1. Descriptive Statistics
Univariate Statistics
Chi Square
ANOVA

2. Descriptive Statistics
Summarization of a collection of data in a clear and understandable way
the most basic form of statistics
lays the foundation for all statistical knowledge

3. Inferential Statistics
Two main methods:
estimation
the sample statistic is used to estimate a population parameter
a confidence interval about the estimate is constructed.
hypothesis testing
a null hypothesis is put forward
Analysis of the data is then used to determine whether to reject it.
Inferential statistics generally require that sampling be random

4. Are you satisfied with your education at U of L?
TYPES OF DATA
Nominal : gender, type of customer (loyalty), flavor/color liked, etc.
Ordinal/Ranking :type of user, preferred brand, brand awareness, etc.
Interval: Attitudinal or satisfaction scales.
Are you satisfied with your education at U of L?
Dissatisfied Satisfied
Ratio: Income, price willing to pay, age, etc. 3 4 5 2 1

5. Proportion (percentage)
Type of
Measurement
Type of
descriptive analysis
Frequency table
Proportion (percentage)
Category proportions
(percentages)
Mode
Two
categories
Nominal
More than
two categories

6. Type of
Measurement
Type of
descriptive analysis
Ordinal
Rank order
Median
Interval
Arithmetic mean
Ratio
means

7. Frequency Tables
The arrangement of statistical data in a row-and-column format that exhibits the count of responses or observations for each category assigned to a variable
How many of certain brand users can be called loyal?
What percentage of the market are heavy users and light users?
How many consumers are aware of a new product?
What brand is the “Top of Mind” of the market?

8. WebSurveyor Bar Chart

9. Bar Graph

11. Measures of Central Location or Tendency
Mean: average value
Mode: the most frequent category
Median: the middle observation of the data

12. The Mean (average value)
sum of all the scores divided by the number of scores.
a good measure of central tendency for roughly symmetric distributions
can be misleading in skewed distributions since it can be greatly influenced by extreme scores in which case other statistics such as the median may be more informative
formula m = SX/N (population)
X = xi/n (sample)
where m/X is the population/sample mean
and N/n is the number of scores.

13. the most frequent category
Mode
the most frequent category
users 25%
non-users 75%
Advantages:
meaning is obvious
the only measure of central tendency that can be used with nominal data.
Disadvantages
many distributions have more than one mode, i.e. are "multimodal
greatly subject to sample fluctuations
therefore not recommended to be used as the only measure of central tendency.

14. number times per week consumers use mouthwash
Median
the middle observation of the data
number times per week consumers use mouthwash
Frequency distribution of
Mouthwash use per week
Heavy user
Light user
Mode
Median
Mean

15. Normal Distributions Curve is basically bell shaped from -  to 
symmetric with scores concentrated in the middle (i.e. on the mean) than in the tails.
Mean, medium and mode coincide
They differ in how spread out they are.
The area under each curve is 1.
The height of a normal distribution can be specified mathematically in terms of two parameters: the mean (m) and the standard deviation (s).
Normal Distributions

16. Normal Distribution 
-    a b
Area between a and b = P(a=X =b)

17. Normal Distributions with different Mean
-   1 2

18. Skewed Distributions
Occur when one tail of the distribution is longer than the other.
Positive Skew Distributions
have a long tail in the positive direction.
sometimes called "skewed to the right"
more common than distributions with negative skews
E.g. distribution of income. Most people make under $40,000 a year, but some make quite a bit more with a small number making many millions of dollars per year
The positive tail therefore extends out quite a long way
Negative Skew Distributions
have a long tail in the negative direction.
called "skewed to the left."
negative tail stops at zero

19. Measures of Dispersion or Variability
Minimum, Maximum, and Range
Variance
Standard Deviation

20. Variance 2 = (x- xi)2/n ¯
The difference between an observed value and the mean is called the deviation from the mean
The variance is the mean squared deviation from the mean
i.e. you subtract each value from the mean, square each result and then take the average.
Because it is squared it can never be negative
2 = (x- xi)2/n

21. Standard Deviation S =  (x- xi)2/n ¯
The standard deviation is the square root of the variance
Thus the standard deviation is expressed in the same units as the variables
Helps us to understand how clustered or spread the distribution is around the mean value.
S =  (x- xi)2/n

22. Measures of Dispersion
Suppose we are testing the new flavor of a fruit punch
Dislike Like Data x
X= 4
2= 1
S = 1
2 = (x- xi)2/n
S =  (x- xi)2/n

23. Measures of Dispersion
Dislike Like Data x
X = 4.6
2=0.26
S = 0.52
2 = (x- xi)2/n
S =  (x- xi)2/n

24. Measures of Dispersion
Dislike Like Data x
X= 3
2=4
S = 2
2 = (x- xi)2/n
S =  (x- xi)2/n

25. Normal Distributions with different SD
-   1 2 3

26. How does the Normal Distribution help to make decisions?
Suppose you are about to introduce new “Guacamole Doritos” to the market.
Need to determine:
Desired flavor intensity (How hot it should be)
Package size offered
Introduction price

27. What do you do in order to answer your questions?
ASK THE CONSUMER
How?
TAKE A SAMPLE
How can you be sure that what you conclude on the sample would be true for the whole population?

28. Suppose you conducted a research study
Took a random sample of n=100 subjects
They tasted the new "Guacamole Doritos”
They rated the flavor of the chip on the following scale:
Too Perfect Too
Mild Flavor Hot 1 2 3 4 5 6 7

29. Suppose you take a series of random samples of n=100 subjects:
Results show : x1 = 2.3 and S1= 1.5
Can you conclude that on average the target population thought the flavor was mild?
Suppose you take a series of random samples of n=100 subjects:
x2 = 3.7 and S2 = 2
x3 = 4.3 and S3 = 0.5
x4 = 2.8 and S4 = .97 .
x50 = 3.7 and S50 = 2

30. The Sampling Distribution
The means of all the samples will have their own distribution called the sampling distribution of the means
It is a normal distribution
The sampling distribution of a proportions is a binomial that approximates a normal distribution in large samples (30+)
The mean of the sampling distribution of the mean =
It equals the population parameter X
= (ΣXi)/n

31. Sampling Distribution
The standard deviation of the sampling distribution is called the sampling error of the mean (or proportion).
The formula for the proportion is
Often the population standard deviation  is unknown and has to be estimated from the sample
=  =  / n X
p= π(1-π)/n
S =   Σ(Xi-X)/n-1

32. Population distribution of the Doritos’ flavor (X) X 
Sample distribution of the x Doritos’ flavor x 1 2 3 4 5 6 7

33. The Central Limit Theorem
What relationship does the Population Distribution have to the Sample Distribution?
The Central Limit Theorem
Let x1, x2….. xn denote a random sample selected from a population having mean  and variance 2. Let X denote the sample mean. If n is large, the X has approximately a Normal Distribution with mean  and variance 2/n.
The Central Limit Theorem does not mean that the sample mean = population mean.
It means that you can attach a probability to that value and decide.

34. Interpretation
The process of making pertinent inferences and drawing conclusions
concerning the meaning and implications of a research investigation
You do not need to know the population distribution in order to take decisions.
In order to draw conclusions n must be “big enough.”
How big?, it DEPENDS

35. Univariate Statistics
Test of statistical significance
Hypothesis testing one variable at a time
Hypothesis
Unproven proposition
Supposition that tentatively explains certain facts or phenomena
Assumption about nature of the world

36. What is a Hypothesis Test?
It is used when we want to make inferences about a population.
Generally we have a particular theory, or hypothesis, about certain events like:
The average age of our regular customers
The average money spent per week on fast food restaurants
The percentage of unsatisfied customers of our store.

37. Basic Concepts
The hypothesis the researcher wants to test is called the alternative hypothesis H1.
The opposite of the alternative hypothesis us the null hypothesis H0 (the status quo)(no difference between the sample and the population, or between samples).
The objective is to DISPROVE the null hypothesis.
The Significance Level is the Critical probability of choosing between the null hypothesis and the alternative hypothesis

38. General Procedure for Hypothesis Test
Formulate H1 and H0
Select appropriate test
Choose level of significance
Calculate the test statistic
Determine the probability associated with the statistic.
Determine the critical value of the test statistic.

39. General Procedure for Hypothesis Test
a) Compare with the level of significance, 
b) Determine if the critical value falls in the rejection region.
Reject or do not reject H0
Draw a conclusion

40. 1. Formulate H1and H0 Null hypothesis represents status quo.
Alternative hypothesis represents the desired result.
Example: One-Sample t-test
The manager of Pepperoni Pizza has developed a new baking method with lower costs and wishes to test it with some customers. He asked customers to rate the difference between both pizzas on a scale from -10 (old style) to +10 (new style)

41. 1. Formulate H1and H0
As a manager you would like to observe a difference between both pizzas
Since the new baking method is cheaper, you would like the preference to be for it.
Null Hypothesis H0 =0
Alternative H1 0 or H1  >0
Two tail
test
One tail
test

42. 2. Select Appropriate Test
The selection of a proper Test depends on:
Scale of the data
categorical
interval
the statistic you seek to compare
proportions
means
the sampling distribution of such statistic
Normal Distribution
T Distribution
2 Distribution
Number of variables
Univariate
Bivariate
Multivariate
Type of question to be answered

43. 3. Choose Level of Significance
Whenever we draw inferences about a population, there is a risk that an incorrect conclusion will be reached
The significance level states the probability of incorrectly rejecting H0. This error is commonly known as Type I error, and we denote the significance level as .
Significance Level selected is typically .05 or .01
In our example the Type I error would be rejecting the null hypothesis that the pizzas are equal, when they really are perceived equal by the customers of the entire population.

44. 3. Choose Level of Significance
We commit Type error II when we incorrectly accept a null hypothesis when it is false. The probability of committing Type error II is denoted by .
In our example, the Type II error would be not rejecting the null hypothesis that the pizzas are equal, when they are perceived to be different by the customers of the entire population.

45. Type I and Type II Errors
Accept null
Reject null
Null is true
Correct-
no error
Type I
error
Null is false
Type II
error
Correct-
no error

46. Which is worse?
Both are serious, but traditionally Type I error has been considered more serious, that’s why the objective of hypothesis testing is to reject H0 only when there is enough evidence that supports it.
Therefore, we choose  to be as small as possible without compromising .
Increasing the sample size for a given α will decrease β

47. 4. Calculate the Test Statistic
Example
If we are testing whether the consumer perceives a difference between the pizzas
We would need a statistic for the mean
We know that X N(, 2/n)
Perceived difference between the pizzas (X) for a given population of size N with mean  and variance estimated from the sample 2/n

48. If we suppose Ho true, then =0 and
X N(0, 2/n)
If we standardized X, we would get
Since we do not know the population value of , we would have to estimate it with the SD of the sample.
X- 0
/n
 N(0, 1)
Z =

49. But…..X no longer has a Normal distribution, now X has a T distribution with n-1 degrees of freedom.
s/n
t =
 T(n-1) - 

50. X= perceived difference between the pizzas
 = real population mean, that equals zero if H0 is true.
x = 3.5, observed sample mean
SD= 2.1, observed sample standard deviation
n=40
=.01
2.1/40
t =
 T (39)
T=.005(39)=2.074
t =10.54

51. 5. Determine the Probability-value (Critical Value)
The p-value is the probability of seeing a random sample at least as extreme as the sample observed given that the null hypothesis is true.

52. For example:
In reference to the null hypothesis, if H0 hypothesized that there would be no difference between the pizzas, a sample mean value of 2.5 would be high, but even more extreme would be a value of 3.5.
If the p-value is 0.03, it would mean that if we take 100 samples we would observe only three samples with an extreme value of 3.5.
It would be concluded that we have enough evidence to reject H0.

53. 7 & 8. Reject or do not reject H0 and draw a conclusion
6. Compare with the level of significance,  and determine if the critical value falls in the rejection region
10.54
2.074
-2.074
/2
1-
Reject H0
Do not Reject H0
7 & 8. Reject or do not reject H0 and draw a conclusion
Since the statistic t falls in the rejection area we reject Ho
and conclude that the perceived difference between the
pizzas is different from zero.

54. Hypothesis Test for Two Independent Samples
Test for mean difference:
Null Hypothesis H0 1= 2
Alternative H1 1 2
Under H0 1- 2 = 0. So, the test concludes whether there is a difference between the parameters or not.
e.g. high income consumers spend more on sports activities than low income consumers
The proportion of brand-loyal users in segment 1 is different from that in segment 2
Can be used for examining differences between means and proportions

55. Test for Means Difference
Suppose X measures the preference for a mouthwash flavor (cool mint) on a scale from 1-dislike to 5-like
We want to know if the flavor preference is different between the type of user (heavy or light) H0: H= L
H1: H L

56. It would be the same to test if the difference is zero or not
It would be the same to test if the difference is zero or not. H0: H-L= 0
So, if we reject H0 we can conclude that the means of the independent samples are different.
t = S
XH-XL
(XH-XL)- (H-L)
 T(nH-nL -2)

57. Test for Variance Difference
Tests if the variance ratio is equal to 1
H0: H/ L= 1
H1: H/ L  1
So, if we reject H0 we can conclude that the variances of the independent samples are different.

58. Test for Variance Difference
The test statistic has an F Distribution:
f = SH 2 SL
 F (nH-1)(nL -1)  F f

59. SPSS Output

60. Test for Proportion Difference
Suppose X measures the number of individuals that preferred the mouthwash flavor cool mint.
We want to know if the proportion of people who preferred cool mint is different between heavy and light users.
H0: H= L
H1: H L

61. number of favorable cases
It would be the same to test if the difference is zero or not. H0: H- L= 0
The sample proportion would be equal to
number of favorable cases
total number of cases
 N(p, pq/n) where q=1-p
How is the difference between two proportions distributed?
p =

62. Therefore, under H0: H- L= 0, the test statistic is as follows:
The difference of two independent sample proportions is distributed as:
1- 2  N(p1-p2, p1q1/n1+ p2q2/n2)
Therefore, under H0: H- L= 0, the test statistic is as follows:
z =
p1-p2-0
 N
p1q1/n1+ p2q2/n2

63. Example
Suppose we are the brand manager for Tylenol, and a recent TV ad tells the consumers that Advil is more effective (quicker) treating headaches than Tylenol.
An independent random sample of 400 people with a headache is given Advil, and 260 people report they feel better within an hour.
Another independent sample of 400 people is taken and 252 people that took Tylenol reported feeling better. Is the TV ad correct?

64. Tylenol vs Advil
We would need to test if the difference is zero or not.
H0: A - T = 0;
H1: A - T  0
pA = 260/400= 0.65
pT = 252/400= 0.63
z =
= 0.66
(.65)(.35)/400+ (.63)(.37)/400

65. Tylenol vs Advil  = 0.10 N(0,1) = 1.64  -1 /2 /2 - -1.64 1.64
0.66 

66. Test for Means Difference on Paired Samples
What is a paired sample?
When observations from two populations occur in pairs or are related then they are not independent
When you want to measure brand recall before and after an ad campaign.
When employing a consumer panel, and comparing whether they increased their consumption of a certain product from one period to another.

67. Test for Means Difference on Paired Samples
Since both samples are not independent we employ the differences as a random sample
di=x1i-x2i i=1,2,…,n
Now we can test this variable to compare it to against any other value.

68. SPSS Output ASSIGNMENT 3
Use the mouthwash file and test whether there is difference between the likelihood of purchase ( WOULD BUY) depending on whether the consumers:
Use Listerine or Scope
Gender
Create a new variable if price willing to ay is less or equal to 4 then =0, if higher than 4 =1. Test whether there is a difference between those groups.

69. Cross Tabulation
and Chi Square Test
for Independence

70. Cross-tabulation
Helps answer questions about whether two or more variables of interest are linked:
Is the type of mouthwash user (heavy or light) related to gender?
Is the preference for a certain flavor (cherry or lemon) related to the geographic region (north, south, east, west)?
Is income level associated with gender?
Cross-tabulation determines association not causality.

71. Dependent and Independent Variables
The variable being studied is called the dependent variable or response variable.
A variable that influences the dependent variable is called independent variable.

72. Cross-tabulation
Cross-tabulation of two or more variables is possible if the variables are discrete:
The frequency of one variable is subdivided by the other variable categories.
Generally a cross-tabulation table has:
Row percentages
Column percentages
Total percentages
Which one is better?
DEPENDS on which variable is considered as independent. .

73. Cross tabulation

74. Contingency Table
A contingency table shows the conjoint distribution of two discrete variables
This distribution represents the probability of observing a case in each cell
Probability is calculated as:
Observed cases
Total cases P=

75. Chi-square Test for Independence
The Chi-square test for independence determines whether two variables are associated or not.
H0: Two variables are independent
H1: Two variables are not independent
Chi-square test results are unstable if cell count is lower than 5

76. Chi-Square Test Estimated cell Frequency
Ri = total observed frequency in the ith row
Cj = total observed frequency in the jth column
n = sample size
Eij = estimated cell frequency
Chi-Square statistic
x² = chi-square statistics
Oi = observed frequency in the ith cell
Ei = expected frequency on the ith cell
Degrees of Freedom
d.f.=(R-1)(C-1)

77. Awareness of Tire Manufacturer’s Brand
Men Women Total
Aware 50/ /
Unaware / /

78. Chi-Square Test: Differences Among Groups Example
X2 with 1 d.f. at .05 critical value = 3.84

79. Chi-square Test for Independence
Under H0, the joint distribution is approximately distributed by the Chi-square distribution (2).
Chi-square
Reject H0 
3.84 2
22.16

80. Analysis of Variance
(ANOVA)

81. What is an ANOVA? One-way ANOVA stands for Analysis of Variance
Purpose:
Extends the test for mean difference between two independent samples to multiple samples.
Employed to analyze the effects of manipulations (independent variables) on a random variable (dependent).

82. Definitions
Dependent variable: the variable we are trying to explain, also known as response variable (Y).
Independent variable: also known as explanatory variables (X).
Therefore, we would like to study whether the independent variable has an effect on the variability of the dependent variable

83. One or More Independent
Continuous Dependent variable
One Independent
Variable
One or More Independent
Variable
Categorical and
Continuous
Categorical
Continuous
Binary
ANOVA
t Test
ANCOVA
Regression
One Factor
More than
one Factor
One-Way
ANOVA
N-Way
ANOVA

84. What does ANOVA tests?
H0 1= 2 = 3 …..= n
H1 1 2  3 …..  n
The null hypothesis tests whether the mean of all the independent samples is equal
The alternative hypothesis specifies that all the means are not equal

85. Comparing Antacids Non comparative ad: Explicit Comparative ad:
Acid-off provides fast relief
Explicit Comparative ad:
Acid-off provides faster relief than Tums
Non explicit comparative ad
Acid-off provides the fastest relief

86. Comparing Antacids Brand Attitude Means Type of Ad Non Comparative
Explicit
Comparative
Non Explicit
Comparative

87. Comparing Antacids Brand Attitude Means Type of Ad Non Comparative
Explicit
Comparative
Non Explicit
Comparative

88. Decomposition of the Total Variation
Independent Variable X
Categories Total Sample
X1 X2 X3 …. Xc
Y1 Y1 Y1 …. Y1 Y1
Y2 Y2 Y2 …. Y2 Y2
Yn Yn Yn …. Yn Yn
Y1 Y2 Y3 Yc Y
Within
Category
Variation
SSwithin
Total
Variation
SSy
Category
Mean
Grand
Mean
Between Category Variation SSbetween

89. Decomposition of the Total Variation
SSy = (Yi- Y)2
SSy =SSbetween + SSwithin
SSy =SSx + SSerror
Between variation:
SSx=  n(Yj- Y)2
Within variation:
SSerror=  (Yij- Yj)2 c j c n j i

90. Measurement of the Effects
We would like to know how strong are the effects of the independent variable (X) on the dependent variable (Y).
SSy =SSx + SSerror
SSx =SSy – SSerror
 SSy – SSerror SSx = =
SSy
SSy

91. ANOVA Test
Under H0 1= 2 = 3 …..= n, SSx and SSy have the same source of variability since the means are equal between categories.
Therefore the estimate of the population variance of Y can be based on either sum of squares:
Sy=
SSx
SSerror
(c-1)
(N-c) =
MSx
MSerror

92. ANOVA Test The null hypothesis would be tested with the F distribution
MSx
f =
MSerror
F distribution
Reject H0 
f(c-1)(N-c)