Download hypothesis testing

Five types of statistical analysis
Descriptive
What are the characteristics of the respondents?
Inferential
What are the characteristics of the population?
Differences
Are two or more groups the same or different?
Associative
Are two or more variables related in a systematic way?
Predictive
Can we predict one variable if we know one or
more other variables?
General Procedure for
Hypothesis Test
1. Formulate H0 (null hypothesis) and H1
(alternative hypothesis)
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.
General Procedure for
Hypothesis Test
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 H0
8 Draw a conclusion
1. Formulate H1and H0
• The hypothesis the researcher wants to test is called
the alternative hypothesis H1.
• The opposite of the alternative hypothesis is 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
2. Select Appropriate Test
• 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.
The 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.
New pizza
Old pizza
-10
0
+10
1. Formulate H1and H0
One-Tailed Versus Two-Tailed Tests
• 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. Formulate H1and H0
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. Formulate H1and H0
• 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 –H0 =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
H1 0
Two tail
test
or
H1  >0
One tail
test
2. Select Appropriate 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, (and saying that they are different or the new
style is better) 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.
3. Choose Level of Significance
Significance Level selected is typically .05 or .01
• i.e 5% or 1%
The ratings of 40 randomly selected customers
produces 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 suggests a slight preference for the new
pizza, (alternative hypothesis) but there is a fair degree of variation.
What we don’t know is whether this preference is significant
4. Calculate the Test Statistic
t=
t  value 
X- 0
s/n
 T(n-1)
2.10  0
4.717 / 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 H0
1-
Reject H0
Reject H0
/2
/2
-2.023
0
2.023 2.816
7. Reject or do not reject H0
Since the statistic falls in the rejection area we reject Ho
and conclude that the perceived difference between the
pizzas is significantly different from zero.
8 Conclusion
• the sample evidence is fairly convincing that
customers, on average, prefer the new-style pizza.
• Should the manager switch to the new-style pizza on
the basis of these sample results?
• Depends. There is no indication that the new-style pizza
costs any more to make than the old-style pizza.
Therefore, unless there are reasons for not switching (for
example, costs) then we recommend the switch.
Comparing Means
• 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
Hypothesis Test for Two
Independent Samples
•Test for mean difference:
– Null Hypothesis
– Alternative
H0 1= 2
H1 1 2
Under H0 1- 2 = 0. So, the test concludes whether there is a
difference between the means or not.
Comparison of means:
Graphically
Are the means equal? Or are the differences simply
due to chance?
2. Select Appropriate Test
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 (eg males)
is more than the proportion in segment II (e.g. females)
The proportion of households with Internet in Canada
exceeds that in USA
• Can be used for examining differences between means and
proportions
2. Select Appropriate Test
The two populations are sampled and the means and
variances computed based on the samples of sizes n1 and n2
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

0
f
Unequal variances: The problem
Tylenol vs Advil
• We would need to test if the difference is zero or not.
H0: A - T = 0;
H 1:  A -  T  0
t =
pA = 260/400= 0.65
pT = 252/400= 0.63
mean 1 – mean 2
Variability of random
means
.65 - .63
= 0.66
z=
(.65)(.35)/400+ (.63)(.37)/400
For large samples the t-distribution approaches the normal
distribution and so the t-test and the z-test are equivalent.
Differences Between Groups
when Comparing Means
• Ratio scaled dependent variables
• t-test
– When groups are small
– When population standard deviation is
unknown
• z-test
– When groups are large
Degrees of Freedom
• d.f. = n - k
• where:
n = n1 + n2
k = number of groups
The degrees of freedom is (n1 + n2 –2)
Tylenol vs Advil
 = 0.10 Critical value = 1.64
 -1
/2
-
/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
Test for Means Difference on Paired Samples
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.
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
H0
There is no difference in importance between brand and
price
H1 One tailed
Price is more important than brand
H1 Two Tailed
There is a difference in importance between
brand and price
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).
What does ANOVA test?
•
The null hypothesis tests whether the mean of all
the independent samples is equal
H0 1= 2 = 3 …..= n
H1 1 2  3 …..  n
•
The alternative hypothesis specifies that all the
means are not equal
Definitions
• 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
Comparing Antacids
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.
Comparing Antacids
Brand
Attitude
Means
Type of Ad
Non
Explicit
Non Explicit
Comparative Comparative 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.
Variance  between  groups
F 
Variance  within  groups
Degrees of Freedom
The F statistic 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
Decomposition of the Total
Variation
Within
Category
Variation
SSwithin
Category
Mean
Independent Variable X
Categories
Total Sample
X1
X2
X3
Xc
….
Y1
Y1
Y1
Y1
Y1
….
Y2
Y2
Y2
Y2
Y2
….
Total
Variation
SSy
Yn
Yn
Yn
Yn
Yn
….
Y1
Y2
Y3
Yc
Y
Between Category Variation SSbetween
Grand
Mean
ANOVA Test
The null hypothesis would be tested with
the F distribution
F distribution
Reject H0 
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