Statistical Reasoning and Type III Error

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Statistical Reasoning and Type I/II Error

• INCO 601
• Scott Titsworth

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An Approximate Normal Distribution

• Any distribution has measures of central
  tendency.
• Any distribution has measures of variance.

µ
s -1
s 1
s 2
s -2
s 3
s -3
s -4
s 4
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An Approximate Normal Distribution of Two Groups
A common question asked in the social sciences is
whether or not two groups differ.
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1
2
3
4
5
6
7
8
9
10
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Statistical Reasoning

• Hypotheses are used to test assumptions about
  data.
• Hypotheses come in variations of two forms.
• The Null Hypotheses
• The Alternate or Research hypothesis

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Example Hypotheses

• Null Hypothesis (H0)
• Means that µ1 µ2
• Students receiving public speaking training will
  have the same level of public speaking anxiety as
  those who do not receive public speaking
  training.

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Example Hypotheses

• Research Hypothesis (H1)
• Means that µ1 ? µ2
• Students receiving public speaking training will
  have different levels of public speaking anxiety
  from those who do not receive public speaking
  training.

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The Logic of Hypothesis Testing

• H0 is tested directly by statistical tools
• If the null hypothesis is rejected then
  researchers conclude that the research hypothesis
  is tenable.
• If the null hypothesis is not rejected the
  researchers conclude that no claim can be made
  about the differences between the groups. Fail
  to reject the null.

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How do we test H0?

• We use statistical analyses to determine if
  unusual patterns exist.
• Unusual is typically defined in the social
  sciences as something that would be unlikely to
  occur more than 5 of the time by chance.
• Specific conditions could alter our working
  definition of unusual being 5.
• This 5 chance corresponds to a probability of
  .05, which is the alpha level commonly used by
  researchers.

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An Approximate Normal Distribution

• The distribution for t-tests represents a normal
  distribution.
• We are only interested in t values that are
  unlikely to occur by chance.
• This shows a two tailed test.

Approximately 2.5 of t values are here.
µ
Approximately 2.5 of t values are here.
s -1
s 1
s 2
s -2
95 of the t values fall just inside 2 standard
deviations.
s 3
s -3
s -4
s 4
t -8.91
t 9.70
t 1.2
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An Approximate Normal Distribution

• An alternative to the two tailed test is the one
  tailed test.
• The one tailed test is used when testing a
  directional hypothesis.
• H1 Students receiving communication skills
  training will have less communication
  apprehension than students not receiving
  communication skills training.

Approximately 5 of t values are here.
µ
s -1
s 1
s 2
s -2
95 of the t values fall just inside 2 standard
deviations.
s 3
s -3
s -4
s 4
t 9.70
t 1.2
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An Approximate Normal Distribution of Two Groups
X 6.4 SD .548
X 5.4 SD .548
This area represents the area hidden by our
alpha level of .05.
T -2.887, p. lt .05
H0 is rejected
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An Approximate Normal Distribution of Two Groups
X 5.0 SD 2.55
X 6.4 SD 1.14
This area represents the area hidden by our
alpha level of .05.
T 1.121, p. gt .05
We fail to reject H0
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Lets See an Example

• Group 1 (Had Class)
• 8
• 7
• 7
• 6
• 7
• 7
• 7
• 7
• M 7
• SD .535

• Group 2 (No Class)
• 6
• 1
• 9
• 2
• 7
• 3
• 6
• 4
• M 4.75
• SD 2.71

X1-X22.5
T 2.30
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Two Issues Facing Researchers

• Type I Error
• Saying that groups differ when they actually do
  not differ.
• Setting the alpha level appropriately protects
  you against this type of error.
• Type II Error
• Saying that the groups do not differ when they
  actually do.
• To protect against Type II Error you need
  adequate power for your statistical test.

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Having Adequate Power

• Power is influenced by 4 things
• The alpha level set by the researcher
• Sample size
• The magnitude of error variance
• Effect size

Modifying the sample size provides the safest
means for controlling Type II error.
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Safe Ground for a Researcher