Chapter 8
Hypothesis Testing I
Significant Differences
 Hypothesis testing is designed to detect
significant differences: differences that did
not occur by random chance.
 This chapter focuses on the “one sample”
case: we compare a random sample against
a population.
 We compare a sample statistic to a
population parameter to see if there is a
significant difference.
Example
 The education department at a large state
university has been accused of “grade
inflation” such that education majors have
higher GPAs than students in general.
 GPAs of all education majors should be
compared with the GPAs of all students.
 There are 1000s of education majors, far too
many to interview.
 How can the dispute be investigated without
interviewing all education majors?
Example
 The average GPA
for all students is
2.70 and the
standard deviation
is 0.70..
 The box reports
the statistical
information for a
random sample of
education majors
Mu = 2.70
=
3.00
s=
0.70
N=
117
X
Example
 There is a difference between the
parameter (2.70) and the statistic (3.00).
 It seems that education majors do have higher
GPAs, at least in our sample.
 However, we are working with a random
sample (not all education majors).
 The observed difference may have been
caused by random chance.
Two Explanations for the
Difference
1. The mean for all education students
is the same as the pop. mean (2.70).
 The difference in the sample mean is
caused by random chance.
2. The difference is real (significant).
 Education majors have higher GPAs than
students in general
Hypotheses
1. Null Hypothesis (H0)
 “The difference is caused by random chance”.
 The H0 always states there is “no significant
difference.”
2. Alternative hypothesis (H1)
 The mean GPA for all education majors is higher than
the mean GPA for all students in the college.
 (H1) always contradicts the H0.
 If we can reject the null hypothesis, our data support
the alternative hypothesis (but don’t prove it)
Testing the two explanations
 Assume that H0 is true.
 What is the probability of getting the
sample mean (3.0) if the H0 is true and
all education majors really have a mean
of 2.7?
 If the probability is less than 0.05,
reject the null hypothesis.
Test the Hypotheses
 This will be a one-tailed test since our alternative
hypothesis is not just that education majors’ GPAs are
different but that they are higher.
 At a 95% confidence level, we put the whole .05 in
the upper tail of the normal sampling distribution.
 Using the Z score formula and Appendix A to
determine the probability of getting the observed
difference, we find that Z(critical) = +1.65.
 If Z(obtained) is at or above 1.65, we can reject the
null hypothesis. The zone of rejection begins at 1.65
and includes any z-score higher than that
Test the Hypotheses
 Substituting the values into formula 8.1, we
calculate a Z score of 4.6. For a large N, we
can use the sample standard deviation
 This is above 1.65. and a difference this
large would be very rare. In fact, if the null
hypothesis were true, the probability of
such an extreme outcome is p = .0001.
 Therefore we can reject the null hypothesis
and say that our result tends to support the
alternative hypothesis, that GPAs in the
education department are higher than for
the university as a whole. .
Interpreting results: significance
 This difference is significant (note the
technical meaning of significant—
namely, that it probably didn’t
happen by chance).
 “Significant” doesn’t necessarily mean
large or even important.
 If N is very large, even a small
difference could be significant.
Review: five step model for
testing hypotheses
1. Make Assumptions and meet test
requirements.
2. State the null and alternative
hypotheses.
3. Select the sampling distribution and
establish the critical region.
4. Compute the test statistic.
5. Make a decision and interpret
results.
The Five Step Model: Do the
steps more formally
Is there a grade inflation
problem in the education dept?
Step 1 Make Assumptions and
Meet Test Requirements
 Random sampling
 Hypothesis testing assumes samples were
selected according to EPSEM.
 The sample of 117 was randomly selected from
all education majors.
 LOM is Interval-Ratio
 GPA is I-R so the mean is an appropriate
statistic.
 Sampling Distribution is normal in shape
 This is a “large” sample (N>100).
Step 2 State the Null
Hypothesis
Null Hypothesis: μ(education
department) = 2.7, the same as the
mean for the university as a whole
 The sample of 117 comes from a
population that has a GPA of 2.7.
 The difference between 2.7 and 3.0 is
trivial and caused by random chance.
Step 2 State the Alternative
Hypothesis
Alternative (or research) hypothesis:
Mean in education is higher than 2.7
 The sample of 117 probably comes from
a population that has a higher GPA than
2.7.
 The difference between 2.7 and 3.0
probably reflects an actual difference
between education majors and other
students.
Step 3 Select Sampling Distribution
and Establish the Critical Region
 Sampling Distribution= Z
 Alpha (α) = .05
 Any difference with a probability less than α
is rare and will cause us to reject the H0.
 Critical Region begins at +1.65
 This is the critical Z score associated with
α = .05, eon-tailed test.
 If the obtained Z score falls in the C.R., reject
the H0.
Step 4
Compute the test statistic
 Using formula 8.1, Z (obtained) = 4.6
Step 5 Make a Decision and
Interpret Results
 The obtained Z score fell in the C.R., so we
reject the H0.
 If the H0 were true, a sample outcome of 3.00
would be unlikely.
 Therefore, the H0 is false and must be rejected.
 GPAs in the Education Department are most
likely significantly different than GPAs in
the college as a whole.
The Five Step Model: Summary
 In hypothesis testing, we try to identify
statistically significant differences that did
not occur by random chance.
 In this example, the difference between the
parameter 2.70 and the statistic 3.00 was
large and unlikely (p < .05) to have
occurred by random chance.
The Five Step Model: Summary
 We rejected the H0 and concluded
that the difference was significant.
 It is very likely that Education majors
have GPAs higher than the general
student body
One-tailed vs. two-tailed
hypothesis testing
 What if our alternative hypothesis were not
that education majors have higher GPAs
but simply that their GPAs are significantly
different than for the student body as a
whole
 In that case, we do a two-tailed test, and
we split the alpha=.05 into both tails (.025
in each)
 In that case, z(crit) = +/- 1.96
 We’d still use formula 8.1 to calculate
Z(obt)
One-tailed vs. two-tailed
hypothesis testing
 What if our null hypothesis were that
education majors had lower mean GPAs
than the student body as a whole.
 In that case, the whole alpha (.05) goes
into the negative tail of the distribution.
 Z(crit) = -1.65
 We’d still use formula 8.1 to calculate
z(obt)
Student’s t distribution
 What if N is smaller than 100? Is there still
a way to test for significance
 Yes, the student’s t distribution does the
trick.
 What’s more, as N gets to 100 or more, the
t-distribution behaves just like the Zdistribution, so we can actually just use the
t-distribution all the time
What if we had a smaller sample of
education majors?
 N = 65
 Sample mean = 2.9
 Sample standard deviation = 0.4
Step 1 Make Assumptions and
Meet Test Requirements
 Random sampling
 Hypothesis testing assumes samples were
selected according to EPSEM.
 The sample of 65 was randomly selected from
all education majors.
 LOM is Interval-Ratio
 GPA is I-R so the mean is an appropriate
statistic.
 Sampling Distribution is a t-distribution
 This is a small sample (N<100).
Step 2 State the Null and
Alternative Hypothess
Null hypothesis: The overall GPA for
Education majors = 2.7, the same as
in the college a a whole.
Alternative hypothesis: The mean GPA
in education is actually higher than
2.7
Step 3 Select Sampling Distribution
and Establish the Critical Region
 Sampling Distribution= student’s t

Alpha (α) = .05
 Degrees of freedom = N-1 (64), 1-tailed
test
 Critical Region begins at +1.671
 If the obtained t score falls in the Critical
Region, at or above 1.671, reject the H0
and support the alternative or research
hypothesis
Step 4
Compute the test statistic
 Using formula 8.2, t (obtained) = 4.
Step 5 Make a Decision and
Interpret Results
 The obtained t score fell in the Critical
Region, so we reject the H0.
 If the H0 were true, a sample outcome of t = 4
would be unlikely.
 Therefore, the H0 must be rejected.
 Our supports the proposition that education
majors have a GPA that is significantly
significantly higher than the GPA of the
general student body
2 types of error
 Type 1 or alpha error: the probability of
rejecting the null hypothesis when it is
actually true. We set up the problem in a
way that minimizes (but does not
eliminate) this possibility.
 Type 2 or beta error: the probability of
failing to reject a null hypothesis when it is
actually false.
 The two types of error are inversely
related; lowering the risk of type 1 error
raises the risk of type 2 error