Hypothesis Testing And Univariate Analysis

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Hypothesis Testing And Univariate Analysis

• Chapter 15

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Preface

• For market researchers, the scientific inquiry
  translates into a desire to ask questions about
  the nature of relationships that affect behavior
  within markets
• The willingness to formulate hypothesis capable
  of being tested to determine
• What relationships exists
• When and where these relationships hold
• This chapter will extend the research process to
  include the
• Testing of relationships
• Formulation of hypotheses
• Making of inferences

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Formulating Hypotheses

• The Objective of the study, with associate
  hypotheses, should be stated as clearly as
  possible and agreed upon at the outset.
• Objectives and hypotheses shape and mold the
  study, they determine
• the kinds of question to be asked
• The measurement scales for the data to be
  collected
• The kinds of analysis that will be necessary
• Actual research projects almost always formulate
  and test new hypotheses during the project. It is
  both acceptable and desirable.
• The new hypothesis can be supported or not
  supported by the data or it can be neither
  supported nor rejected be the data.

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Formulating Hypotheses cont.

• In a typical survey project, the analyst may
  alternate between searching the data and
  formulating hypotheses.
• Three practices of survey analysts (Selvin and
  Stuart 1996)
• Snooping
• -the process of searching through a body of data
  and looking at many relations in order to find
  those worth testing
• Fishing
• -the process of using the data to choose which of
  a number of predesignated variables to include in
  an explanatory model
• Hunting
• -the process of testing each of a predesignated
  set of hypotheses with the data
• This investigative approach is reasonable for
  basic research but my not be practical for
  decisional research.

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Formulating Hypotheses cont.

• What is a hypothesis?
• A hypothesis is an assertion that variables
  (measured concepts) are related in a specific way
  such that this relationship explains certain
  facts or phenomena.
• Outcomes are predicted if a specific course of
  action is followed.
• Hypotheses are often stated as research questions
  when reporting either the purpose of the
  investigation or the findings.
• Hypotheses must be empirically testable.
• Hypotheses may be stated informally as research
  questions, or more formally as a set of
  alternative hypotheses, or in a testable form
  known as a null hypothesis, which states that
  there is no relationship between the variables to
  be examined
• Research questions are not empirically testable,
  but aid in the important task of directing and
  focusing the research effort.

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EXHIBIT 15.1 Development of a Research Question
for Mingles

• Mingles is an exclusive restaurant specializing
  in seafood prepared with a light Italian flair.
  Barbara C., the owner and manager, has attempted
  to create an airy contemporary atmosphere that is
  conducive to conversation and dining enjoyment.
  In the first three months, business has grown to
  about 70 percent of capacity during dinner hours.
  Barbara developed a questionnaire that asked,
  among other things, How would you rate the value
  of Mingles food for the price paid? The response
  form provided five answers with boxes for the
  respondent to check Much Better A Little
  Better About A Little Worse Much Worse Than
  Expected Than Expected Average Than Expected
  Than Expected Customers responses were coded
  using a scale of 2, 1, 0, 1, and 2.
• When tabulated, the average response was found
  to be 0.89 with a sample standard deviation of
  1.43. The research question asks if Mingles is
  perceived as being better than average when
  considering the price and value of the food.
• The research question that Barbara C. has
  developed is How satisfied are Mingles customers
  with the concept, service, food, and value?

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Formulating Hypotheses cont.

• Null hypotheses (H0) are statements identifying
  relationships that are statistically testable and
  can be shown not to hold (nullified).
• The logic of the null hypothesis is that we
  hypothesize no difference, and we reject the
  hypotheses if a difference is found.
• A null hypothesis may also be used to specify
  other types of relationships that are being
  tested, such as the difference between two
  groups, or the ability of a specific variable to
  predict a phenomenon such as sales or repeat
  business.
• Alternative hypotheses may be considered to be
  the opposite of the null hypotheses.
• The alternative hypothesis makes a formal
  statement of expected difference, and may state
  simply that a difference exists or that a
  directional difference exists, depending upon how
  the null hypothesis is stated.

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Making Inferences

• Once the data have been tabulated and summary
  measures calculated, we often will make
  inferences about the nature of the population and
  ask a multitude of questions.
• we would want to know about the underlying
  associated variables that influence preference
  purchase, or use (such as color, ease of opening,
  accuracy in dispensing the desired quantity,
  comfort in handling, etc.)
• The broad objective of testing hypotheses
  underlies all decisional research. Sometimes the
  population as a whole can be measured and
  profiled in its entirety.
• We cannot measure everyone in the population but
  instead must estimate the population using a
  sample of respondents drawn from the population.

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The Relationship Between a Population, a Sampling
Distribution, and a Sample
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The Relationship Between theSample and the
Sampling Distribution
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Acceptable Error in Hypothesis Testing

• A question that continually plagues analysts is,
  What significance level should be used in
  hypothesis testing?
• The significance level refers to the amount of
  error we are willing to accept in our decisions
  that are based on the hypothesis test.
• In hypothesis testing the sample results
  sometimes lead us to reject H0 when it is true.
  This is a Type I error.
• On other occasions the sample findings may lead
  us to accept H0 when it is false. This is a Type
  II error.

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Types of Error in Making a Wrong Decision

• 1. A Type I error occurs when we incorrectly
  conclude that a difference exists. The
  probability of this is expressed as a, the
  probability that we will incorrectly reject H0,
  the null hypothesis, or hypothesis of no
  difference.
• 2. A Type II error occurs when we accept a null
  hypothesis when it is in reality false (we find
  no difference when a difference really does
  exist).
• 3. We correctly retain the null hypothesis (we
  could also say we tentatively accept or that it
  could not be rejected). This is equal to the area
  under the normal curve less the area occupied by
  a, the significance level.
• 4. The power of the test is the ability to reject
  the null hypothesis when it should be rejected
  (when false). Because power increases as abecomes
  larger, esearchers may choose an a of .10 or
  even .20 to increase power. Alternatively, sample
  size may be increased to increase power.
  Increasing sample size is the preferred option
  for most market researchers.

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Power of a Test

• The power of a hypothesis test is defined as 1
  ß, or one minus the probability of a Type II
  error. This means that the power of a test is its
  ability to reject the null hypothesis when it is
  false
• The power of a statistical test is determined by
• 1. acceptable amount of discrepancy between the
  tested hypothesis and the true situation
• 2. Power is also increased by increasing the
  sample size (which decreases the confidence
  interval).

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SELECTING TESTS OF STATISTICAL SIGNIFICANCE

• Tests are performed on interval or ratio data
  using what is known as parametric tests and
  include such techniques as the F, t, and z tests.
• Nonparametric methods are often called
  distribution-free methods because the inferences
  are based on a test statistic whose sampling
  distribution does not depend upon the specific
  distribution of the population from which the
  sample is drawn
• to determine an appropriate test for a particular
  set of data depends on
• the level of measurement of the data
• the number of variables that are involved
• for multiple variables, how they are assumed to
  be related.

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PARAMETRIC AND NONPARAMETRIC ANALYSIS

• The process of making inferences from the sample
  to the populations parameters is called
  parametric analysis.
• Parametric methods rely almost exclusively on
  either interval or ratio scaled data.
• nonparametric methods may be used to compare
  entire distributions that are based on nominal
  data.
• Other nonparametric methods use an ordinal
  measurement scale test for the ordering of
  observations in the data set.
• Problems that may be solved with parametric
  methods may often be solved by a nonparametric
  method designed to address a similar question.

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Univariate Analyses of Parametric Data

• Marketing researchers are often concerned with
  estimating parameters of a population. In
  addition, many studies go beyond estimation and
  compare population parameters by testing
  hypotheses about differences between them.
• Very often, the means, proportions, and variances
  are the summary measures of concern.

The Confidence Interval The confidence interval
is a range of values with a given probability
(.95, .99, etc.) of including the true population
parameter.
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Univariate Hypothesis Testing of Means

• Population Variance Is Known
• The z statistic describes probabilities of the
  normal distribution and is the appropriate tool
  to test the difference between µ, the mean of the
  sampling distribution, and X, the sample mean
  when the population variance is known.
• The z statistic may be used only when the
  following conditions are met

• Individual items in the sample must be drawn in
  a random manner.
• The population must be normally distributed. If
  this is not the case, the sample must be large (gt
  30), so that the sampling distribution is
  normally distributed.
• The data must be at least interval scaled.
• The variance of the population must be known.

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Population Variance Is Known

• The traditional hypothesis testing approach is as
  follows

• The null hypothesis (H0) is specified that there
  is no difference between µ and x -. Any observed
  difference is due solely to sample variation.
• The alpha risk (Type I error) is established
  (usually .05).
• The z value is calculated by the appropriate z
  formula
• The probability of the observed difference having
  occurred by chance is determined from a table of
  the normal distribution (Appendix A, Table A.1).
• If the probability of the observed differences
  having occurred by chance is greater than the
  alpha used, then H0 cannot be rejected and it is
  concluded that the sample mean is drawn from a
  sampling distribution of the population having
  mean µ.

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Univariate Hypothesis Testing of Means

• Population Variance Is Unknown
• Researchers rarely know the true variance of the
  population, and must therefore rely on an
  estimate of s2, namely, the sample variance s2.
  With this variance estimate, we may compute the t
  statistic.
• The appropriate t distribution to use in an
  analysis is determined by the available degrees
  of freedom. In univariate analyses, the available
  degrees of freedom are n 1.

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Univariate Analysis of Categorical dataThe
Chi-Square Goodness of Fit Test

• Chi-square analysis (?2) can be used when the
  data identifies the number of times or frequency
  that each category of a tabulation or
  cross-tabulation appears. Chi-square is a useful
  technique for testing the following
• we compute a measure (chi-square) of the
  variation between actual and theoretical
  frequencies, under the null hypothesis that there
  is no difference between the model and the
  observed frequencies. We may say that the model
  fits the facts.

1. Determining the significance of sample
deviations from an assumed theoretical
distribution that is, determining whether a
certain model fits the data. This is typically
called a goodness-of-fit test. 2. Determining the
significance of the observed associations found
in the cross-tabulation of two or more variables.
This is typically called a test of independence.
If the measure of variation is high, we reject
the null hypothesis at some specified alpha risk.
If the measure is low, we accept the null
hypothesis that the models output is in
agreement with the actual frequencies.
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Univariate Analysis Test of a Proportion

• The univariate test of proportions, like the
  univariate test of means, compares the population
  proportion to the proportion observed in the
  sample. For a sample proportion, p
• where Sp, the estimated standard error of the
  proportion, is given by

z standard normal value p the sample
proportion of successes q (1 - p) the sample
proportion of failures n sample size
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Summary

• This chapter has introduced the basic concepts of
  formulating hypotheses, hypothesis testing, and
  making statistical inferences in the context of
  univariate analysis.
• A hypothesis is a statement that variables
  (measured constructs) are related in a specific
  way. The null hypothesis, H0, is a statement that
  no relationship exists between the variables
  tested or that there is no difference.
• Statistics are based on making inferences from
  the sample of respondents to the population of
  all respondents by means of a sampling
  distribution.
• A Type I Error occurs when a true H0 is rejected
  (there is no difference, but we find there is). A
  Type II Error occurs when we accept a false H0
  (there is a difference, but we find that none
  exists).

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Summary (cont.)

• The power of a test was explained as the ability
  to reject H0 when it should be rejected.
• Selecting the appropriate statistical technique
  for investigating a given relationship depends of
  the level of measurement (nominal, ordinal, or
  interval) and the number of variables to be
  analyzed.
• The choice of parametric versus nonparametric
  analyses depends on the analysts willingness to
  accept the distributional assumptions of
  normality and homogeneity of variances.
• Univariate hypothesis testing was demonstrated
  using the standard normal distribution
• Statistic (z) to compare a mean and proportion to
  the population values.
• The t-test was demonstrated as a parametric test
  for populations of unknown variance and samples
  of small size.
• The chi-square goodness-of-fit test was
  demonstrated as a nonparametric test of nominal
  data that make no distributional assumptions.