Topics in Analysis

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Topics in Analysis
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Figure 16.6 A General Procedure for Hypothesis
Testing
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Formulating the Hypothesis

• Null hypothesis A statement of the status quo,
  one of no difference or no effect. If the null
  hypothesis is not rejected, no changes will be
  made.
• An alternative hypothesis is one in which some
  difference or effect is expected. Accepting the
  alternative hypothesis will lead to changes in
  opinions or actions.
• In marketing research, the null hypothesis is
  formulated in such a way that its rejection leads
  to the acceptance of the desired conclusion. The
  alternative hypothesis represents the conclusion
  for which evidence is sought.

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Selecting the test

• One sample
• Z-test Uses the z distribution used when n 30
  and variance is known.
• t-test Uses the t distribution used when n and variance is unknown.
• Paired sample t-tests are used when data is
  collected on two variables from the same
  respondents.
• With two independent samples, use two sample
  t-test, chi-square, or F-test.
• ANOVA Used when data is gathered from more than
  two samples

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Testing Hypothesis of a Mean Example

• Rex hypothesizes that interns in an insurance
  firm make 2,750 in commissions.
• A survey (n100) shows a sample mean of 2,800
  and a standard deviation of 350.
• Does the survey sample statistic support or fail
  to support Rexs hypothesis?
• That is, is the difference of 50 sufficient
  enough to cast doubt on Rexs estimate?

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Since 1.43z falls between -1.96z and 1.96 z, we
ACCEPT the hypothesis
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The probability that our sample mean of 2,800
came from a distribution of means around a
population parameter of 2,750 is 95.
Therefore, we accept Rexs hypothesis.
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Testing Hypothesis of Proportions Example

• Alamo rent-a-car execs believe that Alamo
  accounts for 50 of all Cadillacs rented. To test
  this belief, a researcher identifies 20 airport
  rental car lots and records the of Cadillacs
  rented in a four hour period.
• 500 were observed and 30 were returned to
  Alamo. What are the implications of this finding
  for Alamo execs belief?

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Relationships

• A relationship is a consistent and systematic
  linkage between two variables.
• Presence
• means that a relationship exists
• determined by statistical tests (e.g.,
  correlation)
• Direction
• whether the relationship is positive or negative
• Strength of Association
• How consistent is the association? (strong,
  moderate, weak, nonexistent)
• determined by statistical methods

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Types of Relationships

• Nonmonotonic two variables are associated, but
  only in a very general sense.
• We know that the presence (or absence) of one
  variable is associated with the presence (or
  absence) of another variable. But, we dont know
  a direction of the relationship.
• Monotonic there is a general association and a
  direction can be discerned
• increasing monotonically -- as one variable
  increases so does the other
• age and cognitive development
• decreasing monotonically -- as one variable
  increases, the other one decreases
• age and parents involvement
• Note The relationships are not defined in terms
  of how much of a change in one variable will
  lead to a certain change in another variable.
• Linear Relationship Two variables have a
  straight line relationship
• Curvilinear Relationships The relationship
  between the two variables can be described using
  some curve.

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Analyze these relationships

• To place advertisements, a sporting goods retail
  store is studying readership of certain
  sections of the Sunday newspaper and age of the
  reader
• PBS, for fund raising purposes, examines
  ownership of a telephone answering machine and
  household income

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Analyze these relationships

• An auto service chain, for providing discounts to
  corporate customers, studies number of miles
  driven in company cars and need for service,
  oil change, etc.
• Ace hardware studies amount of do-it-yourself
  home repair kits and state of economy (e.g.,
  growth, recession)
• A resort in Jamaica studies plan to take a 5-day
  vacation and the exchange rate of the Jamaican
  .

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Cross Tabulations

• A technique for organizing data by groups,
  categories, or classes, thus facilitating
  comparisons.

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Cross-Tabulations

• Did you study for the midterm test? _Yes_ No
• How did you perform on the test?___ Pass___Fail

Do you see a relationship? Do you see the
presence of studying with the presence of
passing? Do you see the absence of passing
with the presence of not studying?
What type of relationship do you see here?
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Correlation

• Correlation coefficients
• an indexed number designed to fall between -1 and
  1.
• The size of the correlation coefficient indicates
  the strength of the association.
• The sign of the correlation coefficient indicates
  the direction.

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Perfect positive (all points fall on a straight
line)                                         
                                  
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More realistic example                       
                                                  
   
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Perfect negative                             
                                              
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More realistic example                       
                                                  
 
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No correlation                                
                                            
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More realistic example                       
                                                  
  
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Correlation
Positive Relationship
Negative Relationship
.81 to 1.0 Strong .61 to .80 Moderate .41 to
.60 Weak .21 to .40 Very Weak 0 to .19 None
-.81 to -1.0 Strong -.61 to -.80 Moderate -.41
to -.60 Weak -.21 to -.40 Very Weak 0 to
-.19 None
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Regression Analysis Uses

• Determine whether the independent variables
  explain a significant variation in the dependent
  variable whether a relationship exists.
• Determine how much of the variation in the
  dependent variable can be explained by the
  independent variables strength of the
  relationship.
• Determine the structure or form of the
  relationship the mathematical equation relating
  the independent and dependent variables.
• Helps in predict the values of the dependent
  variable.
• Control for other independent variables when
  evaluating the contributions of a specific
  variable or set of variables.

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Regression Model

• The basic regression equation is Yi Xi
  ei,
• Y dependent variable
• X independent variable
• intercept of the line
• slope of the line
• ei is the error term associated with the i th
  observation.
• Coefficient of determination. The strength of
  association is measured by the coefficient of
  determination, r 2. It varies between 0 and 1
  and signifies the proportion of the total
  variation in Y that is accounted for by the
  variation in X.