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MBA Statistics 51-651-00 http://www. hec .ca/sites/ cours /51-651-02/
MBAStatistics 51-651-00
http://www.hec.ca/sites/cours/51-651-02/
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What is statistics?"I like to think of statistics as the science of learning from data... Statistics is essential for the proper running of government, central to decision making in industry, and a core component of modern educational curricula at all levels."
Jon Kettenring ASA President, 1997
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What is statistics? American Heritage Dictionary® defines
statistics as: "The mathematics of the collection, organization,
and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling.«
The Merriam-Webster’s Collegiate Dictionary® definition is: "A branch of mathematics dealing with
the collection, analysis, interpretation, and presentation of masses of numerical data."
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Course syllabus Variation. Sampling and estimation. Decision making from statistical
inference. Qualitative data analysis. Simple and multiple linear regression. Forecasting. Statistical process control. Revision.
COURSE # 1
Variation, sampling and estimation.
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Variation"The central problem in management
and in leadership ... is failure to understand the information in variation"
W. Edwards Deming
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Variation"Management takes a major step
forward when they stop asking you to explain random variation"
F. Timothy Fuller
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Variation"Failure to understand variation is a
central problem of management"
Lloyd S Nelson
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Airport ImmigrationOfficer Passengers
processed Alan 9 Barbara 10 Colin 4 Dave 8 Enid 6 Frank 14
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Airport Immigration Management expected their officers to
process 10 passengers during this period The immigration services manager, in
reviewing these figures, was concerned about the performance of
Colin thinking how best to reward Frank
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Debt Recovery When the amount of recovered debt is
much lower than the target recovery level of 80%, the General Manager visits all the District Offices in New Zealand to remind managers of the importance of customers paying on time
What do you think of the GM policy? What would you do?
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Budget Deviations Budget deviations measure the
difference between the amount budgeted and the actual amount, expressed as a percentage of the budgeted amount.
The aim is to have a zero deviation. Most of the variation lies between -3% and 4%.
Illustration of variation
Excel program:beads.xls (Deming)
The red balls are associated with defective products.
Five times a day, 5 technicians select a sample of 50 beads and counts the number of defectives (red).
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Beads History – 17 July 2000
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Beads History - 9 March 2000
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Beads History - 8 March 2001
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Beads History - 5 March 1999
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Beads History - 19 July 1996
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Beads History - 8 March 1996
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Beads History - 10 March 1995
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Beads History - 6 March 1998
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Beads History: 27 Experiments
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Beads Averages
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1 3 5 7 9 11 13 15 17 19 21 23 25 27
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Discussion What is the main difference between
the graph of the 25x27=675 draws and the graph of the 27 averages?
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Two approaches in management Fire-fighting
Scientific
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Fire-fighting approach
Problem Solution
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Scientific Approach
SolutionProblem
Cause
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Scientific Approach Making decisions based on data
rather than hunches. Looking for the root causes of
problems rather than reacting to superficial symptoms.
Seeking permanent solutions rather than quick fixes.
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The Need for Data To understand the process To determine priorities To establish relationships To monitor the process To eliminate causes of variation
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The steps of statistical analysis involve:1. Planning the collection of
information2. Collecting information3. Evaluating information4. Drawing conclusions
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Surveys: Collect information from a carefully
specified sample and extend the results to an entire population.
Sample surveys might be used to: Determine which political candidate is
more popular Discover what foods teenagers prefer
for breakfast Estimate the number of potential clients
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Sampling Definitions
Population
Sample
Parameter
Statistic
choose
calculate
estimate
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Government Operations: Conduct experiments to aid in the
development of public policy and social programs.
Such experiments include: consumer prices; fluctuations in the economy; employment patterns; population trends.
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Scientific Research: Statistical sciences are used to enhance
the validity of inferences in: radiocarbon dating to estimate the risk of
earthquakes; clinical trials to investigate the
effectiveness of new treatments; field experiments to evaluate irrigation
methods; measurements of water quality; psychological tests to study how we reach
the everyday decisions in our lives.
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Business and Industry: predict the demand for products and
services; check the quality of items
manufactured in a facility; manage investment portfolios; forecast how much risk activities
entail, and calculate fair and competitive insurance rates.
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Sampling Our knowledge, our attitudes and
our actions are mainly based on samples.
For example, a person’s opinion of an institution or a company which makes thousands of transactions every day is often determined by only one or two meetings with this institution.
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Census vs Sample Census = reality (True or false?!) The information needed is available
for all individuals of the study population.
Sample = estimation of the reality The information needed is only
available for a subset of the individuals of the study population.
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Advantages of a sample Reduced costs Accrued speed Offers more possibilities
in some cases it may be impossible to have a census (ex: quality control)
Perhaps more precise!Cases where highly qualified personal are necessary for collecting data
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Probabilistic vs non probabilistic samples
s im ple random system atic stra tified cluster(area)
Probabilist
on a vo luntary basis purposive
Non probabilist
Sam pling m ethods
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Sampling errors Random error
• different samples will produce different estimates of the study population characteristics
Systematic error - bias• non probabilistic sample• probabilistic sample with a high rate of
non respondents• biased instrument of measure
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TV Show Poll - March 1998 Should Hamilton be renamed
Waikato City? 4400 dialled the 0900 number 73% were against the change
What type of sample was taken? What conclusions would you draw?
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Bias vs variability Bias is a systematic error, in the
same direction, of successive estimations of a parameter.
Large variability means that repeated values of estimations are scattered; the results of successive sampling cannot be reproduced.
(see …)
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a) large bias, low variability b) low bias, high variability
c) large bias, high variability d) low bias, low variability
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Bias due to non-response
Bias is often caused by non-response in surveys. For example, suppose that the population is
divided in two groups : respondents (60%) and non-respondents (40%).
Within respondents, 65% are in favour of a project et within non-respondents, 20% are in favour.
The real proportion in the population in favour of the project is p = 47% , while a survey will give an estimation of p at about 65% 47%. The bias is 18%.
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How do we make a simple random sample drawing?
1. We need a list. Each element of the population is assigned a number from 1 to N.
2. We use a computer program to select n numbers as randomly as possible (ex: Excel, MINITAB, SAS, C).
3. The corresponding elements form the sample.
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Notes : The results obtained depend on the sample
taken. If the samples are taken according to codes of
practice, the results should all be similar. For a simple random draw, each individual of
the population is as likely to be selected at each draw.
For a simple random draw, there are many different possible samples. All possible samples of the same size have the same chance of being selected.
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Opinion polls The results obtained in a
probabilistic sample will be used to generalize the entire population.
But the fact of using a sample necessarily induces a margin of error that we will try to control.
We will distinguish two types of data: qualitative and quantitative.
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Types of data Qualitative (measurement scale: nominal or ordinal)
(parameter: %)Examples:
• sex (F, M)• political party (PLQ, PQ, ADQ)• preferred brand (Coke, Pepsi, Homemade brand, …)• satisfaction level (Likert scale from 1 to 5)
Quantitative (measurement scale: interval or ratio) (parameter: mean)Examples:
• age• income• temperature (in degrees Celsius)
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Case study
Data in credit.xls represent the credit balance and the total income of 100 randomly chosen families in Quebec.
What is the mean credit balance for a family in Quebec? What is the precision (margin of error) of your estimate?
What about a Canadian family? Assuming that 2 500 000 families use at least one
credit card regularly, what is the total debt of families in Quebec? What is the precision of the estimate?
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Confidence intervals
To estimate the proportion p of individuals with a given characteristic among the population,
or to estimate the mean of a given quantitative variable,
one uses a confidence interval at the (1- ) level.
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Confidence intervals (continued)
It consists of constructing an interval of values which enables one to affirm, with a certain level of confidence (in general: 90%, 95% or 99%), that the true value of the parameter for the population, is included in this interval.
Illustration: Confidence interval applet
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Confidence interval for estimating a proportion p Example: In a sample of 125 college
students who were questioned on their intentions to vote in the next election, 45 answered positively.
Estimate, in a specific way, the proportion of the entire student population of this institution who intend to vote at the next elections.
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Confidence interval for estimating a proportion (continued) Excel program: proportion1.xls
In general, if the sample size n is large enough, the (1 - ) confidence interval to estimate the true proportion p of the studied characteristic in the population is given by:
; - 1 ] z Z z- P[
as,such is on table,distributi normal in the read , z
/2/2
/2
. 1.96 z then 5%, if /2
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Example (continued) : Consequently, a confidence interval of 95%
certainty for the proportion of the entire student population of this institution who intend to vote at the next election is given by:
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Example (continued) : How would we report the results of
this survey in the student newspaper of this college?
36% of the students of this college intend to exercise their voting rights at the next student election. The margin of error is 8.4% with a 95% degree of confidence (or with 95% certainty or 19 times out of 20).
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Notes:This is an approximate formula and applies only for large samples.If all possible random samples of size n are taken and their 95% confidence interval calculated, 95% of them will include the true proportion p of the population, and thus 5% will not include it.
The quantity is called the margin of error and is used to establish the 95% confidence level (19 times out of 20).
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Margin of error at the 95% level
p (%)100 300 500 1000 3000 10000
10 5,9 3,4 2,6 1,9 1,1 0,620 7,8 4,5 3,5 2,5 1,4 0,830 9,0 5,2 4,0 2,8 1,6 0,940 9,6 5,5 4,3 3,0 1,8 1,050 9,8 5,7 4,4 3,1 1,8 1,060 9,6 5,5 4,3 3,0 1,8 1,070 9,0 5,2 4,0 2,8 1,6 0,980 7,8 4,5 3,5 2,5 1,4 0,890 5,9 3,4 2,6 1,9 1,1 0,6
size of the sample n
sample size n and the value of p at the 95% level
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Margin of error at the 90% level
p (%)100 300 500 1000 3000 10000
10 4,9 2,8 2,2 1,6 0,9 0,520 6,6 3,8 2,9 2,1 1,2 0,730 7,5 4,4 3,4 2,4 1,4 0,840 8,1 4,7 3,6 2,5 1,5 0,850 8,2 4,7 3,7 2,6 1,5 0,860 8,1 4,7 3,6 2,5 1,5 0,870 7,5 4,4 3,4 2,4 1,4 0,880 6,6 3,8 2,9 2,1 1,2 0,790 4,9 2,8 2,2 1,6 0,9 0,5
sample size n
Margin of error or precision (in %) according to the sample size n and the value of p at the 90% level
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Calculation of size n to ensure a maximum margin of error If we want to estimate the
proportion p at a (1-) confidence level with a maximum margin of error e, then we have the following relation for the calculation of the sample size n :
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Discussion Take a look at the following survey:
Survey on California recall election
In the light of the recent results, what can you say about the survey.
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Confidence interval for estimating the mean
In a general way, if the sample size n is large enough, the (1 - ) confidence interval for estimating the true mean of the population, is given by:
. - 1 ] t T t- P[
as,such and , on tabledistributiStudent in the read , t
1-n /2,1-n /2,
1-n /2,
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Notes: This is an approximate formula that applies
for small samples when the characteristic is has a normal distribution or from a large sample (n ≥ 30).
When n is very large (n ≥ 100), the values of t/2, n-1 and z/2 coincide.
Excel: Tools/Data Analysis/Descriptivre Statistics or mean-1.xls
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Notes: (continued)Interpretation of a 95% confidence interval for the mean of a characteristic in the population:
If all the random samples of size n are taken and their confidence intervals calculated, 95% of them will include the true mean of the population, and thus 5% will not include it.Recall the confidence interval applet.
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Confidence interval for :Example
In order to know the weekly average cost of the grocery basket for a family of 4 people residing in Sherbrooke, we take a sample of 50 of these families and we note the amount of their grocery for this week. We obtain an average amount of 155$ with a standard deviation estimate of 15$.
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Example (continued) :Estimate the current average cost of the grocery basket for a family of 4 people residing in Sherbrooke using a 95% confidence interval:
By stating that the current average cost of the grocery basket of a family of 4 people residing at Sherbrooke is included in the interval [150.74$; 159.26$], I am 95% certain to be right. My prediction will be true 95% of time.
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Example A company wants to commercialize a new software to get
rid of junk mail. The potential market is 800 000 consumers.
Before starting selling the product, the company realized
a survey from a random sample of 40 families. Six families were interested in buying the new software.
The net gain is 3$ per software and there are fixed costs
of 50 000$. What is the decision?
