 Formulating
hypotheses for testing
 Two
types of hypotheses

 Research
hypothesis (also known as the alternative
hypothesis)
 The
substantive hypothesis of interest we really want to
test.
 EXAMPLE:
Northwestern students are atypical of college
students in ideological attitudes, i.e, they have
DIFFERENT attitudes
 In
saying they are different, we are not specifying that
NU students are either more liberal or more
conservative  only that they are different without
saying HOW MUCH or in WHAT DIRECTION.
 A lack of
specificity in the research hypothesis leads to the
null hypothesis
 We test
the research hypothesis indirectly by testing the
similarity between NU students and other
students
 The
NULL hypothesis states that there is NO DIFFERENCE
between the mean ideological orientation of NU
students and others
 Hence,
it asserts that NU ideology = Population
ideology
 Or,
expressed differently, NU ideology  Population
ideology = 0
 Hence,
this assertion is called the NULL hypothesis, for it
asserts 0 difference.
 Testing
the null hypothesis
 Formalization:
 H_{0}
= NU  population = 0
 H_{1}
= NU  population is not = 0
 If we
disprove H_{0}, we can accept
H_{1}
 Consider
this example:
 Assume
that we create a 5 point scale to measure
conservatism:

1=far
left

2=liberal

3=middleofroad

4=conservative

5=far
right

 American
Council on Education (ACE) Data for all entering
college students in 1994 showed 2.97 as their
mean score on this scale.
 Data
from a sample of 1,184 NU students show a score of
2.87, meaning the sample of NU students is more
liberal than the population of all
students
 But
because we have data from only a sample of NU
students, it is possible that sampling error could
account for the difference of .10 points on the
conservatism scale and that NU students as a whole had
a mean of 2.97 like the population.
 How can
we test to determine the likelihood of observing a
score of 2.87 if in fact NU students as a whole did
score 2.97, just like the national
population?
Testing an
observed sample mean against a hypothesized population
mean
This involves
the Difference of Means Test for a "single
sample"
 Compute
the mean for sample data,
 Subtract
the population mean from the sample
mean,
 Evaluate
the difference (if any) in terms of (i.e., dividing by)
the standard error of the sampling distribution of
means:
 i.e.,
the standard deviation of a hypothetical distribution
of an infinite number of sample means of size
N
 coming
from a population with standard
deviation,
 Remember:
 the
standard deviation of a sampling distribution is
known as
 the
standard error of mean
 s.e. = sigma =
 This
formula applies, when the population standard deviation,
sigma, is known.
 What would
be the likely conservatism score if we took another
sample?
 Factors in
the variability of sample means:
 The
amount of ideological variation in the population of
NU students.
 The
size of the sample N (but not the % the sample is of
all NU students)
 Formula
for standard error of sampling distribution of means:
 This
formula assumes that we know the standard deviation of
the attribute in the population. And we do, it is
.77.

 Computing
the standard error
s.e. of the sampling distribution of means
=

.77 ÷ sqrt(1184) =

.77 ÷ 34.4 = .022


Computing
the TEST STATISTIC: a zscore
zscore = (X  µ) ÷
s.e.

= (2.87  2.97) ÷ .022

= .10 ÷ .022

=  4.5

Given a normal
distribution (and the sampling distribution of means
distributes normally), a zscore with an absolute value of
4.5 (whether it is negative or positive) is highly
unlikely.
 To
interpret a test statistic such as z = 4.5, one
needs some decision rules:
 Set a
level of significant that indicates how
"deviant" or unlikely a test statistic is before we
call it "significant"
 This
level of significance is called the alpha
level.
 It
refers to a chosen probability or significance
level.
 It
expresses the probability of a Type I error,
rejecting a true null hypothesis.
 Convention,
and not much else, often sets alpha at
.05.
 Meaning
of a test statistic significant at the .05 level: such
a test statistic would occur only as often as 5 times
in 100 samples if in fact the population had the
hypothesized mean
 The level
of significance and the alpha value are associated with
the region of rejection delineated on a normal
curve.
 If a
zscore is observed that falls in the region of
rejection, the decision rule is to reject the null
hypothesis.
 The
zscore that marks the region of rejection is called
the critical value.
 Thus,
in essence, the test statistic (observed zscore) is
compared with the critical zscore, and the decision
to accept or reject the null hypothesis depends on the
comparison.


When the
population standard deviation, ,
is NOT known
 This is
the usual case  we don't know EITHER µ or
 Because we
need s to compute the standard error of the mean, we must
estimate ,
which we can call .
 Our best
estimate, ,
of the population standard deviation, ,
is the sample standard deviation, s.
 In our
case, the s.d. for 1,184 NU students was
.81.
 Recall
that the population
was .77.
 Unfortunately,
the formula that we have used up to now to compute the
standard deviation, s, does not yield the correct
estimate of the population standard deviation, ,
for it is a biased estimate.
 You
learned to calculate the standard deviation as
 But
when calculated that way,
 the
standard deviation for samples systematically
underestimates the population standard
deviation.
 Because
it is biased, we must adjust the formula by
dividing the variation by Nl instead of N.
 SPSS
routinely calculates the sample standard deviation, s, to
provide an unbiased estimate of the population standard
deviation, .
 This
formula computes the unbiased sample standard
deviation
 which
we will call s'
 to
distinguish it from s when computed with N in
the formula's denominator.
 The
standard deviation for a sample as calculated by SPSS is
the corrected, unbiased estimate of the population
standard deviation.
 This value
is used to compute the estimated standard deviation of
the sampling distribution of means: which is used to
estimate the known value .
When the
population standard deviation is estimated, rather than
knownnew error enters.
 The
smaller the sample, the greater the error in
estimation.
 The
resulting test statistic no longer "distributes z"
(normally) instead it "distributes t"
 There
is a different t distribution for each degree of
freedom, measured by N  1
 The
smaller the sample, and the fewer the degrees of
freedom, the flatter the t distribution  i.e., the
more spread it has
 When
sample sizes are large (around 100), the normal
and t distributions converge.
 Thus, when
sample sizes are small and s is estimated by s', be sure
to consult the tdistribution rather than the normal
distribution in assessing the test statistic.
