Quantitative Business Methods for Decision Making

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Quantitative Business Methods for Decision Making

• Estimation and Testing of Hypotheses

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Lecture Outlines

• Estimation
• Confidence interval for estimating means
• Confidence interval for predicting a new
  observation
• Confidence interval for estimating proportions

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Lecture Outlines (cont)

• Hypothesis Testing
• Null and alternative hypotheses
• Decision rules (Tests) and their level of
  significance
• Type I and Type II errors
• Tests of hypotheses for comparing means
• Tests of hypotheses for comparing proportions

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Estimating a Population Mean

• Population mean is estimated by , the
  sample mean
• Standard error of , i.e.
• will decrease as n gets large.

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Confidence Interval for Estimating if
is known

• With a 95 degree of confidence is
  estimated within ( )
• written as Or more accurately
• by

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Confidence Interval for if is not known

• Use instead of ,
• remember , and
• t is 95th percentile of the t
• distribution with (n-1) degrees of freedom.

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An Illustration

• Suppose n 26. Then degrees of freedom
• (d.f.) n-1 25.
• A two-sided degree of C.I. is computed
• by
• But, for a one-sided 95 C.I. , t 1.711
  instead
• of 2.064

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Assumptions and Sample Size forEstimation of
the mean

• The population should be normally (at least
• close to) distributed. If skew, then median is
• an appropriate measure of the center than the
• mean.
• To estimate mean with a specified margin of
• error (m.e.), take a random sample of size n
• large size.

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Prediction Interval for a New Observation on X
Prediction Interval for a new observation is
given by
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Confidence Interval for a Population Proportion

• Let denote the proportion of items in a
• population having a certain property
• An estimate of is the binomial
• proportion , What is ?
• For a C.I. for , use

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Confidence Intervals for the Proportion (cont)

• For estimating ,t is the percentile of the
• t-distribution with (equivalently,
• percentile of the standard normal
• distribution), and s.e. of p is

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Hypotheses Testing

• The hypothesis testing is a methodology for
  proving or disproving researchers prior
• beliefs.
• Statements that express prior beliefs are
• framed as alternative hypotheses.
• Complementary statement to an
• alternative hypothesis is called null
• hypothesis.

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Null and Alternative
Ha Researchers belief that are to be tested
(alternate hypothesis) H0 Complement of Ha
(Null hypothesis)
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Statistical Decision
A decision will be either Reject H0 (Ha is
proved) or Do not reject H0 (Ha is not proved)
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Hypothesis Testing Methodology for the mean

• Depending upon what an investigator
• believes a priori, an alternative hypothesis
• is formulated to be one of the followings
• 1.
• 2.
• 3.

one-sided
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A Test Statistic

• Regardless of what an alternative hypothesis
• about the mean is formulated, the decision
• rule is defined by a t- statistic

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Decision Rules for Testing Hypotheses About the
Mean
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Type I and Type II Errors
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Comparing Two Means

• The reference number ? is a specified amount for
  comparing the
• difference between two means. There are two
  distinct practical
• situations resulting in samples on X and Y.

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Two Sampling Designs

• Paired Sample
• Two independent Samples

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Paired Sample

• Two variables X and Y are observed for each unit
  in the sample to measure the same aspect but
  under two different conditions.
• Thus, for n randomly selected units, a sample of
  n pairs (X, Y) is observed.
• Compute differences X1-Y1 d1, X2-Y2 d2,
  etc. and then mean
• Compute Sd of differences

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Paired Samples (cont)

• Compute
• t-statistic

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Paired Samples (cont)

• Reject H0 if absolute value of t-statistic is
  more than
• the desired percentile of the t-distribution.
• Alternatively, find the p value of the
  t-statistics and
• reject H0 if the p value is less than the desired
  
• significance level.

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Two Independent (Unpaired) Samples

• Populations of variables X and Y (for
• example, males salary X and females
• salary Y).
• Take samples independently on X and Y.
• Compute
• Compute pooled standard deviation

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Unpaired Samples (cont)

• Compute
• Finally, compute
• t-statistic
• Use p value to reach a decision

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Comparing Proportions

• To estimate in a 95 C.I.,
• compute,

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Comparing Two Proportions

• For testing hypothesis about the difference
  , compute
• 
  and
• t-statistic