[PDF]Statistics For Business & Economics

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Marcelo Fernandes



Statistics for Business and
Economics



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Statistics for Business and Economics

© 2009 Marcelo Fernandes & Ventus Publishing ApS

ISBN 978-87-7681-481-6



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Statistics for Business and Economics



Contents



Contents



1. Introduction

1 . 1 Gathering data

1.2 Data handling

1.3 Probability and statistical inference



6

7
8
9



2. Data description

2.1 Data distribution

2.2 Typical values

2.3 Measures of dispersion



11

11
13
15



3. Basic principles of probability

3.1 Set theory

3.2 From set theory to probability



18

18
19



4. Probability distributions

4. 1 Random variable

4.2 Random vectors and joint distributions

4.3 Marginal distributions

4.4 Conditional density function

4.5 Independent random variables

4.6 Expected value, moments, and co-moments



36
36

53
56
57
58
60




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Statistics for Business and Economics



Contents



4.7 Discrete distributions

4.8 Continuous distributions



74
87



5. Random sampling

5.1 S ample statistics

5.2 Large-sample theory



95

99
102



6. Point and interval estimation

6.1 Point estimation

6.2 Interval estimation



107

108
121



7. Hypothesis testing

7. 1 Rejection region for sample means

7.2 Size, level, and power of a test

7.3 Interpreting p-values

7.4 Likelihood-based tests



127
131
136
141
142




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Statistics for Business and Economics



Introduction



Chapter 1
Introduction



This compendium aims at providing a comprehensive overview of the main topics that ap-
pear in any well-structured course sequence in statistics for business and economics at the
undergraduate and MBA levels. The idea is to supplement either formal or informal statistic
textbooks such as, e.g., "Basic Statistical Ideas for Managers" by D.K. Hildebrand and R.L.
Ott and "The Practice of Business Statistics: Using Data for Decisions" by D.S. Moore,
G.P. McCabe, W.M. Duckworth and S.L. Sclove, with a summary of theory as well as with
a couple of extra examples. In what follows, we set the road map for this compendium by
describing the main steps of statistical analysis.



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Statistics for Business and Economics Introduction

Statistics is the science and art of making sense of both quantitative and qualitative data.
Statistical thinking now dominates almost every field in science, including social sciences such
as business, economics, management, and marketing. It is virtually impossible to avoid data
analysis if we wish to monitor and improve the quality of products and processes within a
business organization. This means that economists and managers have to deal almost daily
with data gathering, management, and analysis.



1.1 Gathering data



Collecting data involves two key decisions. The first refers to what to measure. Unfortu-
nately, it is not necessarily the case that the easiest-to-measure variable is the most relevant
for the specific problem in hand. The second relates to how to obtain the data. Sometimes
gathering data is costless, e.g., a simple matter of internet downloading. However, there are
many situations in which one must take a more active approach and construct a data set
from scratch.

Data gathering normally involves either sampling or experimentation. Albeit the latter
is less common in social sciences, one should always have in mind that there is no need for a
lab to run an experiment. There is pretty of room for experimentation within organizations.
And we are not speaking exclusively about research and development. For instance, we could
envision a sales competition to test how salespeople react to different levels of performance
incentives. This is just one example of a key driver to improve quality of products and
processes.

Sampling is a much more natural approach in social sciences. It is easy to appreciate

that it is sometimes too costly, if not impossible, to gather universal data and hence it makes

sense to restrict attention to a representative sample of the population. For instance, while

census data are available only every 5 or 10 years due to the enormous cost/effort that it

involves, there are several household and business surveys at the annual, quarterly, monthly,

and sometimes even weekly frequency.
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Statistics for Business and Economics Introduction

1.2 Data handling

Raw data are normally not very useful in that we must normally do some data manipulation
before carrying out any piece of statistical analysis. Summarizing the data is the primary
tool for this end. It allows us not only to assess how reliable the data are, but also to
understand the main features of the data. Accordingly, it is the first step of any sensible
data analysis.

Summarizing data is not only about number crunching. Actually, the first task to trans-
form numbers into valuable information is invariably to graphically represent the data. A
couple of simple graphs do wonders in describing the most salient features of the data. For
example, pie charts are essential to answer questions relating to proportions and fractions.
For instance, the riskiness of a portfolio typically depends on how much investment there
is in the risk-free asset relative to the overall investment in risky assets such as those in
the equity, commodities, and bond markets. Similarly, it is paramount to map the source
of problems resulting in a warranty claim so as to ensure that design and production man-
agers focus their improvement efforts on the right components of the product or production
process.

The second step is to find the typical values of the data. It is important to know, for
example, what is the average income of the households in a given residential neighborhood if
you wish to open a high-end restaurant there. Averages are not sufficient though, for interest
may sometimes lie on atypical values. It is very important to understand the probability
of rare events in risk management. The insurance industry is much more concerned with
extreme (rare) events than with averages.

The next step is to examine the variation in the data. For instance, one of the main
tenets of modern finance relates to the risk-return tradeoff, where we normally gauge the
riskiness of a portfolio by looking at how much the returns vary in magnitude relative to
their average value. In quality control, we may improve the process by raising the average

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Statistics for Business and Economics



Introduction



quality of the final product as well as by reducing the quality variability. Understanding
variability is also key to any statistical thinking in that it allows us to assess whether the
variation we observe in the data is due to something other than random variation.

The final step is to assess whether there is any abnormal pattern in the data. For instance,
it is interesting to examine nor only whether the data are symmetric around some value but
also how likely it is to observe unusually high values that are relatively distant from the bulk
of data.

1.3 Probability and statistical inference



It is very difficult to get data for the whole population. It is very often the case that it is
too costly to gather a complete data set about a subset of characteristics in a population,
either because of economic reasons or because of the computational burden. For instance, it
is impossible for a firm that produces millions and millions of nails every day to check each
one of their nails for quality control. This means that, in most instances, we will have to
examine data coming from a sample of the population.




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Statistics for Business and Economics Introduction

As a sample is just a glimpse of the entire population, it will entail some degree of uncer-
tainty to the statistical problem. To ensure that we are able to deal with this uncertainty, it
is very important to sample the data from its population in a random manner, otherwise
some sort of selection bias might arise in the resulting data sample. For instance, if you wish
to assess the performance of the hedge fund industry, it does not suffice to collect data about
living hedge funds. We must also collect data on extinct funds for otherwise our database
will be biased towards successful hedge funds. This sort of selection bias is also known as
survivorship bias.

The random nature of a sample is what makes data variability so important. Probability
theory essentially aims to study how this sampling variation affects statistical inference,
improving our understanding how reliable our inference is. In addition, inference theory is
one of the main quality-control tools in that it allows to assess whether a salient pattern
in data is indeed genuine beyond reasonable random variation. For instance, some equity
fund managers boast to have positive returns for a number of consecutive periods as if this
would entail unrefutable evidence of genuine stock-picking ability. However, in a universe of
thousands and thousands of equity funds, it is more than natural that, due to sheer luck,
a few will enjoy several periods of positive returns even if the stock returns are symmetric
around zero, taking positive and negative values with equal likelihood.



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10



Statistics for Business and Economics Data description



Chapter 2
Data description



The first step of data analysis is to summarize the data by drawing plots and charts as well
as by computing some descriptive statistics. These tools essentially aim to provide a better
understanding of how frequent the distinct data values are, and of how much variability
there is around a typical value in the data.

2.1 Data distribution

It is well known that a picture tells more than a million words. The same applies to any
serious data analysis for graphs are certainly among the best and most convenient data
descriptors. We start with a very simple, though extremely useful, type of data plot that
reveals the frequency at which any given data value (or interval) appears in the sample. A
frequency table reports the number of times that a given observation occurs or, if based
on relative terms, the frequency of that value divided by the number of observations in the
sample.

Example A firm in the transformation industry classifies the individuals at managerial
positions according to their university degree. There are currently 1 accountant, 3 adminis-
trators, 4 economists, 7 engineers, 2 lawyers, and 1 physicist. The corresponding frequency
table is as follows.



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11



Statistics for Business and Economics



Data description



degree

value

counts

relative frequency



accounting business economics engineering law physics

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13 4 7 2 1



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7/18 1/9 1/1*



Note that the degree subject that a manager holds is of a qualitative nature, and so it is not
particularly meaningful if one associates a number to each one of these degrees. The above
table does so in the row reading 'value' according to the alphabetical order, for instance.

The corresponding plot for this type of categorical data is the bar chart. Figure 2.1 plots
a bar chart using the degrees data in the above example. This is the easiest way to identify
particular shapes of the distribution of values, especially concerning data dispersion. Least
data concentration occurs if the envelope of the bars forms a rectangle in that every data
value appears at approximately the same frequency.



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Statistics for Business and Economics



Data description



In statistical quality control, one very often employs bar charts to illustrate the reasons
for quality failures (in order of importance, i.e., frequency). These bar charts (also known
as Pareto charts in this particular case) are indeed very popular for highlighting the natural
focus points for quality improvement.

Bar charts are clearly designed to describe the distribution of categorical data. In a similar
vein, histograms are the easiest graphical tool for assessing the distribution of quantitative
data. It is often the case that one must first group the data into intervals before plotting a
>>>