Download Five types of statistical analysis General Procedure for Hypothesis

General Procedure for
Hypothesis Test
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
6 a) Compare with the level of significance, α
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 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.
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
• χ2 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.
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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 -- 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
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
• 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 µ=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
µ= mu=population mean
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.
H1
µ≠0
Two tail
test
or
H1 µ >0
One tail
test
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
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.
Accepting the null hypothesis that the pizzas are
equal, when they are really perceived to be different
by the customers of the entire population.
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3. Choose Level of Significance
The ratings of 40 randomly selected customers and
produce the following table and statistics
Significance Level selected is typically .05 or
.01
•I.e 5% or 1%
From the summary statistics, we see that the sample
mean is 2.10 and the sample standard deviation is 4.717
Summary Statistics
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 H0?
5. Determine the Probability-value (Critical Value)
4. Calculate the Test Statistic
X- 0
s/√
√n
t=
t − value =
∼ T(n-1)
2.10 − 0
4.717 / 40
= 2.816
6. Compare with the level of significance, α (.05)and
determine if the critical value falls in the rejection region
Do not Reject H0
• 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.
1-α
α
Reject H0
Reject H0
α/2
α/2
-2.023
0
7. Reject or do not reject H0
2.023 2.816
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.
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Comparing Means
8 Conclusion
• 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.
Hypothesis Test for Two
Independent Samples
• 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
Comparison of means:
Graphically
•Test for mean difference:
– Null Hypothesis
– Alternative
H0 µ1= µ2
H1 µ1≠ µ2
Under H0 µ1- µ2 = 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?
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 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
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
α
0
f
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Unequal variances: The problem
Tylenol vs Advil
• We would need to test if the difference is zero or not.
H0: πA - πT = 0;
H1: πA - πT ≠ 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
• d.f. = n - k
• where:
– When groups are small
– When population standard deviation is
unknown
n = n1 + n2
k = number of groups
• z-test
The degrees of freedom is (n1 + n2 –2)
– When groups are large
Test for Means Difference on
Paired Samples
Tylenol vs Advil
What is a paired sample?
α = 0.10 Critical value = 1.64
α -1
α/2
-∝
α/2
-1.64
0 0.66
Degrees of Freedom
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
–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.
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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
• 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
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).
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
There is no difference in importance between brand and
price
What does ANOVA tests?
•
The null hypothesis tests whether the mean of all
the independent samples is equal
H0 µ1= µ2 = µ3 …..= µn
H1 µ1≠ µ2 ≠ µ3 ….. ≠ µn
•
The alternative hypothesis specifies that all the
means are not equal
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.
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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)
Comparing Antacids
The independent variables are called factors, in this case
type of ad: non-comparative, explicit comparative, non-explicit
comparative
Brand
Attitude
Means
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.
Type of Ad
Non
Explicit
Non Explicit
Comparative Comparative Comparative
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
Variation within the treatment (e.g. within the noncomparative ad etc.)
Variation between the treatments (I.e. between the three
types of ads)
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 α
– One way ANOVA investigates:
– Main effects
• factor has an across-the-board effect
• e.g., type of ad
• Or age
df(c-1)/(N-c)
• or involvement
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– 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
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