Univariate Statistics

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Univariate Statistics

• Business Research Methods
• BUAD 259
• Penwell

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Key Topics

• Hypothesis Testing
• Types of Error
• Decisions regarding type of test to be used
• Three Univariate techniques

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Hypothesis Testing

• Hypothesis - an unproven proposition or
  supposition that tentatively explains certain
  facts, relationships, or phenomena
• Research Hypothesis - A proposition that is
  empirically testable
• Null Hypothesis (H0)- a testable hypothesis that
  presupposes that any differences observed are due
  to random error
• Alternative hypothesis (H1)- the hypothesis that
  observed differences are not due to random error

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Hypotheses
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Hypothesis Testing Procedure

• Question - Does the sample mean deviate from the
  mean of the hypothesized sampling distribution
  by a large enough value to conclude that it
  probably does not belong to that sampling
  distribution, if the null hypothesis is true?
• Significance Level (Alpha) - A probability level,
  set by the researcher, that is considered too low
  to accept the null hypothesis as true.
• ? (Alpha) .05 five times out of one hundred
• ? (Alpha) .01 one time out of one hundred

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Sampling Distribution
Population N 100,000
Samples
Sampling Distribution
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Setting Confidence Levels
Recall that the error term can be estimated by
Or, as it known, the Standard Error of the Mean
A confidence interval can then be estimated by
Where ZCL Z at a specified confidence level
For ? (Alpha) .05 ZCL 1.96 For ? (Alpha)
.01 ZCL 2.58
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Confidence Levels
? can be one-tailed or two-tailed if ? is
two-tailed then ? .025 at each end
ZCL 1.96
ZCL -1.96
?CL1001.96(S/?n)
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Type I and Type II Errors
Beta can be calculated
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Choosing the right technique

• What question are you asking?
• Central tendency
• Frequency
• Dispersion or distribution
• Number of variables involved?
• Scales Used?
• Test for Differences?
• Measure Association?

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Number of variables

• Univariate analysis - assesses the statistical
  significance of a hypothesis about a single
  variable
• Bivariate analysis - used to simultaneously test
  for differences or measures of association
  between two variables at the same time.
• Multivariate analysis - the simultaneous
  investigation of more than two variables

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Measurement Scale
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Parametric Vs. Non-parametric

• Parametric procedures require
• Interval or Ratio Scales
• Population is normally distribution (Bell Curve)
• Non-Parametric procedures required when
• Ordinal or Nominal Scales used
• Normality can not be assumed

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Univariate Parametrics

• Students t test
• When n is less than 30
• Degrees of Freedom n-1
• T-distribution (Table 3 p. 715)
• Set Confidence Intervals
• Compare with sampled observation

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Univariate Non-Parametrics

• Hypothesis test of a proportion (Ratio Scale)
• Used to estimate population proportions (?) based
  on a sample proportion (p).
• Like a t-test can set confidence levels
• Sp Standard error of proportion
• q not p 1 - p
• Requires large sample size

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Univariate Non-Parametrics

• Chi Square Test for Goodness of Fit
• Used to test Frequency Distributions
• Compares Observed (O) to Expected (E)
• Uses (k-1) as degrees of freedom (k think
  categories)
• Use Chi Square Distribution (Table 4, p.716)

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Summary

• Hypothesis testing
• Null Hypothesis v. Alternative Hypothesis
• Error
• Type I (?) Type II (?)
• Decisions
• Number of Variable
• Scale Used
• Parametric v. Non-parametric (Distribution Free)
• Three Univariate Techniques
• T-test
• Chi Square
• Z-test of proportion