Business Statistics for Managerial Decision

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Business Statistics for Managerial Decision

• Farideh Dehkordi-Vakil

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ExampleRetail sales and floor space

• It is customary in retail operations to asses the
  performance of stores partly in terms of their
  annual sales relative to their floor area (square
  feet). We might expect sales to increase linearly
  as stores get larger, with of course individual
  variation among stores of the same size. The
  regression model for a population of stores says
  that
• SALES ?0 ?1? AREA ?

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ExampleRetail sales and floor space

• The slope ?1 is as usual a rate of change it is
  the expected increase in annual sales associated
  with each additional square foot of floor space.
• The intercept ?0 is needed to describe the line
  but has no statistical importance because no
  stores have area close to zero.
• Floor space does not completely determine sales.
  The term ? in the model accounts for difference
  among individual stores with the same floor
  space. A stores location, for example, is
  important.

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Estimation of the variance of the error terms, ?2

• The variance ?2 of the error terms ?i in the
  regression model needs to be estimated for a
  variety of purposes.
• It gives an indication of the variability of the
  probability distributions of y.
• It is needed for making inference concerning
  regression function and the prediction of y.

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Regression Standard Error

• To estimate ? we work with the variance and take
  the square root to obtain the standard deviation.
• For simple linear regression the estimate of ?2
  is the average squared residual.
• To estimate ? , use
• s estimates the standard deviation ? of the error
  term ? in the statistical model for simple linear
  regression.

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Regression Standard Error
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Inference in Regression Analysis

• The simple linear regression which is the basis
  for inference, imposes several conditions. We
  should verify these conditions before proceeding
  to inference.
• The conditions concern the population, but we can
  observe only our sample.
• In doing inference we act as if
• The sample is a SRS from the population.
• There is a linear relationship in the population.
• The standard deviation of the responses about the
  population line is the same for all values of the
  explanatory variable.
• The response varies Normally about the population
  regression line.

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Inference in Regression Analysis

• Plotting the residuals against the explanatory
  variable is helpful in checking these conditions
  because a residual plot magnifies patterns.
• A Normal quantile plot of the residuals can be
  used to check the Normality assumptions.

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Confidence Intervals and Significance Tests

• In our previous lectures we presented confidence
  intervals and significance tests for means and
  differences in means.In each case, inference
  rested on the standard error s of the estimates
  and on t or z distributions.
• Inference for the slope and intercept in linear
  regression is similar in principal, although the
  recipes are more complicated.
• All confidence intervals, for example , have the
  form
• estimate ? t Seestimate
• t is a critical value of a t distribution.

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Confidence Intervals and Significance Tests

• Confidence intervals and tests for the slope and
  intercept are based on the sampling distributions
  of the estimates b1 and b0.
• Here are the facts
• If the simple linear regression model is true,
  each of b0 and b1 has a Normal distribution.
• The mean of b0 is ?0 and the mean of b1 is ?1.
• The standard deviations of b0 and b1 are
  multiples of the model standard deviation ?.

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Confidence Intervals and Significance Tests
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Example Do wages rise with experience?

• Many factors affect the wages of workers the
  industry they work in, their type of job, their
  education and their experience, and changes in
  general levels of wages. We will look at a sample
  of 59 married women who hold customer service
  jobs in Indiana banks. The following table gives
  their weekly wages at a specific point in time
  also their length of service with their employer,
  in month. The size of the place of work is
  recorded simply as large (100 or more workers)
  or small. Because industry, job type, and the
  time of measurement are the same for all 59
  subjects, we expect to see a clear relationship
  between wages and length of service.

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Example Do wages rise with experience?
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Example Do wages rise with experience?
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Example Do wages rise with experience?
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Example Do wages rise with experience?

• Do wages rise with experience?
• The hypotheses are
• H0 ?1 0, Ha ?1 gt 0
• The t statistic for the significance of
  regression is
• The P- value is
• P(t gt 2.85) lt .005
• The t distribution for this problem have n-2 57
  degrees of freedom.
• Conclusion
• Reject H0 There is strong evidence that the
  mean wages increase as length of service
  increases.

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Example Do wages rise with experience?

• A 95 confidence interval for the slope ?1 of the
  regression line in the population of all married
  female customer service workers in Indiana bank
  is
• The t distribution for this problem have n-2
  57 degrees of freedom

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Inference about Correlation

• The correlation between wages and length of
  service for the 59 bank workers is r 0.3535.
  This appears in the Excel out put, where it is
  labeled Multiple R.
• We expect a positive correlation between length
  of service and wages in the population of all
  married female bank workers. Is the sample result
  convincing that this is true?
• This question concerns a new population
  parameter, the population correlation. This is
  correlation between length of service and wages
  when we measure these variables for every member
  of the population.

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Inference about Correlation

• We will call the population correlation?.
• To assess the evidence that ? . 0 in the bank
  worker population, we must test the hypotheses
• H0 ? 0
• Ha ? gt 0
• It is natural to base the test on the sample
  correlation r.
• There is a link between correlation and
  regression slope.
• The population correlation ? is zero, positive,
  negative exactly when the slope ??1 of the
  population regression line is zero, positive, or
  negative.

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Inference about Correlation
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Example Do wages rise with experience?

• The sample correlation between wages and length
  of service is r 0.3535 from a sample of n 59.
  
• To test
• H0 ? 0
• Ha ? gt 0
• Use t statistic

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Example Do wages rise with experience?

• Compare t 2.853 with critical values from the t
  table with n - 2 57 degrees of freedom.
• Conclusion
• P( t gt 2.853) lt .005, therefore we reject H0.
  There is a positive correlation between wages and
  length of service.