Rejecting Chance Testing Hypotheses in Research

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Chapter 21

• Rejecting Chance - Testing Hypotheses in Research

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Thought Questions
1, page 371
In the courtroom, juries must make a decision
about the guilt or innocence of a defendant.
Suppose you are on the jury in a murder trial.
It is obviously a mistake if the jury claims the
suspect is guilty when in fact he or she is
innocent. What is the other type of mistake the
jury could make? Which is more serious?
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Thought Questions
2, page 371
Suppose exactly half, or 0.50, of a certain
population would answer yes when asked if they
support the death penalty. A random sample of
400 people results in 220, or 0.55, who answer
yes. The Rule for Sample Proportions tells us
that the potential sample proportions in this
situation are approximately bell-shaped, with
standard deviation of 0.025. Using the formula
on page 136, find the standardized score the for
observed value of 0.55. Then determine how often
you would expect to see a standardized score at
least that large or larger.
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Thought Question 2 Bell-Shaped Curve of Sample
Proportions (n400)
mean 0.50 S.D. 0.025
2.5
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Thought Questions
3, page 371
Suppose you are interested in testing a claim you
have heard about the proportion of a population
who have a certain trait. You collect data and
discover that if the claim is true, the sample
proportion you have observed falls at the 99th
percentile of possible sample proportions for
your sample size. Would you believe the claim
and conclude that you just happened to get a
weird sample, or would you reject the claim? What
if the result was at the 85th percentile? At the
99.99th percentile?
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Thought Question 3 Bell-Shaped Curve of Sample
Proportions (n400)
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Case Study FingerprintsStudy Results

• 186 heterosexual men and 66 homosexual men were
  studied.
• 26 (14) heterosexual men showed the leftward
  asymmetry.
• 20 (30) of the homosexual men showed the
  leftward asymmetry.

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Case Study FingerprintsThe Question
If there really is no clear relationship, how
likely would we be to observe sample results of
this magnitude or larger, just by chance?
p-value
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The Four Steps of Hypothesis Testing

• Determining the Two Hypotheses
• Collecting and Summarizing the Data
• Determining How Unlikely the Test Statistic is if
  the Null Hypothesis is True
• Making a Decision

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The Null Hypothesis

• Case Study
• There is no relationship between sexual
  orientation and leftward asymmetry in men.
• The population proportions are the same.
• In general
• status quo
• no relationship
• no change
• etc.

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The Alternative Hypothesis or Research Hypothesis

• Case Study
• There is a relationship between sexual
  orientation and leftward asymmetry in men.
• The population proportions are different.
• In general
• not status quo
• relationship exists
• a change occurred

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Summarizing the Data
Chi-Squared 8.70 Statistically Significant
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P-Value

• The p-value is the probability of observing a
  relationship this extreme or more so in a sample
  of size 252, assuming that the null hypothesis is
  true.
• The smaller the p-value, the stronger the
  evidence against the null hypothesis.
• P-value for the chi-square test statistic (based
  on a 2x2 table) for this example is 0.003.

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Decision

• If we think the p-value is too low to believe the
  observed test statistic is obtained by chance
  only, then we would reject chance (reject the
  null hypothesis) and conclude that a
  statistically significant relationship exists
  (accept the alternative hypothesis).
• Otherwise, we fail to reject chance and do not
  reject the null hypothesis of no relationship.

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Typical Cut-off for the P-value

• Commonly, p-values less that 0.05 are considered
  to be small enough to reject chance.
• Some researchers use 0.10 or 0.01 as the cut-off
  instead of 0.05.

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Decision Errors Type I

• If we decide there is a relationship in the
  population
• This is an incorrect decision only if the null
  hypothesis is true.
• The probability of this incorrect decision is
  equal to the cut-off for the p-value.
• If the null hypothesis is true and the cut-off is
  0.05
• There really is no relationship and the extremity
  of the test statistic is due to chance.
• About 5 of all samples from this population will
  lead us to wrongly reject chance.

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Decision Errors Type II

• If we decide not to reject chance and thus allow
  for the plausibility of the null hypothesis
• This is an incorrect decision only if the
  alternative hypothesis is true.
• The probability of this incorrect decision
  depends on
• the magnitude of the true relationship,
• the sample size,
• the cut-off for the p-value.

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Power of a Test

• This is the probability that the sample we
  collect will lead us to reject the null
  hypothesis when the alternative hypothesis is
  true.
• The power is larger for larger departures of the
  alternative hypothesis from the null hypothesis
  (magn. of diff.)
• The power may be increased by increasing the
  sample size.

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Case Study I Chap. 20
Exercise and Pulse Rates

• Null Hypth. Mean resting pulse rate is the same
  for exercisers and nonexercisers.
• Alt. Hypth. Mean resting pulse rate is lower for
  exercisers than for nonexercisers.

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Case Study I Chap. 20
Exercise and Pulse Rates

• Can show that the standard error of the
  difference in sample means is 2.26.
• Thus the standardized difference is 9/2.263.98!
• Observing such a large standardized difference is
  a very rare event.
• The p-value is very small! (0.00003)
• What may we conclude?
• Recall, 95 CI for the difference in means is
  (4.4, 13.6).

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Key Concepts

• Decisions are often made on the basis of
  incomplete information.
• Four steps of hypothesis testing.
• P-values and statistical significance.
• Decision errors.
• Power of a test.