Statistics for Decision Making

The business report briefly analyzes profile of customers buying three different models of car. The three chosen car models are BMW, Lexus and Mercedes. Households are surveyed on the scale of age, income and education years. The business problem of car sellers is addressed in light of statistical tools. In order to reach a firm conclusion about customers’ profile 420 samples are considered. It has been observed that buyers of Mercedes are older, have a relatively higher income and are more educated. The claim that average age, income and education years are independent among the buyers of three different luxury cars are supported based on ANOVA test result. In the chosen sample, 36% people prefer Mercedes and 64% prefer either Lexus or BMW. The fitted binary logit model indicates age, income and education years have significant positive influence on preference of Mercedes cars. The Mercedes car sellers thus should target richer, older and educated people to their car model.

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Statistics for Decision Making
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Executive Summary
The business report briefly analyzes profile of customers buying three different models of car.
The three chosen car models are BMW, Lexus and Mercedes. Households are surveyed on the
scale of age, income and education years. The business problem of car sellers is addressed in
light of statistical tools. In order to reach a firm conclusion about customers’ profile 420 samples
are considered. It has been observed that buyers of Mercedes are older, have a relatively higher
income and are more educated. The claim that average age, income and education years are
independent among the buyers of three different luxury cars are supported based on ANOVA test
result. In the chosen sample, 36% people prefer Mercedes and 64% prefer either Lexus or BMW.
The fitted binary logit model indicates age, income and education years have significant positive
influence on preference of Mercedes cars. The Mercedes car sellers thus should target richer,
older and educated people to their car model.

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Table of Contents
1.Introduction...................................................................................................................................3
1.1 Business Problem...................................................................................................................3
1.2 Statistical Problem.................................................................................................................3
2. Analysis.......................................................................................................................................3
2.1 Analysis of ages of buyers of luxury cars..............................................................................3
2.2 Analysis of incomes of buyers of luxury cars.......................................................................7
2.3 Analysis of education years of buyers of luxury cars..........................................................11
2.4 Hypothesis testing for independence of average age of buyers...........................................15
2.5 Hypothesis testing for independence of average income of buyers....................................16
2.6 Hypothesis testing for independence of average education years of buyers.......................17
2.7 Test of preference of Mercedes cars to BMW or Lexus......................................................18
3. Conclusion.................................................................................................................................21
4. Implication of the study.............................................................................................................21

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1.Introduction
1.1 Business Problem
Designing appropriate business strategy is the central problem of any business. Sellers
should know their buyers well before marketing their problem. The report addresses the business
problem associated with luxury car sellers. The analysis aims to provide an overview of buyers’
profile based upon their age, income and education years. The business objective is to find out
how preference of luxury cars are influenced by age, income and education years of customers.
The analysis of customer profile helps to accomplish primary business goal of profit
maximization.
1.2 Statistical Problem
Different statistical tools are used to analyze profile of the customers. Multiple bar graphs
are used to visually present the data relating to age, income and education years of household
purchasing different luxury cars. To determine the nature of statistical distribution, its shape,
location and variability descriptive statistics involving measures of central tendency, dispersion
and skewness needs to be analyzed. Independence of means of age, income and education years
among the three group of buyers ANIVA test is found to be a useful tool. The binary logit
regression helps to determine the influence of age, income and education years on buying
decision of Mercedes cars.
2. Analysis
2.1 Analysis of ages of buyers of luxury cars
Visual presentation of ages

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In the survey, buyers of the luxury cars belong to the various age group. The buyers have
a minimum age of 35 and a maximum age of 70. In order to analyze ages of buyers, ages are
categorized in six different classes with a class width of 6. The BMW buyers belong to the
youngest age group of 35-40. The middle age group ranged from 47 to 52 has shown their
preferences towards Lexus cars. With increase in ages, preference of buyers gradually shifted
from BMW to Lexus and Mercedes. There are 54 people in the age group of 53-58 preferring
Mercedes. In this age group only 6 people buy BMW while 36 people buy Lexus. In the two
oldest group, 59-64 and 65-70 no one buy BMW. People belonging to these two group mostly
prefer Mercedes or Lexus. This is presented in figure below with the help of a multiple bar
graph.
35-40 41-46 47-52 53-58 59-64 65-70
20
66
38
6
8
26
58
36
8
4
6
24
44
54
20
2
1
2
3
Figure 1: Multiple bar graph for buyers of different age group
Descriptive statistics for ages of buyers
The table below presents descriptive statistics for age of buyers preferring BMW cars
Table 1: Descriptive Statistics of age of BMW buyers

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The mean age of 130 BMW buyers is obtained as 45. The mean age of BMW buyers indicates
that people buying BMW cars have an average age of 45 years. The median age is estimated to
be 45. From the median age it can be said that fifty percent of the chosen population is 45 years.
The modal age is 46. That is most of the BMW buyers are found to be 46 years’ old which is
close to mean and median age. Buyers of BMW can be as young as 21 years and as old as 36
years. Coefficient of variation provides a measure of variability. This is computed as a ratio of
standard deviation to its mean. The standard deviation is estimated to be 4.35. The standard
deviation being less than mean of the distribution implies that the age distribution of BMW
buyers has a relatively less variability. The measure of skewness indicates symmetry of the
distribution. The measured skewness value is 0.50 which implies that the distribution tends to be
skewed positively.
Table 2: Descriptive Statistics of age of Lexus buyers

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The mean age of 140 Lexus buyers is obtained as 50. The mean age of Lexus buyers indicates
that people buying Lexus cars have an average age of 50 years. The median age is estimated to
be 50. From the median age it can be said that fifty percent of the chosen population is 50 years.
The modal age is 55. That is most of the Lexus buyers are found to be 55 years’ old which is
close to mean and median age. Buyers of Lexus can be as young as 36 years and as old as 68
years. Coefficient of variation provides a measure of variability. This is computed as a ratio of
standard deviation to its mean. The standard deviation is estimated to be 6.10. The standard
deviation being less than mean of the distribution implies that the age distribution of Lexus
buyers has a relatively less variability. The measure of skewness indicates symmetry of the
distribution. The measured skewness value is 0.36 which implies that the distribution tends to be
skewed positively.
Table 3: Descriptive Statistics of age of Mercedes buyers

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The mean age of 150 Mercedes buyers is obtained as 52. The mean age of Mercedes buyers
indicates that people buying Mercedes cars have an average age of 52 years. The median age is
estimated to be 53. From the median age it can be said that fifty percent of the chosen population
is 53 years. The modal age is 53. That is most of the Mercedes buyers are found to be 53 years’
old which is equal to the median age. Buyers of Mercedes can be as young as 35 years and as old
as 70 years. Coefficient of variation provides a measure of variability. This is computed as a ratio
of standard deviation to its mean. The standard deviation is estimated to be 6.74. The standard
deviation being less than mean of the distribution implies that the age distribution of BMW
buyers has a relatively less variability. The measure of skewness indicates symmetry of the
distribution. The measured skewness value is -0.02. As the skewness measure is close to zero,
this indicates an almost symmetrical distribution.
2.2 Analysis of incomes of buyers of luxury cars
Visual representation

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Income of the chosen sample varied between 46068 dollars and 334823 dollars. Incomes
are divided into three broad class with a class width of 100000. In the first income group ranged
from 46068 dollars to 146067 dollars, there are 72 people who buy BMW. In this group 58
people buy Lexus and 34 people buy Mercedes. Among 420 samples, 244 people belong to the
income second income class having a lower limit of 146068 dollars and upper limit of 246067
dollars. In this income group, 106 people buys Mercedes, 80 people buy Lexus and 58 people
buy BMW. The highest income class ranged from 246068 dollars to 346067 dollars. Here, no
people buy BMW, 2 people buy Lexus and 10 people buy Mercedes. The comparative analysis
of buyers’ preference according based on their income is presented in the figure below
46068-146067 146068-246067 246068-346067
72
5858
80
2
34
106
10
1
2
3
Figure 2: Multiple bar graph for buyers of different income group
Descriptive Statistics
The nature of income distribution of buyers of the three luxury cars are analyzed
separately. The descriptive statistics for each of the buyers group is presented and discussed
below.

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Table 4: Descriptive Statistics of income of BMW buyers
From the summary statistics, average income of buyers of BMW cars found to be 139271
dollars. The median income representing position of fifty percent observation in the group is
obtained as 138512 dollars. The modal income indicating income of maximum number of buyers
in this group is obtained as 109568. Range of the distribution is 170652 with a maximum income
of 216720 dollar and minimum income of 46068 dollars. The standard deviation measure
indicates distance of the observations from the mean income. The descriptive statistics gives
standard deviation of the distribution as 33154.54. The average income exceeds standard
deviation of the distribution indicating a smaller variability. The income distribution for BMW
buyers is almost symmetric as the skewness measure is close to zero (-0.04).
Table 5: Descriptive Statistics of income of Lexus buyers

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The average income of buyers of Lexus cars found to be 154187 dollars. The median
income representing position of fifty percent observation in the group is obtained as 154492
dollars. The modal income indicating income of maximum number of buyers in this group is
obtained as 179617. Range of the distribution is 152065 with a minimum income of 96069 dollar
and maximum income of 248134 dollars. The standard deviation measure indicates distance of
the observations from the mean income. The descriptive statistics gives standard deviation of the
distribution as 30248.02. The average income exceeds standard deviation of the distribution
indicating a smaller variability. The positive value of skewness (0.69) indicates that distribution
of income for Lexus buyers is positively skewed.
Table 6: Descriptive Statistics of age of Mercedes buyers

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The descriptive statistics gives average income of buyers of Mercedes cars as 184424
dollars. The median income representing position of fifty percent observation in the group is
obtained as 186070 dollars. The modal income indicating income of maximum number of buyers
in this group is obtained as 161590 dollars. Range of the distribution is 284882 with a maximum
income of 334823 dollar and minimum income of 49941 dollars. The standard deviation measure
indicates distance of the observations from the mean income. The descriptive statistics gives
standard deviation of the distribution as 47095.22. The average income exceeds standard
deviation of the distribution indicating a smaller variability. The income distribution for BMW
buyers is positively skewed as the skewness measure is estimated to be positive.
2.3 Analysis of education years of buyers of luxury cars
Visual representation
Education years of buyers are discussed in terms of three different classes having a class
width of 4. People having education years of 11 to 14 years mostly buy Lexus. There are 48
people in this class who buy Lexus car, 38 people with the same education years buy BMW

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while only 6 people buy Mercedes. People with education years of 15 to 18 on mostly prefer to
buy Mercedes following BMW and Lexus. The most educated people belong to the class ranged
between 19 and 22 years of education tend to buy Mercedes more. Among the total 68 people
belonging to this group, 38 people buy Mercedes, 24 people buy Lexus and only 6 people buy
BMW. This can be better understood from the graphical representation of the data set.
11-14 15-18 19-22
34
90
6
48
68
24
8
104
38
1
2
3
Figure 3: Multiple bar graph for buyers of different education group
Descriptive Statistics
The detailed information regarding the distribution of education is discussed by
computing descriptive statistics of education years of buyers of three luxury cars.
Table 7: Descriptive Statistics of education years of BMW buyers

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Average education years of total 130 BMW buyers is 16 years. Half of the people buying
BMW cars has 16 years of education as obtained from the median education years. This is also
the modal education year for this group. People here have education as much as 19 years and as
less as 11 years giving range as 8 years. Obtained standard deviation is 1.83 which is smaller
than mean education years. This implies less variability of the distribution. The skewness
measure is -0.4345 implying the distribution to be negatively skewed.
Table 8: Descriptive Statistics of education years of Lexus buyers

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Average education years of total 140 Lexus buyers is 16 years. Half of the people buying
Lexus cars has 16 years of education as obtained from the median education years. This is also
the modal education year for this group. People here have education as much as 21 years and as
less as 12 years giving range as 9 years. Obtained standard deviation is 2.14 which is smaller
than mean education years. This implies less variability of the distribution. The skewness
measure is 0.17, implying the distribution to be positively skewed.
Table 9: Descriptive Statistics of education years of Mercedes buyers

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Average education years of total 150 Mercedes buyers is 17 years. Half of the people
buying Mercedes cars has 17 years of education as obtained from the median education years.
This is also the modal education year for this group. People here have education as much as
22years and as less as 13 years giving range as 9 years. Obtained standard deviation is 1.73
which is smaller than mean education years. This implies less variability of the distribution. The
skewness measure is 0.04 implying the distribution is almost symmetric.
2.4 Hypothesis testing for independence of average age of buyers
The suitable statistical tool to test the independence of average ages of three different
luxury cars buyers is Analysis of Variance (ANOVA). The null and alternative hypothesis
corresponding to the ANOVA test is as follows
Null Hypothesis: Mean ages of buyers of three different luxury cars are equal.
Alternative Hypothesis: Mean ages of buyers of three different luxury cars are not equal
Result of the ANOVA test is presented in the table given below

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Table 10: Test result of independence of average age of three group of buyers
The decision rule for ANOVA test suggests rejection of null hypothesis at 5% level of
significance if the computed F value is higher than critical F value at the chosen level of
significance. From the table showing ANOVA result, computed F value is obtained as 49.80.
The critical F value at 5% level of significance with degrees of freedom of 419 is 3.01. As the
obtained F value exceeds the critical F value the null hypothesis can be rejected. Therefore,
conclusion can be drawn in favor of independence of average ages of three group of buyers.
2.5 Hypothesis testing for independence of average income of buyers
The independence of average income of three different luxury cars buyers is tested by
conducting the test of Analysis of Variance (ANOVA). The null and alternative hypothesis
corresponding to the ANOVA test is as follows
Null Hypothesis: Mean income of buyers of three different luxury cars are equal.
Alternative Hypothesis: Mean income of buyers of three different luxury cars are not equal
Result of the ANOVA test is presented in the table given below

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Table 11: Test result of independence of average age of three group of buyers
From the table showing ANOVA result, computed F value is obtained as 52.17. The critical F
value at 5% level of significance with degrees of freedom of 419 is 3.01. As the obtained F value
exceeds the critical F value the null hypothesis can be rejected. Therefore, it can be said that
mean income of buyers of three different luxury cars are not equal.
2.6 Hypothesis testing for independence of average education years of buyers
The Analysis of Variance (ANOVA) technique is used for testing the independence of average
education years of buyers of luxury cars. The following hypotheses are framed to conduct the
test of independence of mean education years
Null Hypothesis: Mean education years of buyers of three different luxury cars are equal.
Alternative Hypothesis: Mean education years of buyers of three different luxury cars are not
equal
The table below presents the result of ANOVA test corresponding to education years
Table 12: Test result of independence of average education years of three group of buyers

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The ANOVA result gives computed F value as 25.93. The critical F value at 5% level of
significance with degrees of freedom of 419 is 3.01. As the obtained F value exceeds the critical
F value the null hypothesis can be rejected. This implies that mean education years differ
significantly among the three group of buyers.
2.7 Test of preference of Mercedes cars to BMW or Lexus
Some car dealers claim that older people who has a higher income and higher education
years as well prefer Mercedes to BMW or Lexus. Regression analysis is the most appropriate
tool to test this claim. However, given the discrete nature of the dependent variable (choice of
luxury cars), binary logistic regression has been conducted. The table below represents the
proportion of people preferring Mercedes and Lexus or BMW.
Table 13: Sample proportion of preference of two categories of luxury cars

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In the above table 0 represent preference for Lexus or BMW while 1 represents
preference for Mercedes. In the chosen sample, 36% of the household prefer Mercedes while
64% prefer BMW or Lexus.
The result related to parameters of the model (age, income and education years) is
presented in the following table.
Table 14: Coefficient of logistic regression model
From the above table, coefficient of all the parameters are found to be positive. This
implies with an increase in age, income and years of education the probability of buying
Mercedes cars increases as well. The P value of all the model parameters are less than 5 percent
significance level. This indicates the coefficients are statistically significant and thus have a
positive significant influence on choice of luxury cars.
The goodness of the fitted model is presented in the following table. This has similar
implication as R square has in simple linear regression model
Table 15: Goodness of fit

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Table 16: Logistic regression test result
The overall significance of the model depends on the significance of log likelihood function. The
probability of the Chi square log ratio is less than 5% level of significance. This indicates overall
significance of the fitted model. This in turn implies that age, income and education years
together have significant influence on buyers’ preference for luxury cars.
Finally, the equation for logistic regression is obtained as
Preference of luxury cars= 1
1+e (− (−14.857+0.098 × Age +0.00002× Annual Income+0.326 × Education ) )
3. Conclusion

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The business report summarizes the result of survey conducted by automobile association
regarding the preference of three luxury cars namely BMW, Lexus and Mercedes. The three
factors that are considered to influence the choice include age, income and years of education.
From the survey is has been observed that among the 420 samples, 130 people buy BMW, 140
people buy Lexus and 150 people buy Mercedes. Most BMW buyers belong to the age group of
41 to 46 years. The dominant age group for Lexus and Mercedes cars are 47 to 52 years and 53
to 58 years. The BMW buyers mostly belong to the income group of 46068 dollars to 146067
dollars. In the highest income group (246068-346067), people buy either Mercedes or Lexus. For
the education years, the domain group is 15 to 18 years of education. There are total 262 people
in the sample belong to this age group with higher preference towards Mercedes. The average
age, income and education years exceed that of BMW and Lexus years. A statistical
independence relation is obtained among the mean age, income and education years of three
different luxury cars. The binary logistic regression result supports the claim that household with
older people, higher income and more years of education show a greater probability of buying
Mercedes over the other two types of cars.
4. Implication of the study
The business report has useful implication in determining the marketing strategy of
respective automobile sellers of BMW, Lexus and Mercedes. This provides an overall idea
regarding the age, income and education years of buyers buying different models of cars. The
sellers should select their targeted customer based on their age, income and education years.
Younger buyers with a relatively low income and less years of education should be the target of
BMW sellers. People belonging to the group of older age, higher income and more education
years are the targeted buyers of Mercedes cars.

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