Descriptive
Inferential
Differences
Associative
Predictive
What are the characteristics of the respondents?
What are the characteristics of the population?
Are two or more groups the same or different?
Are two or more variables related in a systematic way?
Can we predict one variable if we know one or more other variables?

1. Formulate H
0
(null hypothesis) and H
(alternative hypothesis)
1
2. Select appropriate test
3. Choose level of significance
4. Calculate the test statistic (SPSS)
5. Determine the probability associated with the statistic.
• Determine the critical value of the test statistic.

6 a) Compare with the level of significance,
 b) Determine if the critical value falls in the rejection region . (check tables)
7 Reject or do not reject H
0
8 Draw a conclusion

1
0
• The hypothesis the researcher wants to test is called the alternative hypothesis H
1
.
• The opposite of the alternative hypothesis is the null hypothesis H
0
(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

•
The selection of a proper Test depends on:
– Scale of the data
• nominal
• interval
– the statistic you seek to compare
•
Proportions (percentages)
• 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

Testing for Differences Between Mean of the
Sample and Mean of the Population
• The manager of
Pepperoni Pizza Restaurant has recently begun experimenting with a new method of baking its pepperoni pizzas.
• He believes that the new method produces a better-tasting pizza, but he would like to base a decision on whether to switch from the old method to the new method on customer reactions.
• Therefore he performs an experiment.

• For 40 randomly selected customers who order a pepperoni pizza for home delivery, he includes both an old style and a free new style pizza in the order.
• All he asks is that these customers rate the difference between pizzas on a -10 to +10 scale, where -10 means they strongly favor the old style, +10 means they strongly favor the new style, and 0 means they are indifferent between the two styles.
Old pizza
New pizza
-10 0
+10

1
0
• The form of the alternative hypothesis can be either a one-tailed or two-tailed , depending on what you are trying to prove.
• A one-tailed hypothesis is one where the only sample results which can lead to rejection of the null hypothesis are those in a particular direction, namely, those where the sample mean rating is positive.
• A two-tailed test is one where results in either of two directions can lead to rejection of the null hypothesis.

1
0
One-Tailed Versus Two-Tailed Tests -- continued
• Once the hypotheses are set up, it is easy to detect whether the test is one-tailed or two-tailed.
• One tailed alternatives are phrased in terms of “>” or
“<“ whereas two tailed alternatives are phrased in terms of “  ”
• The real question is whether to set up hypotheses for a particular problem as one-tailed or two-tailed.
• There is no statistical answer to this question. It depends entirely on what we are trying to prove.

1
0
• As the manager you would like to observe a difference between both pizzas
• If the new baking method is cheaper, you would like the preference to be for it.
– Null Hypothesis –
H
0

=0 (there is no difference between the old style and the new style pizzas) (The difference between the mean of the sample and the mean of the population is zero )
– Alternative

= mu=population mean
H
1

0 or H
Two tail
1

>0
One tail test test

What we want to test is whether consumers prefer the new style pizza to the old style. We assume that there is no difference (i.e. the mean of the population is zero) and want to know whether our observed result is significantly (I.e. statistically) different.
The one-sample t test is used to test whether the mean of the sample is equal to a hypothesized value of the population from which the sample is drawn.

Type I Error
Rejecting the null hypothesis that the pizzas are equal, when they really are perceived equal by the customers of the entire population.
Type II error
Accepting the null hypothesis that the pizzas are equal, when they are really perceived to be different by the customers of the entire population.

Significance Level selected is typically .05 or
.01
•
I.e 5% or 1%

The ratings of 40 randomly selected customers and produce the following table and statistics
From the summary statistics, we see that the sample mean is 2.10 and the sample standard deviation is 4.717

The positive sample mean provides some evidence in favor of the alternative hypothesis, but given the rather large standard deviation does it provide enough evidence to reject H
0
?

t =
X- 0 s/
 n

T(n-1)


2 .
10
4 .
717 /

0
40

2 .
816

5. Determine the Probability-value (Critical Value)
• We use the right tail because the alternative is one-tailed of the “greater than” variety
• The probability beyond this value in the right tail of the t distribution with n-1 = 39 degrees of freedom is approximately 0.004
• The probability, 0.004, is the p -value for the test. It indicates that these sample results would be very unlikely if the null hypothesis is true.

6. Compare with the level of significance,

(.05)and determine if the critical value falls in the rejection region
Do not Reject H
0
1-

Reject H
0

/2
Reject H
0

/2
-2.023
0
7. Reject or do not reject H
0
2.023
2.816
Since the statistic falls in the rejection area we reject H o and conclude that the perceived difference between the pizzas is significantly different from zero.

8
• Should the manager switch to the new-style pizza on the basis of these sample results?
• We would probably recommend “yes”. There is no indication that the new-style pizza costs any more to make than the old-style pizza, and the sample evidence is fairly convincing that customers, on average, will prefer the new-style pizza.
• Therefore, unless there are reasons for not switching
(for example, costs) then we recommend the switch.

• Suppose you are the brand manager for
Tylenol , and a recent TV ad tells the consumers that Advil is more effective (quicker) at 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? Or, in other words, is there a difference between the means of the two samples

•
Test for mean difference:
–
Null Hypothesis
– Alternative
H
0

1
=

2
H
1

1
 
2
Under H
0

1
-

2
= 0. So, the test concludes whether there is a difference between the parameters or not.

Are the means equal? Or are the differences simply due to chance?

In this example we have two independent samples
Other examples
• populations of users and non-users of a brand differ in perceptions of the brand
• high income consumers spend more on the product than low income consumers
• The proportion of brand-loyal users in Segment 1 is more than the proportion in segment II
• The proportion of households with Internet in Canada exceeds that in USA
•
Can be used for examining differences between means and proportions


The two populations are sampled and the means and variances computed based on the samples of sizes n
1 and n
2

If both populations are found to have the same variance then a t-statistic is calculated.

The comparison of means of independent samples assumes that the variances are equal.

If the variances are not known an F-test is conducted to test the equality of the variances of the two populations.
F
 f
0

• We would need to test if the difference is zero or not .
H
0
:

A
-

T
H
1
:

A
-

T
= 0;

0 p
A
= 260/400= 0.65
p
T
= 252/400= 0.63
t = mean 1 – mean 2
Variability of random means
.65 - .63
z =

(.65)(.35) /400+ (.63)(.37) /400
= 0.66
For large samples the t-distribution approaches the normal distribution and so the t-test and the z-test are equivalent.

• Ratio scaled dependent variables
• t-test
– When groups are small
– When population standard deviation is unknown
• z-test
– When groups are large

• d.f. = n - k
• where: n = n
1 + n
2 k = number of groups
The degrees of freedom is (n
1
+ n
2
–2)


/2

= 0.10 Critical value = 1.64

-1

/2
-

-1.64
0 0.66
1.64

Since 0.66 is less than the critical value of 1.64 we accept the null hypothesis: there is no difference between Advil and Tylenol users

What is a paired sample?
– When two sets of observations relate to the same respondents

When you want to measure brand recall before and after an ad campaign.

Shoppers consider brand name to be more important than price

Households spend more money on pizza than on hamburgers
 The proportion of a bank’s customers who have a checking account exceeds the proportion who have a savings account
– Since it is the same population that is being sampled the observations are not independent.

• Since both samples are not independent we employ the differences as a random sample d i
=x
1i
-x
2i i=1,2,…,n
• The appropriate test is a paired-t-test

Example
Q1. When purchasing golf clubs rate the importance 1-5 of price
Q2. When purchasing golf clubs rate the importance 1-5 of brand
H
0
There is no difference in importance between brand and price
H
1
One tailed Price is more important than brand
H
1
Two Tailed
There is a difference in importance between brand and price

• 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).

• The null hypothesis tests whether the mean of all the independent samples is equal
H
0

1
=

2
H
1

1
 
2
=

3
 
3
…..=  n
…..   n
• The alternative hypothesis specifies that all the means are not equal

• Dependent variable: the variable we are trying to explain, also known as response variable (Y).
• Independent variable: also known as explanatory variables or Factors (X).
• Research normally involves determining whether the independent variable has an effect on the variability of the dependent variable

The maker of Acid-off, an antacid stomach remedy wants to know which type of ad results in the most positive brand attitude among consumers.
• Non 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
Three groups of people are exposed to one type of ad and asked to rate their attitude towards the ad.

Brand
Attitude
Means
Type of Ad
Non
Comparative
Explicit
Comparative
Non Explicit
Comparative


The dependent variable (denoted by Y) is called the response variable and in this case it is brand attitude (I.e. we want to know what effect ad type has on attitude toward the brand)

The independent variables are called factors, in this case type of ad: non-comparative, explicit comparative, non-explicit comparative

The different levels of the factor are called treatments. In this case the treatments are the different ratings for each of the three types of ads.

There will be two sources of variation.

Variation within the treatment (e.g. within the noncomparative ad etc.)

Variation between the treatments (I.e. between the three types of ads)

The whole idea behind the analysis of variance is to compare the ratio of between group variance to within group variance.
If the variance caused by the interaction between the samples is much larger when compared to the variance that appears within each group, then it is because the means are different.
F

Variance
Variance

 between within

 groups groups
Degrees of Freedom
The F statistics has DF for both numerator (between group) and denominator (within group)
DF between group = (c-1) where c=number of groups
DF within group = (N-c) where N is sample size

Within
Category
Variation
SS within
Independent Variable X
Categories Total Sample
X
1
Y
1
Y
2
X
2
Y
1
Y
2
X
3
Y
1
Y
2
….
….
….
X c
Y
1
Y
2
Y
Y
1
2
Total
Variation
SS y
Y n
Y n
Y n
….
Y n
Y n
Category
Mean
Y
1
Y
2
Y
3
Y c
Y
Between Category Variation SS between
Grand
Mean

The null hypothesis would be tested with the F distribution
F distribution
Reject H
0
 df
(c-1)/(N-c)

– One way ANOVA investigates:
– Main effects
• factor has an across-the-board effect
• e.g., type of ad
• Or age
• or involvement

– A
TWO-WAY ANOVA investigates:
–
INTERACTIONS
• effect of one factor depends on another factor
• e.g., larger advertising effects for those with no experience
• importance of price depends on income level and involvement with the product