Introduction to Hypothesis Testing

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Chapter 8

• Introduction to Hypothesis Testing

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Name of the game

• Hypothesis testing
• Statistical method that uses sample data to
  evaluate a hypothesis about a population

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Experimental Hypotheses

• Experimental hypotheses describe the predicted
  outcome we may or may not find in an experiment.

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Order of Procedure

• State hyp about pop
• Before selecting a sample..use hyp to predict
  characteristics that sample should have
• Obtain random sample
• Compare sample data w/ prediction made in hyp

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Of interest to the researcher

• Did the treatment have any effect on the
  individuals
• Must be large (significant) differences in means

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New Statistical Notation

• The symbol for greater than is gt.
• The symbol for less than is lt.
• The symbol for greater than or equal tois .
• The symbol for less than or equal tois .
• The symbol for not equal to is ?.

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The Role of Inferential Statistics in Research
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Sampling Error

• Remember
• Sampling error results when random chance
  produces a sample statistic that does not equal
  the population parameter it represents.

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Setting up Inferential Procedures
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Null Hypothesis H0

• The null hypothesis describes the population
  parameters that the sample data represent if the
  predicted relationship does not exist.

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Alternative Hypothesis H1

• The alternative hypothesis describes the
  population parameters that the sample data
  represent if the predicted relationship exists.

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A Graph Showing the Existence of a Relationship
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A Graph Showing That a Relationship Does Not
Exist
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Interpreting Significant Results

• When we reject H0 and accept H1, we do not prove
  that H0 is false
• While it is unlikely for a mean that lies within
  the rejection region to occur, the sampling
  distribution shows that such means do occur once
  in a while

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Failing to Reject H0

• When the statistic does not fall beyond the
  critical value, the statistic does not lie within
  the region of rejection, so we do not reject H0
• When we fail to reject H0 we say the results are
  nonsignificant. Nonsignificant indicates that the
  results are likely to occur if the predicted
  relationship does not exist in the population.

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Interpreting Nonsignificant Results

• When we fail to reject H0, we do not prove that
  H0 is true
• Nonsignificant results provide no convincing
  evidenceone way or the otheras to whether a
  relationship exists in nature

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Errors in Statistical Decision Making
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Type I Errors

• A Type I error is defined as rejecting H0 when H0
  is true
• In a Type I error, there is so much sampling
  error that we conclude that the predicted
  relationship exists when it really does not
• The theoretical probability of a Type I error
  equals a

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Alpha a

• Probability that the test will lead to a Type I
  error
• Alpha level determines the probability of
  obtaining sample data in the critical region even
  though the null hypo is true

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Type II Errors

• A Type II error is defined as retaining H0 when
  H0 is false (and H1 is true)
• In a Type II error, the sample mean is so close
  to the m described by H0 that we conclude that
  the predicted relationship does not exist when it
  really does
• The probability of a Type II error is b

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Power

• The goal of research is to reject H0 when H0 is
  false
• The probability of rejecting H0 when it is false
  is called power

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Possible Results of Rejecting or Retaining H0
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Parametric Statistics

• Parametric statistics are procedures that require
  certain assumptions about the characteristics of
  the populations being represented. Two
  assumptions are common to all parametric
  procedures
• The population of dependent scores forms a normal
  distribution
• and
• The scores are interval or ratio.

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Nonparametric Procedures

• Nonparametric statistics are inferential
  procedures that do not require stringent
  assumptions about the populations being
  represented.

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Robust Procedures

• Parametric procedures are robust. If the data
  dont meet the assumptions of the procedure
  perfectly, we will have only a negligible amount
  of error in the inferences we draw.

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Predicting a Relationship

• A two-tailed test is used when we predict that
  there is a relationship, but do not predict the
  direction in which scores will change.
• A one-tailed test is used when we predict the
  direction in which scores will change.

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The One-Tailed Test
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One-Tailed Hypotheses

• In a one-tailed test, if it is hypothesized that
  the independent variable causes an increase in
  scores, then the null hypothesis is that the
  population mean is less than or equal to a given
  value and the alternative hypothesis is that the
  population mean is greater than the same value.
  For example
• H0 m 50
• Ha m gt 50

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A Sampling Distribution Showing the Region of
Rejection for a One-tailed Test of Whether Scores
Increase
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One-Tailed Hypotheses

• In a one-tailed test, if it is hypothesized that
  the independent variable causes a decrease in
  scores, then the null hypothesis is that the
  population mean is greater than or equal to a
  given value and the alternative hypothesis is
  that the population mean is less than the same
  value. For example
• H0 m 50
• Ha m lt 50

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A Sampling Distribution Showing the Region of
Rejection for a One-tailed Test of Whether Scores
Decrease
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Choosing One-Tailed Versus Two-Tailed Tests

• Use a one-tailed test only when confident of the
  direction in which the dependent variable scores
  will change. When in doubt, use a two-tailed test.

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Performing the z-Test
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The z-Test

• The z-test is the procedure for computing a
  z-score for a sample mean on the sampling
  distribution of means.

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Assumptions of the z-Test

1. We have randomly selected one sample
2. The dependent variable is at least approximately
   normally distributed in the population and
   involves an interval or ratio scale
3. We know the mean of the population of raw scores
   under some other condition of the independent
   variable

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Setting up for a Two-Tailed Test

1. Choose alpha. Common values are 0.05 and 0.01.
2. Locate the region of rejection. For a two-tailed
   test, this will involve defining an area in both
   tails of the sampling distribution.
3. Determine the critical value. Using the chosen
   alpha, find the zcrit value that gives the
   appropriate region of rejection.

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A Sampling Distribution for H0 Showing the
Region of Rejection for a 0.05 in a Two-tailed
Test
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Two-Tailed Hypotheses

• In a two-tailed test, the null hypothesis states
  that the population mean equals a given value.
  For example, H0 m 100.
• In a two-tailed test, the alternative hypothesis
  states that the population mean does not equal
  the same given value as in the null hypothesis.
  For example, Ha m ? 100.

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Computing z
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Rejecting H0

• When the zobt falls beyond the critical value,
  the statistic lies in the region of rejection, so
  we reject H0 and accept Ha
• When we reject H0 and accept Ha we say the
  results are significant. Significant indicates
  that the results are too unlikely to occur if the
  predicted relationship does not exist in the
  population.

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Interpreting Significant Results

• When we reject H0 and accept Ha, we do not prove
  that H0 is false
• While it is unlikely for a mean that lies within
  the rejection region to occur, the sampling
  distribution shows that such means do occur once
  in a while

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Failing to Reject H0

• When the zobt does not fall beyond the critical
  value, the statistic does not lie within the
  region of rejection, so we do not reject H0
• When we fail to reject H0 we say the results are
  nonsignificant. Nonsignificant indicates that the
  results are likely to occur if the predicted
  relationship does not exist in the population.

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Interpreting Nonsignificant Results

• When we fail to reject H0, we do not prove that
  H0 is true
• Nonsignificant results provide no convincing
  evidenceone way or the otheras to whether a
  relationship exists in nature

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Summary of the z-Test

1. Determine the experimental hypotheses and create
   the statistical hypothesis

• Set up the sampling distribution
• Compare zobt to zcrit