INTRODUCTION TO HYPOTHESIS TESTING

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INTRODUCTION TO HYPOTHESIS TESTING

• From R. B. McCall, Fundamental Statistics for
  Behavioral Sciences, 5th edition, Harcourt Brace
  Jovanovich Publishers, New York 1990

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OUTLINE

• Population parameters - sample statistics
• How to test hypotheses - Null Hypothesis H0
• 2 examples to illustrate - the normal curve
• General Procedures - Assumptions
• General Procedures - Hypotheses
• General Procedures - Significance level
• General Procedures - Decision rules
• General Procedures - Reject/not reject H0
• General Procedures - Possible errors

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POPULATION PARAMETERS

• Assume that we know that, in the entire
  population, non-dragged subjects can correctly
  recall, on average, 7 of 15 learned nouns, with a
  standard deviation of 2.
• Thus, population parameters ?7, ?2
• We also have sample statistics For a sample of n
  subjects, we have the mean X and the standard
  deviation s.

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QUESTION HYPOTHESES

• Will subjects perform differently if they are
  given the drug physostigmine?
• Null Hypothesis H0 Drug will have no effect on
  performance.
• Alternative Hypothesis H1 Drug will have some
  effect on performance (either positive or
  negative two-tailed test).

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EXAMPLE 1 SINGLE CASE

• Assume one subject took drug and correctly
  recalled 11 nouns.
• Reject H0 if the subjects score falls into the
  most extreme ?5 of the distribution of
  non-drugged subjects. Score could fall in the
  extreme low ?/22.5 or extreme high 2.5, so
  test against ?/22.5 probability.

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EXAMPLE 1 (continued)

• Translate his/her score into a z score (z score
  is called the standard normal deviate has mean0
  and standard deviation1)
• z(Xi-?)/? (11-7)/2 2.00
• In this case, the z score is 2 standard
  deviations above ?.
• Look up the Table for Normal Curve to determine
  if H0 can be rejected.

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EXAMPLE 1 (continued)

• Here is a pictorial representation of the
  situation

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Proportions of area under Normal Curve
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2
... ... ... ... ... ... ... ... ... ...
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1
... ... ... ... ... ... ... ... ... ... ...
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
P(zlt2) 0.5 0.4772 0.9772 P(z2) 0.0228
lt 0.025. Thus, we must reject H0.
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EXAMPLE 2 SINGLE GROUP

• Assume 20 subjects took drug and correctly
  recalled 8.3 nouns.
• Reject H0 if the subjects score falls into the
  most extreme ?5 of the distribution of
  non-drugged subjects. Score could fall in the
  extreme low 2.5 or extreme high 2.5, so test
  against 2.5 probability.

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EXAMPLE 2 (continued)

• Translate groups score into a z score
• z(X-?) / ?X (X-?)/?/vn
• (8.32-7)/(2/v20) 2.95
• In this case, z score is 2.95 standard deviations
  above ?.
• Look up the Table for Normal Curve to determine
  if H0 can be rejected.

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Proportions of area under Normal Curve
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2
... ... ... ... ... ... ... ... ... ...
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1
... ... ... ... ... ... ... ... ... ... ...
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
P(zlt2.95) 0.5 0.4984 0.9984 P(z2.95)
0.0016 lt 0.025. Thus, reject H0.
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ASSUMPTIONS

• Assumptions are statements of circumstances in
  the population and the samples that the logic of
  the statistical process requires to be true, but
  that will not be proved or decided to be true.
• In Example 2, two assumptions were made
• The 20 subjects that the drug was administered to
  were randomly and independently selected from the
  non-drugged population.
• The sampling distribution of the mean is normal
  in form.

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HYPOTHESES

• Hypotheses are statements of circumstances in the
  population and the samples that the statistical
  process will examine and decided their likely
  truth or validity.
• Null Hypothesis H0 The observed sample mean is
  computed on a sample drawn from a population with
  ?7 that is, the drug has no effect.
• Alternative Hypothesis H1 The observed sample
  mean is computed on a sample drawn from a
  population with ??7 that is, drug has some
  effect (either positive or negative two-tailed
  test).

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SIGNIFICANCE LEVEL

• The significance level (or critical level),
  symbolized by ? (alpha), is the probability value
  that forms the boundary between rejecting and not
  rejecting the null hypothesis.
• Usually, ?0.05. If H0 can be rejected, the
  result is said to be significant at the 0.05
  level.
• This is sometimes written plt0.05, where p
  stands for the probability that H0 is true.

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DECISION RULES
Decision rules are statements, phrased in terms
of the statistics to be calculated, that dictate
precisely when the null hypothesis H0 will be
rejected and when it will not. In our case, we
used the following If the observed sample mean
deviated from the population mean to an extent
likely to occur in the non-drugged population
less than 5 of the time, we reject H0. Notice
that we decided on a two-tailed test BEFORE we
took a sample of 20 subjects. From the Table,
p lt 0.05 corresponds to an extreme tail area of
0.025, or a cumulative area of (1-0.025 0.975)
0.500 0.475, which corresponds to ? z? 1.96.
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
... ... ... ... ... ... ... ... ... ...
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
... ... ... ... ... ... ... ... ... ... ...
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YOU CAN ONLY REJECT OR NOT REJECT H0

• The statistical procedure is designed to test
  only one hypothesis, H0. Thus, depending on the
  results, you can only reject, or not reject, the
  Null Hypothesis H0.
• If pgt?, the only thing you can do is not reject
  H0. You cannot accept the Null Hypothesis H0. In
  our examples, the drug may still have an effect,
  but the effect may be very small. You may need a
  larger sample to observe an effect.
• If p?, the only thing you can do is reject H0.
  You cannot accept the Alternative Hypothesis H1.
  The decision to reject H0 is not equivalent to
  accepting H1.

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POSSIBLE ERRORS
ACTUAL --gt SITUATION DECISION H0 IS TRUE H0 IS FALSE
REJECT H0 Type I error p ? Correct rejection p 1 - ? (POWER)
DO NOT REJECT H0 Correct non-rejection p 1 - ? Type II error p ?
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SUMMARY - I

• Adopt appropriate decision rules.
• State, examine and justify the assumptions.
• State the Null Hypothesis H0 and the Alternative
  Hypothesis H1.
• Translate the raw scores into z scores.
• Look up the normal distribution Tables to reject
  or not reject H0.

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SUMMARY - II

• This lecture introduces the difference between
  population parameters and sample statistics, the
  difference between assumptions and hypotheses,
  and the basic ideas behind designing experiments
  to test the null hypothesis. Two concrete
  examples were used to illustrate decision rules,
  significance level, and possible errors in
  interpreting the data.