7-1
l Chapter 7 l
Hypothesis Tests
Irwin/McGraw-Hill
7.1 Developing Null and Alternative
Hypotheses
7.2 Type I & Type II Error
7.3 Population mean
7.4 Population Proportion
7.5 Inferences About the Difference
Between Two Population Means
7.6 Inferences About the Difference
Between Two Population
Proportions
© Andrew F. Siegel, 1997 and 2000
7-2
Hypothesis Testing
Hypothesis testing
for population
mean
Hypothesis
testing
Hypothesis testing
for population
proportion
Irwin/McGraw-Hill
One population
mean
Two population
mean
One population
proportion
Two population
proportions
© Andrew F. Siegel, 1997 and 2000
7-3
Hypothesis Testing


Hypothesis testing is an inferential statistics method used to determine
something about a population, based on the observation of a sample.
Information about a population mean in the hypothesis statements will be
presented in a form of miu (  ).


e.g. “the mean travel time was 40 minutes…”, μ = 40 minutes
Assumptions

Assumptions needed for validity of Hypothesis Testing



The data should be at the interval or ratio level of measurement.
Data are a random sample from the population of interest
(So that the sample can tell you about the population)
The sample mean, x is approximately NORMAL
(Either the data are figuratively normal or large enough sample size, n, distribution not too skewed)
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-4
Hypothesis Testing

Hypothesis and test procedure (in general)

A statistical test of hypothesis consist of :






The Null hypothesis,
The Alternative hypothesis,
P-value (SPSS) @ Test statistics (manual),
α value (SPSS) @ Critical value (manual),
The rejection region,
The conclusion
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-5
Hypothesis Testing

Hypothesis and Test Procedures (SPSS)
1.
2.
3.
4.
State the null hypothesis, H 0 and alternative hypothesis, H1
Compare a P-value (generated by SPSS) with the given value of
Make an initial decision (whether to reject or not to reject H 0 )
Make the statistical decision and state the managerial conclusion

Differences

Hypothesis and Test Procedures (manual calculation)
1.
2.
3.
4.
5.
6.
State the null hypothesis, H 0 and alternative hypothesis, H1
Calculate the value of the test statistic
Find critical value (using Table Standard Normal III)
Compare steps 2 and 3
Make an initial decision (whether to reject or not to reject H 0)
Make the statistical decision and state the managerial conclusion
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-6
7.1 Developing Null and Alternative Hypotheses


It is not always obvious how the null and alternative hypothesis should be
formulated.
When formulating the null and alternative hypothesis, the nature or purpose of
the test must also be taken into account. We will examine:
1.
2.
3.
4.
5.

The claim or assertion leading to the test.
The null hypothesis to be evaluated.
The alternative hypothesis.
Whether the test will be two-tail or one-tail.
A visual representation of the test itself.
In some cases it is easier to identify the alternative hypothesis first. In other
cases the null is easier.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-7
7.1 Developing Null and Alternative Hypotheses

Null Hypothesis as an Assumption to be Challenged




We might begin with a belief on assumption that a statement about the value of a population
parameter is true.
We then using a hypothesis test to challenge the assumption and determine if there is
statistical evidence to conclude that the assumption is incorrect.
In these situations, it is helpful to develop the null hypothesis first.
When trying to identify the population parameter needed for your solution,
look for phrases such as:




“It is known that…”
“Previous research shows…”
“The company claims that…”
“A survey showed that…”
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-8
7.1 Developing Null and Alternative Hypotheses

When writing the Null Hypothesis, make sure it includes an equality symbol. It
may look like one of the following:




e.g. H 0 :   40
e.g. H 0 :   40
e.g. H 0 :   40
Example:
The label on a soft drink bottle states that it contains at least 500 millilitres.


Null Hypothesis:   500 ml (The label is correct)
Alternative Hypothesis:   500 ml (The label is incorrect)
Average tire life is 56000 kilometres.
 Null Hypothesis:   56000 km
 Alternative Hypothesis:   56000 km
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-9
7.1 Developing Null and Alternative Hypotheses

Alternative Hypothesis as a Research Hypothesis




Many applications of hypothesis testing involve an attempt to gather evidence in support of a
research hypothesis.
In such cases, it is often best to begin with the alternative hypothesis and make it the
conclusion that the researcher hopes to support.
The conclusion that the research hypothesis is true is made if the sample data provide
sufficient evidence to show that the null hypothesis can be rejected.
When trying to identify the information needed for your Alternative
Hypothesis statement, look for the phrases such as:




“Is it reasonable to conclude…”
“Is there enough evidence to substantiate…”
“Does the evidence suggest…”
“Has there been a significant…”
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-10
7.1 Developing Null and Alternative Hypotheses

There are three possible symbols to use in the Alternate Hypotheses, depending
on the wording of the question

Use “≠” when the question uses words/phrases such as:



Use “<” when the question uses words/phrases such as:




“is there a decrease…?”
“is there less…?”
“are there fewer…?”
Use “>” when the question uses words/phrases such as:



“is there a difference...?”
“is there a change...?”
“is there a increase…?”
“is there more…?”
When writing the Alternative Hypothesis, make sure it never includes an “=”
symbol. It should look similar to one of the following:
H 0 :   40
Irwin/McGraw-Hill
H 0 :   40
H 0 :   40
© Andrew F. Siegel, 1997 and 2000
7-11
7.1 Developing Null and Alternative Hypotheses

Determining null and alternative hypotheses (rule of thumb).
H0
Two-tailed test
Left-tailed test
Right-tailed test
=


H1

<
>
Refer previous examples
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-12
7.1 Developing Null and Alternative Hypotheses



H 0 and only H 0 will be either rejected or fail to be rejected.
Rejecting H 0 means that H1 is correct and vice versa.
This can be decided by comparing Z and Z (manual calculation) or by
comparing P-value and α (SPSS calculation).
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-13
7.2 Type I & Type II Error


Type I – Rejecting H 0 when it is correct ( ) .
Type II – Accepting H 0 when it is incorrect (  ) .
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-14
7.3 Population mean (One sample t-test
using SPSS)

Example (ExamplePM1S.sav)
A major oil company developed a petrol
additive that was supposed to increase engine
efficiency. 22 cars were test driven both with
and without the additive and the number of
kilometres per litre was recorded. Whether
the car was automatic or manual was also
recorded and coded as 1= manual and 2 =
automatic. During an earlier trial 22 cars were
test driven without using the additive. The
mean number of kilometres per litre was
10.5. Are the cars that were test driven with
additive running more than those without
additive? Test the appropriate hypothesis
using α=0.05.
Probability of committing Type I error
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-15
7.3 Population mean (One sample t-test using
SPSS)

Analysis in SPSS
Move “withadd” into Test Variables(s)
box. Change the Test Value to 10.5.
Next click “Options”
CI % = (1-α)*100. Continue and ok to
get an output
Important values to
draw conclusion
Analyze  Compare Means  One 
Sample T Test
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-16
7.3 Population mean (One sample t-test using
SPSS)

Example (ExamplePM1S.sav – solution)
1. State the null hypothesis, H 0 and alternative hypothesis, H1
H 0 :   10.5
H1 :   10.5
2. Compare a P-value (generated by SPSS) with the given value of α
P  value 
0.000
   0.05
2
p  value
One- tail: Reject H0 if
< α
2
Two- tail: Reject H0 if p-value < α
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-17
7.3 Population mean (One sample t-test using
SPSS)

Example (ExamplePM1S.sav – solution)
3. Make an initial decision (whether to reject or not to reject H 0 )
Reject H 0
4. Make the statistical decision and state the managerial conclusion.
Hence, it can be concluded that the cars with additive running more than 10.5km/litre, which
is more efficient than those without additive.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-18
7.3 Population mean (One sample t-test using
SPSS)
Exercise (ExercisePM1S.sav)
You have been asked to determine whether hypnosis improves memory. 40 men
and women are given five minutes to attempt to memorise a list of unrelated
words. They are then asked to recall as many as possible while under hypnosis.

You performed a study last year with another sample, so you have access to
descriptive statistics from a similar group of adults. The mean words recalled in
the earlier study, without hypnosis, was 34.6. Determine whether the participants
in the present study are comparable with those in the earlier study in terms of
recall in normal state. Test at α=0.05.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-19
7.3 Population mean (One sample t-test using
SPSS)
Extra Exercise
The average monthly salary for women in managerial and professional positions
is RM2400. Do men in the same positions have average monthly salary that are
higher than those for women?

A random sample of n  40 men in managerial and professional position were
interviewed and the SPSS result are as followed:
Test the appropriate hypothesis using 0.01 significance level.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-20
7.6 Population mean (two sample t-test using
SPSS)




In the previous slides, a one-sample t-test was used to determine whether a single
sample of scores was likely to have been drawn from a hypothesized population.
Next, we are going to test whether two sets of scores using two-sample t-test.
There are two kind of two-sample t-test available in SPSS known as the repeated
measure t-test and the independent groups t-test.
Repeated measure t-test (also known as dependent or paired sample t-test) is used when
you have data of two different conditions from only one group of participants (withinsubject). It has one additional assumption:


The difference between scores for each participant should be normal (applied for small
sample size)
Independent groups t-test is used when there are data of two different groups that have
performed in each of the different conditions. It has two additional assumption:
1.
2.
Independence of groups – participants should appear in only one group and the groups
should be unrelated.
Homogeneity of variance – being determined through another hypothesis testing (hypothesis
testing is conducted in the middle of Independent groups t-test analysis.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-21
7.6 Population mean (independent sample
t-test using SPSS)

Example (ExamplePM1S.sav)
On our previous analysis, it has been proven
that cars that use fuel additive will experience
engine efficiency. But another question arise.
Is there any differences in the engine
efficiency level of manual and automatic cars
that use fuel additive? Test the appropriate
hypothesis using α=0.05.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-22
7.6 Population mean (independent sample
t-test using SPSS)

Analysis in SPSS
Move “withadd” into Test Variables(s)
box and “cartype” into Grouping
Variable. Next, click “Define Groups
(In Option) CI % = (1-α)*100.
Continue and ok to get an output
Analyze  Compare Means 
Independen t  Sample T Test
Input “1” in Group 1 and “2” in
Group 2. Click Continue
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-23
7.6 Population mean (independent sample
t-test using SPSS)

Analysis in SPSS (Cont.)
Important values to
draw conclusion
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-24
7.6 Population mean (independent sample
t-test using SPSS)

Example (ExamplePM1S.sav – solution)
1. State the null hypothesis, H 0 and alternative hypothesis, H1
H 0 : m  a  0
H1 :  m   a  0
2. Compare a P-value (generated by SPSS) with the given value of α. However, variance
equality need to be checked first through Levene’s test ( 2nd assumption - Homogeneity of
variance).
H 0 : Equal variance is assumed
H1 : Equal variance is not assumed
P  value  0.080    0.05
P  value  0.407    0.05
Fail to reject H 0
Hence, equal variance is assumed
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-25
7.6 Population mean (independent sample
t-test using SPSS)

Example (ExamplePM1S.sav – solution)
3. Make an initial decision (whether to reject or not to reject H 0 )
Fail to reject H 0
4. Make the statistical decision and state the managerial conclusion.
Hence, it can be concluded that there are no difference in the engine efficiency level of
manual and automatic cars that use fuel additive.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-26
7.6 Population mean (independent sample
t-test using SPSS)
Exercise (ExercisePM1S.sav)
You have been asked to determine whether hypnosis improves memory. 40 men
and women are given five minutes to attempt to memorise a list of unrelated
words. They are then asked to recall as many as possible while under hypnosis.

Determined whether men and women recall equal numbers of words when under
hypnosis. Test at α=0.05.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
7-27
7.6 Population mean (independent sample
t-test using SPSS)
Extra Exercise
A university conducted an investigation to determine whether car ownership affects academic
achievement based on two random samples of 100 male students and their data were input into SPSS.
Male students who owned a car were coded as 1 and others as 2 (did not owned a car). Do the data
present sufficient evidence to indicate a difference in the mean achievements between car owners and
non owners of cars? Test using α = 0.05

Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000