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HYPOTHESIS TESTING

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In making inference from data analysed, there is the need to subject the results ... Drawing meanings from data in this way is called Inferential Statistics ... – PowerPoint PPT presentation

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Title: HYPOTHESIS TESTING


1
HYPOTHESIS TESTING
2
Introduction
  • In making inference from data analysed, there is
    the need to subject the results to some rigour.
  • Drawing meanings from data in this way is called
    Inferential Statistics
  • Hypothesis testing is one of the important tools
    in inferential statistics.

3
Introduction
  • The issues to be considered involves
  • identifying the null and alternative hypotheses
  • deciding on an appropriate significance level
  • issues to do with
  • Type I and Type II errors,
  • one or two-tailed tests,
  • power.
  • existence of multiple tests of significance.

4
Introduction
  • A hypothesis is a conjecture
  • In the sciences, the hypothesis is the beginning
    of a theory
  • The hypothesis is formulated , tested and if
    acceptable the, becomes a theory.

5
Types of Hypothesis
  • We need to distinguish between a research
    hypothesis and a statistical hypothesis
  • Research hypothesis does not require statistical
    tests to validate
  • Statistical hypothesis requires statistical
    tests to validate

6
Statistical Hypothesis
  • There are generally two forms of a statistical
    hypothesis
  • null (typically represented as H0 pronounced "H
    naught)
  • alternative (typically symbolised as H1or Ha- the
    one we are really interested in showing support
    for).

7
Example of Statistical Hypotheses
8
  • hypotheses are usually formally stated in terms
    of the population parameters about which the
    inference is to be made.
  • Because our interest is in making an inference
    from sample information to population
    parameter(s),

9
  • We use two forms of hypotheses to set the stage
    for a logical decision.
  • If we amass enough evidence to reject one
    hypothesis, the only other state of affairs which
    can exist is covered by the remaining hypothesis.
  • Thus, the two hypotheses (null and alternative)
    are set up to be mutually exclusive and
    exhaustive of the possibilities.

10
Direction of Hypotheses
  • Directional (one tail)
  • Non directional (two tail test)
  • Examples

11
Direction of Hypotheses
  • Directional
  • Also termed one tail test
  • Use lt or gt
  • where the direction of deviation from the null
    value is clearly specified
  • a specific predicted outcome is stated.
  • one-tailed tests, should only be used in the
    light of strong previous research, theoretical,
    or logical considerations.
  • Hence a "claim", "belief", or "hypothesis", or
    "it was predicted that" is not sufficient to
    justify the use of a one-tailed test.

12
Direction of Hypotheses- One tail
  • A directional hypothesis only considers one tail
    (the other tail is ignored as irrelevant to H1),
    thus all of can be placed in that one tail.

13
Direction of Hypotheses- Non-Directional
  • Non Directional
  • Also termed two tailed test
  • Use
  • the direction of deviation of the alternative
    case is not specified
  • A two-tailed test requires us to consider both
    sides of the normal distribution, so we split
    and place half in each tail.

14
Level of Significance
  • Also known as alpha ( ) level
  • specifies the probability level for the evidence
    to be an unreasonable estimate
  • Unreasonable means that the estimate should not
    have taken its particular value unless some
    non-chance factor(s) had operated to alter the
    nature of the sample such that it was no longer
    representative of the population of interest.
  • The researcher has complete control over the
    value of this significance level.

15
  • The choice is for levels up to 10 in social
    sciences
  • The lower the level of significance the stronger
    the effect or phenomenon being studied.
  • The -level should be considered in light of the
    research context and in light of your own
    personal convictions about how strong you want
    the evidence to be, before you will conclude that
    a particular estimate is reasonable or
    unreasonable.

16
  • In some exploratory contexts (perhaps, in an area
    where little previous research has been done),
    you might be willing to be more liberal with a
    decision criterion and relax the level of
    significance to 0.10 or even 0.20.
  • Thus, less extreme values of a statistic would be
    required for in order to conclude that non-chance
    factors had operated to alter the nature of the
    sample.

17
  • On the other hand, there are research contexts in
    which one would want to be more conservative and
    more certain that an unreasonable estimate had
    been found.
  • In these cases, the significance level might be
    lowered to 0.001 (0.1) where more extreme values
    of a statistic would be required before
    non-chance factors were suspected

18
  • SPSS output usually under heads it as "Sig." or
    "Two-tailed Sig.", or "Prob." for the probability
    (or "p-value") of your results being a real
    difference or a real relationship.
  • This can then be the probability you quote as
    being the "level of significance" associated with
    your results.
  • Note that it is normally required that this
    p-value to be less than or equal to 0.05 to make
    the claim of significant.

19
  • It is then up to your discussion to
    explain/justify/interpret this level of
    significance to your reader
  • As a general guide, treat the p-value as a
    measure of the confidence or faith in the results
    being real (and not being due to chance
    fluctuations in sampling).

20
  • The -level is the probability or p-value you
    are willing to accept as significant.
  • Ideally, this -level.
  • The -level can also be interpreted as the
    chance of making a Type I error.

21
Type I Error and Type II Error
  • When alpha is set at a specified level (say,
    0.05) it indicates automatically specify how much
    confidence (0.95) is placed in the decision to
    "fail to reject Ho if it really is the true state
    of affairs.

22
Type I Error and Type II Error
  • Consider doing the same experiment exactly 100
    times, each time using a different random sample
  • If alpha is set to 0.05 (and consequently
  • 1 - 0.95), then in the 100 experiments, it
    should be expected to make an incorrect decision
    in 0.05 x 100 or 5 of these experiments ( 5
    chance of error), and a correct one 95 of the
    time if Ho is really true.

23
Type I Error and Type II Error
24
Type I Error and Type II Error
  • Bear in mind that you are merely making a
    decision concerning what you believe about the
    truth or falsity of the hypothesis you are not
    really ascertaining whether the hypothesis is
    true or false.
  • In other words, if you decide to reject H0, that
    means "I have decided to believe that H0 is
    false" it does not necessarily mean that H0 is
    actually false.

25
Type I Error and Type II Error
  • Similarly, if the decision is to accept (or, more
    precisely, not to reject) H0, that means "I have
    decided to believe that H0 is true" it does not
    necessarily mean that H0 is actually true.

26
Type I Error and Type II Error
  • Thus, states what chance of making an error (by
    falsely concluding that the null hypothesis
    should be rejected) we, as researchers, are
    willing to tolerate in the particular research
    context.

27
Type I Error and Type II Error
  • When we consider the case where Ho is not the
    true state of affairs in the population (i.e., Ho
    is false), we move into an area of statistics
    concerned with the power of a statistical test.
  • If Ho is false, we want to have a reasonable
    chance of detecting it using our sample
    information.

28
Type I Error and Type II Error
  • Of course, there is always the chance that we
    would fail to detect a false Ho, which yields the
    Type II or error
  • However, error is generally considered less
    severe or costly than an error
  • We must be aware of the power of a statistical
    test the test's ability to detect a false null
    hypothesis because we want to reject Ho if it
    really should be rejected in favour of H1

29
Type I Error and Type II Error
  • Hence we focus on 1 - alpha which is the
    probability of correctly rejecting Ho.

30
Type I Error and Type II Error
  • It can be helpful to think of the Type I error as
    "rejecting the null hypothesis when it is true"
    and the Type II error as "failing to recognize
    that the null hypothesis is false when it is
    false
  • Another way of putting it is that the Type I
    error amounts to "disbelieving the truth" the
    Type II error, to "believing an untruth

31
Power of Test
  • Lets depict a hypothesis testing situation
    graphically using two normal distributions
  • one to represent the sampling distribution for
    the null hypothesis
  • the other to represent the sampling distribution
    for the alternative hypothesis
  • the sampling distribution of the mean from any
    population looks more and more normal as sample
    size, N, is increased central limit theorem
    that is why the normal distribution is used.

32
Power of Test
  • Thus, if a reasonable sample size, is assumed it
    is justified to use normal distributions to
    represent the situation.
  • In this illustration we are using sampling
    distributions (in this case for the sample mean)
    as standards against which to compare our
    computed sample statistic.

33
  • In general terms, suppose we evaluate the
    specific hypothesis that µ has one value (µ Ho)
    or another value (µ H1) using a sample of size N
  • Ho µ µHo versus H1 µ µH1

34
Power of Test
35
Power of Test
  • Four factors interact when we consider setting
    significance levels and power
  • 1. Power 1- (probability of correctly
    concluding "Reject Ho")
  • 2. Significance level (probability of falsely
    concluding "Reject Ho")
  • 3. Sample size N
  • 4. Effect size e (the separation between the
    null hypothesis value and a particular value
    specified for the alternative hypothesis).

36
  • One way to increase power is to relax the
    significance level ( ) if e and N remain
    constant

                                                
                                Note how the
37
  • Another way to increase power is to increase the
    sample size

38
  • Yet another way to increase power is to look only
    for a larger effect size if and N remain
    constant.

39
  • Finally, power can be increased if a directional
    hypothesis can be stated (based on previous
    research findings or deductions from theory).

 
40
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