asked for donations. This year’s average contribution with the flyer ended up being $55, with a standard deviation of $12.
Direction of Alternative Hypothesis
Population
Information
Sample
Information

5. Translating Information into Null and Alternative Hypotheses
Set up Alternative Hypothesis First
Null is exact opposite of Alternative
Null + Alternative must include all possibilities
Hence, we say ‘less than or equal to’ rather than just ‘less than’
Ho (Null Hypothesis):
A flyer does not increase contribution to NPR: Flyer$ ≤ Last Year$
H1 (Alternative Hypothesis):
A flyer increases contributions to NPR: Flyer$ > Last Year$
Subscript =
Dependent Variable
(what you are comparing them on)
Groups, Conditions,
or Levels of the
Independent Variable

6. On One-Tailed (Directional) vs. Two-Tailed (Non-Directional) Hypotheses

7. Basics on the Normal Distribution
Positive
Values
Negative
68%
95%
99%

8. One-Tailed Hypothesis (H1: Condition 1 > Condition 2)
Ho (Null Hypothesis):
A flyer does not increase contribution to NPR: Flyer$ ≤ Last Year$
H1 (Alternative Hypothesis):
A flyer increases contributions to NPR: Flyer$ > Last Year$
In H1, b/c Flyer > Last Year
Alpha region is on right side
“Alpha Region”
α = .05, 1-tailed
(positive)
t-critical

9. One-Tailed Hypothesis (H1: Condition 1 < Condition 2)
Ho (Null Hypothesis):
A poster does not decrease lbs of litter in park: Posterlbs ≤ Last Yearlbs
H1 (Alternative Hypothesis):
A poster decreases lbs of litter in park: Posterlbs < Last Yearlbs
In H1, b/c Poster < Last Year
Alpha region is on left side
“Alpha Region”
α = .01, 1-tailed
(negative)
t-critical

10. Two-Tailed Hypothesis (H1: Condition 1 ≠ Condition 2)
Ho (Null Hypothesis):
People are willing to pay the same for Nike vs. Adidas: NikeWTP = AdidasWTP
H1 (Alternative Hypothesis):
People not willing to pay same for Nike vs. Adidas: NikeWTP ≠ AdidasWTP
In H1, b/c NikeWTP ≠ AdidasWTP
Alpha region is on both sides;
Half of .05 goes on each side
“Alpha Region”
α = .025
“Alpha Region”
α = .025
(positive)
t-critical
(negative)
t-critical

11. T-Tests Single Sample Paired Samples (Correlated Groups) Independent Samples

12. Single Sample T-Test (Example 1)
Comparing a sample mean to an existing population mean

13. Gleaning Information from the Statement
Example:
Let’s say we are interested in whether a flyer increases contributions to National Public Radio. We know that last year the average contribution was $52. This year, we sent out a flyer to 30 people explaining the benefits of NPR and asked for donations. This year’s average contribution with the flyer ended up being $55, with a standard deviation of $12.
Direction of Alternative Hypothesis
Single Sample T-test:
df = N-1 = 30-1 = 29
Population
Information
Sample
Information
Use alpha = .05
How do we get a t-critical value? 

14. Critical T-Table
For single sample t-test, df = N-1

15. One-Tailed Hypothesis (H1: Condition 1 > Condition 2)
Ho (Null Hypothesis):
A flyer does not increase contribution to NPR: Flyer$ ≤ Last Year$
H1 (Alternative Hypothesis):
A flyer increases contributions to NPR: Flyer$ > Last Year$
In H1, b/c Flyer > Last Year
Alpha region is on right side
“Alpha Region”
α = .05, 1-tailed
t-critical = 1.699
If t-obtained > t-critical, reject Ho (i.e., if t-obtained falls in the critical region, reject Ho).

16. Computation of Single Sample T-test
“Alpha Region”
α = .05, 1-tailed
t-critical = 1.699
Decision?
Because t-obtained (1.37) < t-critical (1.699), retain Ho.
Conclusion?
The flyer did not increase contributions to NPR.
t-obtained = 1.37

17. Single Sample T-Test (Example 2)
Comparing a sample mean to an existing population mean

18. Gleaning Information from the Statement
Example:
Let’s say we are interested in whether a poster decreases amount of litter in city parks. We know that last year the average amount of litter in city parks was 115 lbs. This year, we placed flyers in 25 parks that said “Did you know that 95% of people don’t litter? Join the crowd.” Later, when we weighed the litter, the average amount of litter was 100 lbs, with a standard deviation of 10 lbs.
Direction of Alternative Hypothesis
Single Sample T-test:
df = N-1 = 25-1 = 24
Population
Information
Sample
Information
Use alpha = .01
How do we get a t-critical value? 

19. Critical T-Table
For single sample t-test, df = N-1

20. One-Tailed Hypothesis (H1: Condition 1 < Condition 2)
Ho (Null Hypothesis):
A poster does not decrease lbs of litter in park: Posterlbs ≤ Last Yearlbs
H1 (Alternative Hypothesis):
A poster decreases lbs of litter in park: Posterlbs < Last Yearlbs
In H1, b/c Poster < Last Year
Alpha region is on left side
“Alpha Region”
α = .01, 1-tailed
t-critical =

21. Computation of Single Sample T-test
Decision?
Because t-obtained (-7.50) < t-critical (-2.492), reject Ho.
Conclusion?
The signs did decrease lbs of trash in the park.
“Alpha Region”
α = .01, 1-tailed
t-critical =
t-obtained = -7.50

22. Comparing two scores from the same Individual (or unit of analysis)
Paired Samples T-Test
Comparing two scores from the same
Individual (or unit of analysis)

23. Gleaning Information from the Statement
Example:
Let’s say we are interested in whether a brand name (Nike vs. Adidas) affects willingness to pay for a sweatshirt. To explore this question, we take 9 people and have them indicate their WTP for a Nike sweatshirt and for an Adidas sweatshirt. The only difference between the sweatshirts is the brand name.
Non-Directional (Two-Tailed) Alternative Hypothesis; doesn’t say “is higher” or “is lower”; just says “affects”
Paired Samples T-test:
df = N-1 = 9-1 = 8
Paired Scores
From Same Person
Use alpha = .05
How do we get a t-critical value? 

24. Two-Tailed Hypothesis (H1: Condition 1 ≠ Condition 2)
Ho (Null Hypothesis):
People are willing to pay the same for Nike vs. Adidas: NikeWTP = AdidasWTP
H1 (Alternative Hypothesis):
People not willing to pay same for Nike vs. Adidas: NikeWTP ≠ AdidasWTP
In H1, b/c NikeWTP ≠ AdidasWTP
Alpha region is on both sides;
Half of .05 goes on each side
“Alpha Region”
α = .025
“Alpha Region”
α = .025
(negative)
t-critical
(positive)
t-critical

25. Critical T-Table
For paired samples t-test, df = N-1

26. Two-Tailed Hypothesis (H1: Condition 1 ≠ Condition 2)
Ho (Null Hypothesis):
People are willing to pay the same for Nike vs. Adidas: NikeWTP = AdidasWTP
H1 (Alternative Hypothesis):
People not willing to pay same for Nike vs. Adidas: NikeWTP ≠ AdidasWTP
In H1, b/c NikeWTP ≠ AdidasWTP
Alpha region is on both sides;
Half of .05 goes on each side
“Alpha Region”
α = .025
“Alpha Region”
α = .025
t-critical =
t-critical =

27. Defining Symbols in Paired T-test_
D = average difference score.
D = difference score (eg., time 1 vs. time 2; midterm vs. final; husband vs. wife)
N = # Paired Scores (not the # of numbers in front of you).
 = average difference score in the Null Hypothesis Population (most often = 0)
SSD = Sum of Squared Deviations for the Difference Scores = D2 – [(D)2/N]
tobt = the t statistic which is compared to tcrit with N-1 df

28. Paired Samples T-test Nike vs. Adidas Sweatshirt Example
Difference
Nike
Adidas D D2 25 27 -2 4 29 16 33 30 3 9 34 36 81 40 11
121 45 5 42 38
Sum 37
281
Mean D
4.11
First, Compute SSD
Then, Compute t

29. Decision and Conclusion?
“Alpha Region”
α = .025
t-critical =
t-critical =
t-obtained = 3.07
Decision?
Because t-obtained (3.07) < t-critical (2.306), reject Ho.
Conclusion?
People willing to pay more for Nike than for Adidas.
We know this, because the average difference score was positive.
(Nike – Adidas)

30. Independent Samples T-Test
Comparing means
of two conditions or groups

31. Gleaning Information from the Statement
Example:
Let’s say we are interested in how consumers respond to service failures, so we decide to run an experiment. We ask people to read about a hypothetical service failure scenario (e.g., delayed service at a restaurant). Then we randomly assign half of the subjects to the “apology” condition (we’ll call this Group 1), and the other half to a “control” condition (we’ll call this Group 2). Those in the apology condition read that the restaurant owner offered a sincere apology for having to wait so long. After this, we assess subjects’ self-reported anger (1 = not at all angry, 11 = fuming mad). We hypothesize that subjects will report less anger in the apology condition.
Independent Samples t-test:
df = N-2 = 20-2 = 18
Directional (One-Tailed) Alternative Hypothesis
Scores come from two
Independent groups
Use alpha = .05
How do we get a t-critical value? 

32. One-Tailed Hypothesis (H1: Condition 1 < Condition 2)
Ho (Null Hypothesis):
An apology does not decrease anger: ApologyAnger ≥ ControlAnger
H1 (Alternative Hypothesis):
Anger will be lower in the Apology Condition: ApologyAnger < ControlAnger
In H1, b/c Aplogy < No Apology
Alpha region is on left side
“Alpha Region”
α = .05, 1-tailed
(negative)
t-critical

33. Critical T-Table
For independent t-test, df = N-2

34. One-Tailed Hypothesis (H1: Condition 1 < Condition 2)
Ho (Null Hypothesis):
An apology does not decrease anger: ApologyAnger ≥ ControlAnger
H1 (Alternative Hypothesis):
Anger will be lower in the Apology Condition: ApologyAnger < ControlAnger
In H1, b/c Aplogy < No Apology
Alpha region is on left side
“Alpha Region”
α = .05, 1-tailed
t-critical =

35. Defining Symbols in Independent T-test
_ _
X1 and X2 = the means of X1 and X2 (our two conditions), respectively
SS1 and SS2 = Sum of Squared Deviations for X1 and X2
where…SS= X2 – [(X)2/N] for each group
n = the number of subjects in each conditions. n1 + n2 = N. In other words, n  N!
tobt = the t statistic which is compared tcrit with N-2 df.

36. Independent Samples T-test Apology vs. No Apology Example
NA2 5 10 25
100 3 7 9 49 2 4 6 36 1 11
121 16
SumA
SumNA
SumA2
SumNA2 32 60
142
434
Mean
3.2
6.0 SS
39.6 74
First, Compute SS for Each Condition
Then, Compute t

37. Decision and Conclusion?
t-critical =
“Alpha Region”
α = .05, 1-tailed
Decision?
Because t-obtained (-2.50) < t-critical (-1.734), reject Ho.
Conclusion?
People report less anger after an apology
t-obtained = -3.07

38. Analysis of Variance (ANOVA)
Comparing means
of three or more conditions or groups

39. The F-Ratio: A Ratio of Variances Between and Within Groups

40. Between Groups Variance (Numerator of F-ratio)
Within
Group 3
Variance
Within
Group 1
Variance
Within
Group 2
Mean = 9
Mean = 3
Mean = 5
Between Groups Variance
(Numerator of F-ratio)

41. F-Distribution Probability distribution
All values positive (variance ratio)
Positively skewed
Median = 1
Shape varies with degrees of freedom (within and between)
“Alpha Region”
α = .05 1

42. F-critical Table: If we have 3 conditions, N = 14, alpha =
F-critical Table: If we have 3 conditions, N = 14, alpha = .05; F-crit = 3.98
Alpha
Level
df numerator = K-1
df denominator = N-K

43. Null and Alternative Hypotheses
Let’s say a marketing researcher is interested in the impact of music on sales at a new clothing store targeted to tweens. She sets up a mock store in her university’s research lab, gives each subject $50 spending money, and then randomly assigns subjects to one of three conditions. One third of the subjects browse the mock store with no music. One third browse the store with soft music. And the final third browse the store with loud music. The sales figures are shown below. Assume the researcher decides to use an alpha level of .01.
Null Hypothesis (Ho): All of the means are equal (ucontrol = usoft music = uloud music)
Alternative (H1): At least two means are different
F-critical (based on alpha = .01; df-numerator = 2; df-denominator = 9): 8.02
Decision Rule: If Fobt ≥ Fcritical, then reject Ho. Otherwise, retain Ho

44. The Data: Sales as a Function of Music Condition

45. Subtract Group Mean from Each Score
Then Square and Add Up
This gives you the SS for that group

46. Do this for each of the three conditions

47. (See Statistics Notes Packet)
Environment
SS Control
SS Soft Music
SS Loud Music
No Music
Soft Music
Loud Music
(x-mean)
(x-mean)2 22 27 39 -4 16 1
1.0 26 23 36 -2
4.0 28 2 4 30 41 3 9
9.0
Mean
Sum 38 24 18
Grand Mean
30.333
SS-between
df-between
MS-between
SS-within 68
df-within
MS-within
7.556 F
23.471
Source SS df MS
Between
177.33
23.47
Within
68.00
7.56
Total
422.67 11
(See Statistics Notes Packet)
Summarize in a Source Table

48. ANOVA - Source Table Source SS df MS F Between 354.667 2 177.33 23.47
Within
68.00 9
7.56
Total
422.67 11

49. F-critical in our example = 8.02
K = 3
Alpha = .01

50. Decision Rule and Conclusion?
Reject Null Hypothesis
At least two means are different
“Alpha Region”
α = .01
F-critical
8.02
F-obtained
23.47

51. Correlation

52. Differences Between Correlation and Regression
Correlation (r)
assessing direction (+ or -) and degree (strong, medium, weak) of relationship between two variables
Linear Regression (slope, y-intercept)
assessing nature of relationship between an outcome variable and one or more predictors
making predictions for Y (cfc) based on X (impuss)

53. Reading Scatterplots Negative Correlation Zero Correlation PositiveX
(Predictor)
(Criterion) Y X
(Predictor)
(Criterion) Y X
(Predictor)
(Criterion) Y

55. Two Interpretations of Correlation Coefficient
Direction & Degree of Relationship Between Two Variables
Range from –1 to +1
Stronger correlations at the extremes
r = -1 (perfect negative relationship)
r = 0 (no relationship)
r = +1 (perfect positive relationship)
Variance Explained
r2, Ranges from 0 to + 1.0
What percent of the variance in Y is explained by X?
Model Comparison Approach

56. Problem Statement - A
Let’s say we survey 5 shoppers about their level of satisfaction with the service they received from a furniture store (X = satisfaction w/service) and their intention to return to the store in the future (Y = future intentions). Presumably, there should be a positive correlation between these variables.
Null Hypothesis (Ho): Satisfaction with service and future shopping intentions are not positively correlated
Alternative (H1): Satisfaction with service and future shopping intentions are positively correlated (this is a directional hypothesis)
r-critical (based on alpha = .05(one-tailed),df = N-2 = 3): r-critical = .8054
Decision Rule (in this example, because r is predicted to be positive): If robt ≥ rcritical, then reject Ho. Otherwise, retain Ho

57. r-critical Table If alpha = .05 (1-tailed), N = 5, df = 3, r-critical = .8054
Decision Rule (when r is predicted to be positive):
If robt ≥ rcritical, then reject Ho. Otherwise, retain Ho
Decision Rule (when r is predicted to be negative):
If robt ≤ rcritical, then reject Ho. Otherwise, retain Ho
Decision Rule (when H1 is non-directional):
If |robt| ≥ |rcritical|, then reject Ho. Otherwise, retain Ho

58. Data and Scatterplot
Data
Scatterplot

59. Raw Score Formula for Pearson’s r Correlation

60. Computing Pearson’s r (and variance explained)
Compute SSx and SSy
Then compute r
r2= (.313*.313) = .098
So, satisfaction explains 9.8% of variance in future intentions

61. Regression

62. Problem Statement - B
Let’s use the data we just worked with for correlation. Five shoppers were asked their satisfaction with the service they received and their intention to shop at the store in the future.
Regression would be used to make predictions for future shopping intentions (Y) based on people’s satisfaction with service (X).
For example, what would we predict if a shopper rated their satisfaction with service at a 3?
First need to compute regression equation, then use it to make a prediction

63. (change in Y for 1 unit change in X)
Slope and Y-Intercept X
(Predictor)
(Criterion) Y
Y-intercept (bo)
(value of Y, when X = 0)
Slope (b1)
(change in Y for 1 unit change in X)

64. Raw Score Formula for Slope and Y-Intercept

65. Computing Regression Equation
(Predictor)
(Criterion)
Satisfaction
Future
Customer
w/Service
Intentions X Y X2 Y2 XY
Hector 5 6 25 36 30
Marge 2 4 16 8
Fredrick 3 9 15
Susie
Gwen
Sum X
Sum Y
Sum X2
Sum Y2
Sum XY 22 20
106 90 91
[(Sum X)*(Sum Y)]/N 88
Numerator
SS X
9.2
b1 (the slope)
0.326
mean y
4.0
mean x
4.4
bo (y-intercept)
2.57
First compute slope
Then compute y-intercept
So, the regression equation is:

66. Data and Scatterplot
Data
Scatterplot

67. Using the Regression Equation to Make a Prediction
Let’s say a customer rates their satisfaction as a ‘3’ on our 7-point scale.
What is their predicted future intention of shopping at the store in the future?
So, a person who gives a ‘3’ on the satisfaction scale has a predicted future intention score of 3.544

68. Y-Predicted & Residuals
If Satisfaction (X) = 3
Predicted Intention (Y) = 3.54
Y predicted = 3.54
Residual =
(Y-Y predicted)
When r is strong,
residuals are small
X = 3

69. Chi-Square

70. One-Way vs. Two-Way Chi-Square
Chi-square is appropriate when our data are frequency (count) data
In “one-way chi-square”, we have one categorical variable (type of shoe) with several levels (Adidas, Asics, Nikes, Pumas) and we want to know whether the frequency of observations differs between the groups (or conditions, or levels)
In “two-way chi-square”, we have two categorical variables (Gender x Support for New Stadium) and we want to know if these two variables are related 1 2

71. One-way Chi-Square
Where…

72. Problem Statement
Let’s say we ask 100 people to pick their favorite brand of shoes among four types. The data are shown below. Clearly, the frequencies are not equal (25 in each). Here, 15 pick Adidas, 30 pick Asics, 45 pick Nike, and 10 pick Puma. The question is whether these frequencies are significantly different.
Null Hypothesis (Ho): Frequencies of people choosing different brands is equal
Alternative (H1): Not all the frequencies are equal (doesn’t mean they’re all different)
X2-critical (based on alpha = .05; df = K-1 = 4-1=3): X2-critical = (X2 critical always positive) In df, K stands for the number of groups. (see critical table next page)
Decision Rule is always as follows (b/c chi-square is always positive): If X2obt ≥ X2critical, then reject Ho. Otherwise, retain Ho

73. Chi-Square Critical Table (for one-way chi-square)
df = K-1, where K = # groups

74. Formula and Frequency Expected
Frequency Observed
(Actual Frequencies)
Frequency Expected
(Typically = Total N/K)
To compute chi-square, we need to know fe = expected frequency. Typically, we’ll just assume this represents an equal distribution across the conditions (Total N/K). So, we have a total of 100 people and 4 conditions (brands of shoe). Based on chance alone, an equal distribution across the conditions would mean 25 people would select each type of shoe. So, here we’ll assume fe = 25.

75. Computation Decision: Reject Ho, because
X2obt (30) ≥ X2critical (7.815).
Conclusion: People do not show an equal preference among the four brands of shoes.

76. Two-Way Chi-Square
Where…

77. Problem Statement
Let’s say we’re interested in whether males and females differ in their support for building a new football stadium. We survey 40 people (10 men, and 30 women) and we ask them a simple (categorical) yes/no question: Do you support building a new football stadium? Now we want to know if there is a relationship between gender (male/female) and support for the stadium (yes/no).
Null Hypothesis (Ho): There is no relationship between gender and support for football stadium
Alternative (H1): There is a relationship between gender and support for football stadium
X2-critical (based on alpha = .05; df = (Rows-1)*(Columns-1)=(2-1)*(2-1)=1: X2-critical = (X2 critical always positive) (see critical table next page)
Decision Rule is always as follows (b/c chi-square is always positive): If X2obt ≥ X2critical, then reject Ho. Otherwise, retain Ho
Of the 10 men surveyed, 8 supported it, and 2 didn’t
Of the 30 women surveyed, 5 supported it and 25 didn’t

78. Data Of the 10 men surveyed, 8 supported it, and 2 didn’t
Of the 30 women surveyed, 5 supported it and 25 didn’t

79. Chi-Square Critical Table (works for two-way chi-square)
df = (Row-1)*(Columns-1)

80. Formula and Frequency Expected
Frequency Observed
(Actual Frequencies)
Frequency Expected
(Row N*Column N)/Total N
To compute chi-square, we need to know fe = expected frequency. Typically, we’ll just assume this represents an equal distribution across the conditions (Row N*Column N)/Total N. The next slide illustrates the computation of frequency expected.

81. Computing Frequency Expected and Chi-Square
Cell 1
Cell 2
Cell 3
Cell 4
Decision: Reject Ho, because X2obt (13.7) ≥ X2critical (3.841).
Conclusion: Gender is related to support for football stadium (men > women)