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Overview of Statistical Hypothesis Testing: The z-Test

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Title: Basic Statistics for the Behavioral Sciences Author: JAN User Last modified by: Windows User Created Date: 10/19/2010 7:21:36 PM Document presentation format – PowerPoint PPT presentation

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Title: Overview of Statistical Hypothesis Testing: The z-Test


1
Chapter 7
  • Overview of Statistical Hypothesis Testing The
    z-Test

2
Going Forward
  • Your goals in this chapter are to learn
  • Why the possibility of sampling error causes
    researchers to perform inferential statistics
  • When experimental hypotheses lead to either
    one-tailed or a two-tailed tests
  • How to create the null and alternative hypotheses
  • When and how to perform the z-test

3
Going Forward
  • How to interpret significant and nonsignificant
    results
  • What Type I errors, Type II errors, and power are

4
The Role of Inferential Statistics in Research
5
Inferential Statistics
  • Inferential statistics are used to decide whether
    sample data represent a particular relationship
    in the population.

6
Parametric Statistics
  • Parametric statistics are inferential procedures
    requiring certain assumptions about the raw score
    population being represented by the sample
  • Two assumptions are common to all parametric
    procedures
  • The population of dependent scores should be at
    least approximately normally distributed
  • The scores should be interval or ratio

7
Nonparametric Procedures
  • Nonparametric statistics are inferential
    procedures not requiring stringent assumptions
    about the populations being represented.

8
Setting up Inferential Procedures
9
Experimental Hypotheses
  • Experimental hypotheses describe the possible
    outcomes of a study.

10
Predicting a Relationship
  • A two-tailed test is used when we do not predict
    the direction in which dependent scores will
    change
  • A one-tailed test is used when we do predict the
    direction in which dependent scores will change

11
Designing aOne-Sample Experiment
  • To perform a one-sample experiment, we must
    already know the population mean for participants
    tested under another condition of the independent
    variable.

12
Alternative Hypothesis
  • The alternative hypothesis (Ha) describes the
    population parameters the sample data represent
    if the predicted relationship exists in nature.

13
Null Hypothesis
  • The null hypothesis (H0) describes the population
    parameters the sample data represent if the
    predicted relationship does not exist in nature.

14
A Graph Showing the Existence of a Relationship
15
The Logic
  • When a relationship is indicated by the sample
    data, it may be because
  • The relationship operates in nature and it
    produced our data
  • OR
  • We are misled by sampling error

16
A Graph Showing a Relationship Does Not Exist
17
Performing the z-Test
18
The z-Test
  • The z-test is the procedure for computing a
    z-score for a sample mean on the sampling
    distribution of means.

19
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 another condition of the independent
    variable
  4. We know the true standard deviation of the
    population described by the null
    hypothesis

20
Setting up for a Two-Tailed Test
  • Create the sampling distribution of means from
    the underlying raw score population that H0 says
    our sample represents
  • Choose the criterion, symbolized by a (alpha)
  • Locate the region of rejection which, for a
    two-tailed test, involves defining an area in
    both tails
  • Determine the critical value by using the chosen
    a to find the zcrit value resulting in the
    appropriate region of rejection

21
Two-Tailed Hypotheses
  • In a two-tailed test, the null hypothesis states
    the population mean equals a given value. For
    example, H0 m 100.
  • In a two-tailed test, the alternative hypothesis
    states the population mean does not equal the
    same given value as in the null hypothesis. For
    example, Ha m 100.

22
A Sampling Distribution for H0 Showing the Region
of Rejection for a 0.05 in a Two-tailed Test
23
Computing z
  • The z-score is computed using the same formula as
    before
  • where

24
Comparing Obtained z
  • In a two-tailed test, reject H0 and accept Ha if
    the z-score you computed is
  • Less than the negative of the critical z-value
  • OR
  • Greater than the positive of the critical z-value
  • Otherwise, fail to reject H0

25
Interpreting Significant andNonsignificant
Results
26
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
    the results are unlikely to occur if the
    predicted relationship does not exist in the
    population.

27
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 the
    results are likely to occur if the predicted
    relationship does not exist in the population.

28
Nonsignificant Results
  • When we fail to reject H0, we do not prove H0 is
    true
  • Nonsignificant results provide no convincing
    evidence the independent variable does not work

29
Summary of the z-Test
  1. Determine the experimental hypotheses and create
    the statistical hypothesis
  2. Select a, locate the region of rejection, and
    determine the critical value
  3. Compute and zobt
  4. Compare zobt to zcrit

30
The One-Tailed Test
31
One-Tailed Hypotheses
  • In a one-tailed test, if it is hypothesized the
    independent variable causes an increase in
    scores, then the null hypothesis states the
    population mean is less than or equal to a given
    value and the alternative hypothesis states the
    population mean is greater than the same value.
    For example

32
One-Tailed Hypotheses
  • In a one-tailed test, if it is hypothesized the
    independent variable causes a decrease in scores,
    then the null hypothesis states the population
    mean is greater than or equal to a given value
    and the alternative hypothesis states the
    population mean is less than the same value. For
    example

33
A Sampling Distribution Showing the Region of
Rejectionfor a One-tailed Test of Whether Scores
Increase
34
A Sampling Distribution Showing the Region of
Rejectionfor a One-tailed Test of Whether Scores
Decrease
35
Choosing One-Tailed Versus Two-Tailed Tests
  • Use a one-tailed test only when it is the
    appropriate test for the independent variable.
    That is, when the independent variable can work
    only if scores go in one direction.

36
Errors in Statistical Decision Making
37
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 we conclude the predicted relationship
    exists when it really does not
  • The theoretical probability of a Type I error
    equals a

38
Type II Errors
  • A Type II error is defined as retaining H0 when
    H0 is false (and Ha is true)
  • In a Type II error, the sample mean is so close
    to the m described by H0 we conclude the
    predicted relationship does not exist when it
    really does

39
Power
  • Power is
  • The probability of rejecting H0 when it is false
  • The probability of not making a Type II error
  • The probability that we will detect a
    relationship and correctly reject a false null
    hypothesis (H0)

40
Example
  • Use the following data set and conduct a
    two-tailed z-test to determine if m 11 and the
    population standard deviation is known to be 4.1

14 14 13 15 11 15
13 10 12 13 14 13
14 15 17 14 14 15
41
Example
  1. Choose a 0.05
  2. Reject H0 if zobt gt 1.965 or if zobt lt -1.965.

42
Example
  • Since zobt lies within the rejection region, we
    reject H0 and accept Ha. Therefore, we conclude m
    does not equal 11.
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