Title: HYPOTHESIS TESTING
1HYPOTHESIS TESTING
2Introduction
- 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. -
3Introduction
- 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.
4Introduction
- 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.
5Types 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
6Statistical 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).
7Example 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.
10Direction of Hypotheses
- Directional (one tail)
- Non directional (two tail test)
- Examples
11Direction 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.
12Direction 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. -
13Direction 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.
14Level 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.
21Type 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.
22Type 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.
23Type I Error and Type II Error
24Type 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.
25Type 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.
26Type 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.
27Type 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.
28Type 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
29Type I Error and Type II Error
- Hence we focus on 1 - alpha which is the
probability of correctly rejecting Ho.
30Type 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
31Power 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.
32Power 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
34Power of Test
35Power 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).
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