Download Lecture Notes

INTRODUCTION TO
HYPOTHESIS TESTING
From R. B. McCall, Fundamental Statistics for
Behavioral Sciences, 5th edition, Harcourt Brace
Jovanovich Publishers, New York 1990
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
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.
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).
EXAMPLE 1: SINGLE CASE
• Assume one subject took drug and correctly
recalled 11 nouns.
• Reject H0 if the subject’s score falls into the
most extreme =5% of the distribution of nondrugged subjects. Score could fall in the
extreme low /2=2.5% or extreme high 2.5%,
so test against /2=2.5% probability.
EXAMPLE 1 (continued)
• Translate his/her score into a z score (z
score is called the standard normal deviate;
has mean=0 and standard deviation=1):
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.
EXAMPLE 1 (continued)
• Here is a pictorial representation of the situation:
H0
/2
1-
/2
Proportions of area under Normal Curve
QuickTime™ and a
TIFF (LZW) decompressor
are needed to s ee this picture.
.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
...
…
...
...
...
...
...
...
...
...
...
2.0
0.4772
0.4778
0.4783
0.4788
0.4793
0.4798
0.4803
0.4808
0.4812
0.4817
...
...
...
...
...
...
...
...
...
...
...
2.9
0.4981
0.4982
0.4982
0.4983
0.4984
0.4984
0.4985
0.4985
0.4986
0.4986
0.2
2.1
P(z<2) = 0.5 + 0.4772 = 0.9772;
P(z≥2) = 0.0228 < 0.025. Thus, we must reject H0.
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.
EXAMPLE 2 (continued)
• Translate group’s score into a z score:
z=(X-) / X = (X-)/[/√n] =
(8.32-7)/(2/√20) = 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.
Proportions of area under Normal Curve
QuickTime™ and a
TIFF (LZW) decompressor
are needed to s ee this picture.
.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
...
…
...
...
...
...
...
...
...
...
...
2.0
0.4772
0.4778
0.4783
0.4788
0.4793
0.4798
0.4803
0.4808
0.4812
0.4817
...
...
...
...
...
...
...
...
...
...
...
2.9
0.4981
0.4982
0.4982
0.4983
0.4984
0.4984
0.4985
0.4985
0.4986
0.4986
0.2
2.1
P(z<2.95) = 0.5 + 0.4984 = 0.9984;
P(z≥2.95) = 0.0016 < 0.025. Thus, reject H0.
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.
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).
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 “p<0.05”, where p
stands for the probability that H0 is true.
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 < 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
...
...
...
...
...
...
...
...
...
...
...
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 p>, 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.
POSSIBLE ERRORS
H0
/2
H1
1-
1-

/2
ACTUAL -->
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=
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.
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.