Download Lecture 7 Correlation and Regression Many marketing research

Lecture 7
Correlation and Regression
Many marketing research projects are carried out to study relationships- relationship between
sales revenues and advertising expenses, relationship between market share and the number of
salespeople, relationship between sales and distance to the nearest competing stores.
Some association between categories obtained from nominal data has already been
discussed in chi-square. Association between two interval-scaled data is measured through
correlation. Correlations are various types and used for various purposes such as
nonmonotonic, monotonic, linear, curvilinear, rank order etc.
Let us discuss some of them.
Non-monotonic Relationship
A non-monotonic relationship is one where presence or absence of one variable is
systematically associated with the presence or absence of another variable but there is no
discernable direction of relationship. However the relationship exists.
For example in a fast food restaurant, the experience tells us that morning customers buy tea
but afternoon customers typically purchase soft drink. This relationship is no way exclusive.
There is no guarantee that morning customer would always order tea and afternoon customers
always order soft drink. However a relationship does exist.
Monotonic Relationship
A monotonic relationship is one when a researcher can assign only a general direction
association between two variables. Monotonic relationship can be either increasing or
decreasing. If one variable increases and the other also increases, it is called monotonic
increasing relationship. On the other hand, monotonic decreasing relationship would be when
one variable increases but the other decreases. Monotonic means that the direction of the
relationship is described. However exact amount of change in one variable as the other
variable changes cannot be indicated.
LINEAR CORRELATION
Correlation or linear correlation is a measure of nature and degree of association or covariation between two variables. These variables are measured by interval or ratio scales. Covariation is defined as an amount of change in one variable systematically associated with a
change in another variable. Coefficient of Correlation(r) is the index number that
communicates the strength of association between two variables say X and Y. Strength is
indicated by the absolute size of correlation. Here is a rule of thumb about the strength of
1|Page
©St. Paul’s University
correlation (after it is proved that correlation is statistically significant)
Range of Coefficient
00 to + 20
+ 21 to + 40
+ 41 to + 80
+ 81 to +100
Strength of Association
None
Very weak
Moderate
Strong
Correlation coefficients that are closer to zero indicate that there is no systematic
association between the two variables and the ones which are closer to + 1.0 show that there
is some systematic association between two variables.
Direction of association is indicated by the sign (+ or -) of the size of the coefficient. If
the sign is positive, it means there is positive direction of the co-variation between the two
variables, that is, both variables change in the same direction. In this case if variable X increases,
Y would also increase. On the other hand a negative sign in correlation indicates a negative covariation which means if one variable increases the other would decrease and vice versa.
Example
If we find that number of years of education (variable X) and hours in reading news paper
(variable Y) have a correlation of 0.87, it means they are positively correlated. It will be
interpreted like this: people with more education spend more time on reading newspaper. But if
we find a correlation of -0.81 between smoking cigarettes and education, it means more
educated people smoke less.
Product Moment Correlation
Product Moment correlation is the statistic widely used in determining the size or strength of
association between two intervals or ratio scaled variables. It is also known as Pearson
Correlation Coefficient as it was originally proposed by Karl Pearson. Sometimes it is also called
as bivariate correlation, simple correlation or correlation coefficient. Product Moment Correlation
is computed with this formula.
2|Page
©St. Paul’s University
Example 1
A company wants to find out if the number of sales persons in a sales territory has some
relationship with the sales revenue. From a random sample of 10 territories following data were
obtained
Territory #
# of Sales Persons
Sales in 000 Rupees
1
2
3
4
5
6
7
8
9
10
5
7
6
9
12
8
11
16
13
7
261
288
381
412
440
317
567
572
428
317
3|Page
©St. Paul’s University
Example II
A researcher is interested to find out whether the high price of large refrigerators compensates in
terms of saving of energy. He collected data on price and annual cost of consumption of electricity
in terms of rupees from random sample of 8 brands of refrigerators. The data are presented below.
Refrigerator Brand
Price Rs. 000
Annual cost of Consumption
on electricity Rs. 000
A
85
4.8
B
76
5.4
C
90
5.8
D
87
6.6
E
110
7.7
F
80
6.6
4|Page
©St. Paul’s University
G
65
7.0
H
75
8.1
Find coefficient of correction r. What do you say about it?
There is a strong positive correlation between the two variables.
Rank Order Correlation
5|Page
©St. Paul’s University
We use Rank Order Correlation coefficient in data analysis to measure the monotonic relationship
between two variables measured on an ordinal scale. That is when the data are ordered into ranks.
Spearman's Rank Order Correlation which measures the direction and degree of association
between two sets of rankings is found with the help of following formula.
Where
rs = Spearman’s rank order correlation
d = difference of ranks in the paired
ranking n = number of items ranked
Ranking of various tooth paste brands according to consumer perception on their decay prevention
and whitening ability.
6|Page
©St. Paul’s University
7|Page
©St. Paul’s University
8|Page
©St. Paul’s University
A Spearman rank order correlation of -.86 shows a fairly strong relationship between the two
rank order i.e. decay prevention and whitening ability for the tooth paste brands under study.
Negative sign indicates that the relationship is negative which means if the ranks are higher on
decay prevention; these are lower on whitening ability. These are the perceived rankings by the
respondent.
Let us take another example.
Example: Spearman’s Rank Order Correlation
Two judges were asked to rate six brands of tissue papers. Their ratings are given below. Find
out the Spearman’s rank order correlation between the ratings assigned by two judges
Tissue Brand
Judge I Rank
Judge II Rank
G
2
6
H
3
3
I
6
5
J
1
4
K
5
2
L
4
1
9|Page
©St. Paul’s University
Testing the difference between the Means
Marketing research may be facing the problem testing the hypothesis relating to the difference
between the means. Such means may be the sample mean and population mean or two
independent means.
Chi-square is one of the procedures we have studied to test the significance of difference of
data derived from nominal data. For ordinal data, we may use Spearman’s correlation.
We use different statistics to test the difference between means based on interval data. These
statistical tests are classified whether one or two samples are involved. One mean generated
from a sample may be compared with a mean by hypothesized to exist in the population; or two
means generated from independent samples.
z-test and the t-test are appropriate to test the difference between the means. The choice is made
on the researcher’s knowledge about the standard deviation of the population and the sample
size. It is appropriate to use these test in the following situations;
If sample size is more than 30 & population standard deviation is unknown, use z-test, and;
If sample size is less than 30 and population is unknown, use t-test.
10 | P a g e
©St. Paul’s University
Example I
11 | P a g e
©St. Paul’s University
A shoe company is investigating the desirability of adding a new design of shoes to its shop.
The company has decided that it will add this design only if sells 100 pairs per week in each
store. It was put on 40 different shops randomly and data on their sale was calculated. The data
revealed an average sale of 106 pairs per week per store. The standard deviation was worked out
to be 13.8. Should the company introduce new design?
Solution
H0
µ≤100
H1
µ>100
Mean
= 106
S
= 13.8
Sx
= 2.18
Since the sample is more than 30 and variance of the population is unknown, we will use ztest, z will be calculated as:
z = (106-100)/2.18
=
6/2.18
=
2.75
As the calculated value of z (2.75) exceeds the critical value in the table of 1.96 at 0.05
confidence level so the null hypothesis is rejected. The company should go ahead and introduce
the new shoes.
Example of Z-Test for Two Independent Means
Let us take the example of z-test for two means. Suppose two independent samples of 50 each of
the two cola beverages A and B yield the following data on average weekly per household
consumption in Gulberg area of Lahore.
Cola beverage A mean consumption=6.3liters
Cola beverage A standard deviation =2.1
Cola beverage B mean consumption=5.7liters
Cola beverage B standard deviation =1.3
12 | P a g e
©St. Paul’s University
Solution
13 | P a g e
©St. Paul’s University
14 | P a g e
©St. Paul’s University