Five types of statistical analysis

1
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?
2
General Procedure for Hypothesis Test

• Formulate H0 (null hypothesis) and H1
  (alternative hypothesis)
• Select appropriate test
• Choose level of significance
• Calculate the test statistic (SPSS)
• Determine the probability associated with the
  statistic.
• Determine the critical value of the test
  statistic.

3
General Procedure for Hypothesis Test

• a) Compare with the level of significance, ?
• b) Determine if the critical value falls in
  the rejection region. (check tables)
• Reject or do not reject H0
• Draw a conclusion

4
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

5
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

6

• 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.

7
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.

8
One-Tailed Versus Two-Tailed Tests
1. Formulate H1and H0

• 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.

9
One-Tailed Versus Two-Tailed Tests -- continued
1. Formulate H1and H0

• 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
  gt or lt 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.

10
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
• Alternative

• 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)

H1 ??0 or H1 ? gt0
Two tail test
One tail test
? mupopulation mean
11
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.
12
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.
13
3. Choose Level of Significance

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

14

• 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 dont know is
whether this preference is significant
15
4. Calculate the Test Statistic
16
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.

17
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
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.
18
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.

19
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

20
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.
21
Comparison of means Graphically
Are the means equal? Or are the differences
simply due to chance?
22
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

23
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.

24
Unequal variances The problem
25
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
0.66
?(.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.
26
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

27
Degrees of Freedom

• d.f. n - k
• where
• n n1 n2
• k number of groups

The degrees of freedom is (n1 n2 2)
28
Tylenol vs Advil
? 0.10 Critical value 1.64
? -1
?/2
?/2
1.64
-1.64
-?
?
0
0.66
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
29
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 banks 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

30
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 H1 Two Tailed
There is no difference in importance between
brand and price
Price is more important than brand
There is a difference in importance between brand
and price
31
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).

32
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

33
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

34
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.
35
Comparing Antacids
Brand Attitude
Means
Non Comparative
Explicit Comparative
Non Explicit Comparative
Type of Ad
36

• 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
  non-comparative ad etc.)
• Variation between the treatments (I.e. between
  the three types of ads)

37
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.
Degrees of Freedom The F statistic has DF for
both numerator (between group) and denominator
(within group) DF between group (c-1) where
cnumber of groups DF within group (N-c) where
N is sample size
38
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
39
ANOVA Test

• The null hypothesis would be tested with the F
  distribution

F distribution
?
Reject H0
df(c-1)/(N-c)
40

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

41

• 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