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Chapter 5: Introduction to
Hypothesis Testing: The
One-Sample z Test
• Suppose we wish to know whether children
who grow up in homes without access to
television have higher IQs than children in the
general population.
– Assume that IQ is normally distributed in the
general population, with μ = 100 and σ = 15
points.
• We draw a random sample of (N =) 25
children from homes without television (no
more than one child per household) and
measure each child’s IQ.
• The mean IQ for our sample turns out to be
103.5. Can we conclude that children without
TV are indeed smarter?
• It is possible that our sample mean exceeds
100 due entirely to chance factors involved in
drawing our random sample. We return to this
problem after we have developed some tools
for statistical inference.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
1
Review: Sampling Distributions
–
Population distributions are composed of
individual scores; however, psychologists
commonly perform their studies on groups.
Therefore, we need to understand distributions that are composed of statistics from
groups (all of which are the same size).
–
Such distributions are called sampling
distributions. If we are looking specifically at
the mean of each sample, the distribution is
called the sampling distribution of the mean
(SDM).
–
–
What will the SDM look like?
The Central Limit Theorem tells us that
as the size of the samples increases, the
SDM becomes closer in shape to the normal
distribution (e.g., less skewed), regardless
of the shape of the original population
distribution.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
2
Review: The Sampling
Distribution of the Mean
– The mean of the SDM is the same as
the mean of the population from which
the samples are being randomly drawn.
– The standard deviation of the SDM is
called the Standard Error of the Mean
(SEM). It is found from the standard
deviation of the population and the
sample size, according to this formula:
X 
– The SEM:

N
• Is larger when the standard deviation of the
population distribution is larger.
• Is smaller than the standard deviation of the
population distribution. The larger the samples,
the smaller the SEM (i.e., as N increases, the
SEM decreases.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
3
The z Score for Sample Means
–
In summary, the sampling distribution
of the mean can, in most cases, be
assumed to be a normal distribution,
with a mean equal to µ (the mean of the
population being sampled), and a
standard deviation equal to σ divided by
the square root of N.
–
To determine whether the mean of a
particular sample is unusual, we can use
the methods of the previous chapter, and
calculate a z score for a sample mean.
The formula needed for this is just like
the z score for individuals, except that
the raw score is a sample mean, and the
standard deviation is the SEM:
z
Chapter 5
X 
X
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
4
The z Score and p Value for the
TV/IQ Example
–
Let us now find the z score for the
example in the first slide to determine
whether drawing a random sample of 25
children would frequently yield a sample
mean as far from 100 as the mean in our
example. First, note that the SEM for our
example is:

15
15
X 


 3.0
N
–
5
Therefore, the z score for our sample
mean is:
z
–
25
X 
X
103.9  100 3.9


 1.3
3.0
3.0
The (one-tailed) p value corresponding
to this z score is the area beyond z = 1.3,
which is (from Table A) 50.00 – 40.32 =
9.68 / 100 = .0968.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen.
5
Null Hypothesis Testing
• Finding the p value for the non-TV sample of
children is a major step toward using null
hypothesis testing (NHT) to decide whether we
can conclude that an entire population of
children without TV would be any smarter than
the current population of children with nearly
universal access to TV.
• The major steps of NHT can be summarized
as follows:
Step 1: Assume that the worst-case scenario—
i.e., the null hypothesis (H0)— is true. In this
example, assume that the mean of the non-TV
population is exactly the same as for the TV
population— that is, µ0 = µ = 100.
Step 2: Set an alpha (α) level such that if p is less
than α you will reject (H0) as implausible. If a
two-tailed test is decided on, double your p
value before comparing it to α.
Step 3: Find the p value with respect to the null
hypothesis distribution (NHD). For the onesample case, the NHD is just the sampling
distribution of the mean.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
6
The Logic of Null Hypothesis
Testing
• It is only for the null hypothesis that we can
easily find the distribution of possible results,
and then determine the probability of
obtaining results as extreme as ours when H0
is really true. We hope that our results will
look unlikely enough when H0 is true that p
will be less than α, and we can therefore
reject (H0), and declare our results to be
statistically significant.
• The alternative hypothesis (HA) is simply
the complement of H0; if H0 is that µ = 100,
then HA is that µ ≠ 100 (anything but 100).
• NHT involves what is called an indirect
proof. By casting doubt on the hypothesis we
hope is not true (i.e., H0), we lend some
support to the opposite hypothesis, HA, which
is more consistent with our research
hypothesis.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen.
7
Statistical Decisions
If your p value is greater than your alpha
level (normally .05), you need not “accept”
the null hypothesis, but you cannot reject it.
Some researchers say that they have to
“retain” H0, whereas others prefer to say that
they have “failed to reject” H0, or that they
have insufficient evidence for rejecting it.
Type I and Type II Errors
– Type I error — Rejecting the null hypothesis when
it is in fact true. This is the error rate that we
control directly by choosing the level for alpha.
– Type II error — Failing to reject the null hypothesis
when it is in fact false. This error rate (beta) is not
directly controlled, but it does increase if alpha is
made smaller.
Researcher’s
Decision
Accept the Null
Hypothesis
Reject the Null
Hypothesis
Chapter 5
Actual Situation
Null Hypothesis is
Null Hypothesis is
True
False
Correct Decision
Type II Error
(prob = 1 – α)
(prob = β)
Type I Error
Correct Decision
(prob = α)
(prob = 1– β) (power)
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
8
More About Type I and
Type II Errors
– A Type I error is like a false alarm, because you
are saying that the mean of your sample is not
consistent with the mean of the larger population,
when in reality your sample mean is unusual due
to chance factors alone. Alpha, the Type I error
rate, can be defined as the percentage of null
(ineffective) experiments that nonetheless attain
statistical significance.
– A Type II error is a “miss,” because you are saying
that the mean of your sample is consistent with the
mean of the larger population, when in reality it is
not, but the difference you found happened not to
be large enough to attain statistical significance.
– You could reduce your Type I error rate by using a
smaller alpha level (e.g., .01), but that would
increase the rate of Type II errors (all else equal).
– You could reduce your Type II error rate by using a
one-tailed rather than a two-tailed test, but you
would have to convince your audience that there is
no chance you would have tested the other tail.
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
9
Try This Example…
National data shows that AP scores are normally
distributed, with μ = 40 and σ = 5. For the past
semester, N = 9 students were privately tutored for
their AP psychology tests. The mean performance
of the 9 students was 44. Is the mean of tutored
students different from the mean of all students?
1.
2.
3.
4.
5.
6.
State the hypotheses.
Select the statistical test (the z score
for groups, in this case) and the
significance (alpha) level.
Select the sample and collect the data.
Find the region of rejection.
Calculate the test statistic.
Make a statistical decision with respect
to the null hypothesis. (Find the p value
corresponding to your test statistic and
compare to alpha.)
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
10
Assumptions Underlying the
One-Sample z Test
–
The dependent variable was measured
on an interval or ratio scale.
–
The sample was drawn randomly from
the population of interest.
–
The dependent variable follows a
normal distribution in the population.
•
–
Because of the Central Limit Theorem, this
assumption is not critical when the size of the
sample is about 30 or more, unless the population
distribution is nearly the opposite of the normal
distribution.
The standard deviation for the sampled
population is the same as that of the
comparison population. (This allows the
SEM to be found from the known
population standard deviation.)
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
11
• Null Hypothesis Testing in
Terms of Bayes’s Theorem
pH 0  pS H 0 
pH 0 S  
pH 0  pS H 0   pH A  pS H A 
“Ineffective”
(null) experiments
α
“Effective”
experiments
1–β
Total number of
significant results
Chapter 5
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
12