Linear regression is a procedure that identifies relationship between independent variables and a dependent variable.

1
Chapter 16

• Linear regression is a procedure that identifies
  relationship between independent variables and a
  dependent variable.
• This relationship helps reduce the unexplained
  variation of the dependent variable behavior,
  thus provide better predictions of its future
  values.

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The Simple linear regression model

• The model is

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The Simple linear regression model

• The model is
• We try to estimate the deterministic part of it
  by developing the line with the best fit.
• Best fit is defined as the minimum sum of squared
  errors.
• An error is the difference between the line value
  and the actual value for a given x.
• The analysis yields

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Problem 6

• Are the costs of welding machines breakdowns
  related to their age?
• From the data answer the following
• Find the sample regression line
• What is the coefficient of determination.
  Interpret.
• Are machine age and monthly repair costs linearly
  related?
• Is the fit good enough to use the model to
  predict the monthly repair costs of a 120 months
  old machine?
• Make the prediction.

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Problem 6
Data

• From Excel we get
• The Cov(age,cost)936.82Mean age (x) 113.35.
  378.77Mean cost (y) 395.21.
  4094.79.
• b1 cov(x,y)/ 936.82/378.77 2.4733b0
  y-b0x 395.21-2.4733(113.35) 114.86The
  regression line

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Problem 6

• Coefficient of determination.
• In this case
• 56.59 of the variation in costs, are explained
  by the model (by the different ages).

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Problem 6

• Is there a linear relationship between monthly
  costs and machine age?
• We need to test if b1 is not equal to zero.
• H0 b1 0H1 b1¹ 0
• In this case
• t 2.47-0/.5106 4.837
• The rejection region is tgtta/2 ortlt -ta/2 with
  n-2 degrees of freedom

Can be calculatedseparately
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Problem 6
Data
The p value lt alpha
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Problem 6

• We need to forecast the expected cost for a 120
  months old machine.
• The equation provides a point predictionCost
  114.862.4733(120) 411.65The prediction
  interval (use data analysis plus) LCL 318.12
  UCL 505.18
• Whats the prediction for the average monthly
  repair cost for all the machines 120 months old?
• To answer this question construct the confidence
  interval (notice, not the prediction interval!)

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Chapter 17

• The multiple regression model allows more than
  one independent variable explain the values of
  the dependent variable.
• We assess the model as before using
• t test for linear relationships between the
  independent variables and the dependent variable
  (tested one at a time)
• F test for the over usefulness of the model
• Coefficient of determination for the fit.

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Problem 7

• When a company buys another company it is not
  unusual that some workers are terminated.
• A buyout contract between Laurier Comp and the
  Western Comp required that Laurier provides a
  severance package to Western workers fired,
  equivalent to packages offered to Laurier
  workers.
• It is suggested that severance is determined by
  three factors Age, length of service, pay.
• Bill smith, a Western employee, is offered a 5
  weeks severance pay when his employment is
  terminated.
• Based on the data provided by Laurier
  (Xr19-05.xls) about severance offered to 50 of
  its employees in the past, answer the following
  questions

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Problem 7 - continued

• Determine the regression equation. Interpret the
  coefficients.
• Comment on how well the model fits the data.
• Do all the independent variables belong in the
  model?
• Does Laurier meet its obligation to Bill Smith?

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Problem 8

• A linear regression model for life longevity
• Insurance companies are interested in predicting
  life longevity of their customers.
• Data for 100 deceased male customers was
  collected, and a regression model run.
• The model studied was
• Longevity b0b1MotherAgeb2FatherAgeb3GrandMb
  4GrandFe

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Problem 8
Coefficient of determination
The equation
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Problem 8
Overall usefulness H0 all bi 0 H1 At least
one bi 0 F Significance p value
4.86(10-27) Reject H0. The model is useful.
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Problem 8
Mothers age and fathers age at death have
strong linear relationships to an Individuals
age at death. Grandparents age at death are not
good predictors of an individuals age at
death.
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Chapter 18.2

• Dummy variables help include qualitative data in
  a regression model.
• If qualitative data can be categorized by n
  categories, there are n-1 dummy variables needed
  to express all the categories.
• Dummy variables take on the values 0 or 1.
• Xi 0 if the data point in question does not
  belong to category i
• Xi 1 if the data point in question belongs to
  category i.

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Problem 9

• In problem 6 we studied the relationship between
  age of welding machines and breakdown costs.
• This study was expanded. It is now including also
  lathe machine and stamping machines. See Data
  file. Code for machine type 1Welding 2Lathe
  3Stamping
• Answer the following
• Develop a regression model
• Interpret the coefficient
• Can we conclude that welding machines cost more
  to repair than stamping machine.
• Predict the monthly cost to repair an 85 month
  old lathe machine

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Problem 9

• First we need to prepare the input data

Original data
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Problem 9

• Run the multiple regression

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Problem 9
Cost119.252.538Age-11.755W-199.37L Repair cost
increase on the average by 2.53 a month. The
monthly repair cost for a welding machine is
11.75 lower than for a stamping machine of the
same age. However, this result is not significant
p value.55). There is insufficient evidence in
the sample to support the hypothesis that there
is any difference between repair costs of welding
machines and stamping machines. The monthly
repair cost for a lathe machine is 199.37 lower
than for a stamping machine of the same age. This
result is significant.

• Run the multiple regression

Note the reference line (for the stamping
machine) Cost119.252.538Age
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Chapter 15

• We test the hypotheses that a set of data belongs
  to certain distributions
• The multinomial distribution
• The normal distribution
• We also study whether two variables are dependent
  or not.
• We apply a tool called a Chi-squared test

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The multinomial experiment

• The multinomial experiment is an extension of the
  binomial experiment.
• Characteristics
• There are n independent trials.
• Each trial can result in one of k possible
  outcomes.
• There is a probability of a type k success (pk)
  in each trial.
• We test whether the sample gathered support the
  hypothesis that p1, p2,,pk are equal to
  specified values. The test is called The
  goodness of fit test.

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Problem 1

• To determine whether a single die is balanced, or
  fair, the die was rolled 600 times. (See
  Xr15-09.xls).
• Is there sufficient evidence at 5 significance
  level to allow you to conclude that the die is
  not fair?

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Problem 1

• The hypothesis
• H0 p1 p2 p6 1/6H1 At least one p is not
  1/6.
• Build a rejection Region
• In our case c2gtc2a,5

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Problem 1

• We calculate c2 as follows
• In our casee1e2e6600(1/6)100

From the file we havef1114 f292
f384f4101 f5107 f6103
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Contingency table

• Here we test the relationship between two
  variables. Are they dependent?
• We build a contingency table and a Chi-Square
  statistic

Variable/ Category 2
c columns
Variable/ Category 1
r rows
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A Sample ProblemContingency table

• Type of music vs. geographic location
• A group of 30-years-old people is interviewed to
  determined whether the type of music is somehow
  related to the geographic location of their
  residence.
• From the data presented can we infer that music
  preference is affected by the geographic
  location? Use (a.10).
• H0 Type of music and geographic location are
  independent.
• H1 Type of music and geographic location are
  dependent.

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A Sample problem contd.
Rock R B Country Classical
Northeast 140 32 5 18
South 134 41 52 8
West 154 27 8 13
195 235 202
428 100 65
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632

• e11(195)(428)/632129.59 e12(195)(100)/63230.
  85
• e23(235)(65)/63224.16
• c2 (140-129.59)2/129.59(52-24.16)2/24.1664
  .92
• c2.10,(3-1)(4-1) 10.64 64.92gt10.64.
• Reject the null hypothesis. Type of music and
  geographic location are not independent.

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A Sample problem contd.

• Using data analysis Plus

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Chi squared test for normality

• Hypothesize on m and s (mm0 and ss0).
• Divide the Z interval into equal size
  sub-intervals. i.e. (2, 1) (-1,0) (0,1)
  (1,2)
• Determine the corresponding probabilities covered
  by each subinterval. i.e. p1P(Zlt-2)
  p2P(-2ltZlt-1)
• Translate the Z scores to the associated X
  values. i,.e. x1m0(-2)s0 x2m0(-1)s0
• Find the actual frequency for each subinterval
  i.e. f1 - for the interval below x1 f2 - for
  the interval (x1,x2)
• Calculate the expected frequency for each
  interval
• e1 np1 e2 np2
• Build a Chi squared statistic and perform the
  test