Jackson S.L. Research Methods and Statistics: A Critical Thinking Approach
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CHAPTER 6
MULTIPLE-CHOICE SELF TEST
Select the single best answer for each of the following
questions. If you have trouble answering any of the
questions, restudy the relevant material.
1. The magnitude of a correlation coefficient is to
as the type of correlation is to .
a. slope; absolute value
b. sign; absolute value
c. absolute value; sign
d. none of the above
2. Strong correlation coefficient is to weak correla-
tion coefficient as
is to .
a. 1.00; 1.00
b. 1.00; .10
c. 1.00; 1.00
d. .10; 1.00
3. Which of the following correlation coefficients
represents the variables with the weakest degree
of relationship?
a. .89
b. 1.00
c. .10
d. .47
4. A correlation coefficient of 1.00 is to
as a correlation coefficient of 1.00 is to
.
a. no relationship; weak relationship
b. weak relationship; perfect relationship
c. perfect relationship; perfect relationship
d. perfect relationship; no relationship
5. If the points on a scatterplot are clustered in a pat-
tern that extends from the upper left to the lower
right, this would suggest that the two variables
depicted are:
a. normally distributed.
b. positively correlated.
c. regressing toward the average.
d. negatively correlated.
6. We would expect the correlation between height
and weight to be
, whereas we would
expect the correlation between age in adults and
hearing ability to be .
a. curvilinear; negative
b. positive; negative
c. negative; positive
d. positive; curvilinear
7. When we argue against a statistical trend based
on one case we are using a:
a. third-variable.
b. regression analysis.
c. partial correlation.
d. person-who argument.
8. If a relationship is curvilinear, we would expect
the correlation coefficient to be:
a. close to 0.00.
b. close to 1.00.
c. close to 1.00.
d. an accurate representation of the strength of
the relationship.
9. The
is the correlation coefficient that
should be used when both variables are meas-
ured on an ordinal scale.
a. Spearman rank-order correlation coefficient
b. coefficient of determination
c. point-biserial correlation coefficient
d. Pearson product-moment correlation
coefficient
10. Suppose that the correlation between age and
hearing ability for adults is .65. What propor-
tion (or percent) of the variability in hearing abil-
ity is accounted for by the relationship with age?
a. 65%
b. 35%
c. 42%
d. unable to determine
11. Drew is interested is assessing the degree of
relationship between belonging to a Greek organ-
ization and the number of alcoholic drinks con-
sumed per week. Drew should use the
correlation coefficient to assess this.
a. partial
b. point-biserial
c. phi
d. Pearson product-moment
12. Regression analysis allows us to:
a. predict an individual’s score on one variable
based on knowing the individual’s score on
another variable.
b. determine the degree of relationship between
two interval/ratio variables.
c. determine the degree of relationship between
two nominal variables.
d. predict an individual’s score on one variable
based on knowing that the variable is inter-
val/ratio in scale.
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Hypothesis Testing and
Inferential Statistics
7
CHAPTER
Hypothesis Testing
Null and Alternative Hypotheses
One- and Two-Tailed Hypothesis Tests
Type I and II Errors in Hypothesis Testing
Statistical Significance and Errors
Single-Sample Research and Inferential Statistics
The z Test: What It Is and What It Does
The Sampling Distribution
The Standard Error of the Mean
Calculations for the One-Tailed z Test
Interpreting the One-Tailed z Test
Calculations for the Two-Tailed z Test
Interpreting the Two-Tailed z Test
Statistical Power
Assumptions and Appropriate Use of the z Test
Confidence Intervals Based on the z Distribution
The t Test: What It Is and What It Does
Student’s t Distribution
Calculations for the One-Tailed t Test
The Estimated Standard Error of the Mean
Interpreting the One-Tailed t Test
Calculations for the Two-Tailed t Test
Interpreting the Two-Tailed t Test
Assumptions and Appropriate Use of the Single-Sample t Test
Confidence Intervals based on the t Distribution
The Chi-Square (
2
) Goodness-of-Fit Test: What It Is and What It Does
Calculations for the
2
Goodness-of-Fit Test
Interpreting the
2
Goodness-of-Fit Test
Assumptions and Appropriate Use of the
2
Goodness-of-Fit Test
Correlation Coefficients and Statistical Significance
Summary
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CHAPTER 7
Learning Objectives
• Differentiate null and alternative hypotheses.
• Differentiate one- and two-tailed hypothesis tests.
• Explain how Type I and Type II errors are related to hypothesis testing.
• Explain what statistical significance means.
• Explain what a z test is and what it does.
• Calculate a z test.
• Explain what statistical power is and how to make statistical tests more
powerful.
• List the assumptions of the z test.
• Calculate confidence intervals using the z distribution.
• Explain what a t test is and what it does.
• Calculate a t test.
• List the assumptions of the t test.
• Calculate confidence intervals using the t distribution.
• Explain what the chi-square goodness-of-fit test is and what it does.
• Calculate a chi-square goodness-of-fit test.
• List the assumptions of the chi-square goodness-of-fit test.
I
n this chapter, you will be introduced to the concept of hypothesis
testing—the process of determining whether a hypothesis is supported
by the results of a research project. Our introduction to hypothesis test-
ing will include a discussion of the null and alternative hypotheses, Type I
and Type II errors, and one- and two-tailed tests of hypotheses, as well as
an introduction to statistical significance and probability as they relate to
inferential statistics.
In the remainder of this chapter, we will begin our discussion of
inferential statistics—procedures for drawing conclusions about a popula-
tion based on data collected from a sample. We will address three different
statistical tests: the z test, the t test, and the chi-square (
2
) goodness-of-fit
test. After reading this chapter, engaging in the critical thinking checks,
and working through the problems at the end of the chapter, you should
understand the differences between these tests, when to use each test, how
to use each to test a hypothesis, and the assumptions of each test.
Hypothesis Testing
Research is usually designed to answer a specific question—for example—
Do science majors score higher on tests of intelligence than students in the
general population? The process of determining whether this statement is
supported by the results of the research project is referred to as hypothesis
testing.
hypothesis testing
The process of determin-
ing whether a hypothesis is
supported by the results of a
research study.
hypothesis testing
The process of determin-
ing whether a hypothesis is
supported by the results of a
research study.
inferential statistics
Procedures for drawing
conclusions about a population
based on data collected from
a sample.
inferential statistics
Procedures for drawing
conclusions about a population
based on data collected from
a sample.
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© 2005 Sidney Harris, Reprinted with permission.
Suppose a researcher wants to examine the relationship between the type
of after-school program attended by a child and the child’s intelligence level.
The researcher is interested in whether students who attend after-school
programs that are academically oriented (math, writing, computer use)
score higher on an intelligence test than students who do not attend such
programs. The researcher will form a hypothesis. The hypothesis might be
that children in academic after-school programs have higher IQ scores than
children in the general population. Because most intelligence tests are stand-
ardized with a mean score () of 100 and a standard deviation () of 15, the
students in academic after-school programs must score higher than 100 for
the hypothesis to be supported.
Null and Alternative Hypotheses
Most of the time, researchers are interested in demonstrating the truth of
some statement. In other words, they are interested in supporting their
hypothesis. It is impossible statistically, however, to demonstrate that
something is true. In fact, statistical techniques are much better at demon-
strating that something is not true. This presents a dilemma for research-
ers. They want to support their hypotheses, but the techniques available
to them are better for showing that something is false. What are they
to do? The logical route is to propose exactly the opposite of what they
want to demonstrate to be true and then disprove or falsify that hypoth-
esis. What is left (the initial hypothesis) must then be true (Kranzler &
Moursund, 1995).
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Let’s use our sample hypothesis to demonstrate what we mean. We
want to show that children who attend academic after-school programs
have different (higher) IQ scores than those who do not. We understand
that statistics cannot demonstrate the truth of this statement. We therefore
construct what is known as a null hypothesis (H
0
). Whatever the research
topic, the null hypothesis always predicts that there is no difference
between the groups being compared. This is typically what the researcher
does not expect to find. Think about the meaning of null—nothing or
zero. The null hypothesis means we have found nothing—no difference
between the groups.
For the sample study, the null hypothesis is that children who attend
academic after-school programs have the same intelligence level as other
children. Remember, we said that statistics allow us to disprove or falsify
a hypothesis. Therefore, if the null hypothesis is not supported, then our
original hypothesis—that children who attend academic after-school pro-
grams have different IQs than other children—is all that is left. In statistical
notation, the null hypothesis for this study is
H
0
:
0
1
, or
academic program
general population
The purpose of the study, then, is to decide whether H
0
is probably true or
probably false.
The hypothesis that the researcher wants to support is known as the
alternative hypothesis (H
a
), or the research hypothesis (H
1
). The statistical
notation for H
a
is
H
a
:
0
1
, or
academic program
general population
When we use inferential statistics, we are trying to reject H
0
, which means
that H
a
is supported.
One- and Two-Tailed Hypothesis Tests
The manner in which the previous research hypothesis (H
a
) was stated
reflects what is known statistically as a one-tailed hypothesis, or a
directional hypothesis—an alternative hypothesis in which the researcher
predicts the direction of the expected difference between the groups. In this
case, the researcher predicted the direction of the difference—namely, that
children in academic after-school programs will be more intelligent than
children in the general population. When we use a directional alternative
hypothesis, the null hypothesis is also, in some sense, directional. If the
alternative hypothesis is that children in academic after-school programs
will have higher intelligence test scores, then the null hypothesis is that
being in academic after-school programs either will have no effect on intel-
ligence test scores or will decrease intelligence test scores. Thus, the null
hypothesis for the one-tailed directional test might more appropriately be
written as
H
0
:
0
1
, or
academic program
general population
null hypothesis (H
0
)
The hypothesis predicting that
no difference exists between the
groups being compared.
null hypothesis (H
0
)
The hypothesis predicting that
no difference exists between the
groups being compared.
alternative hypothesis (H
a
),
or research hypothesis
(H
1
) The hypothesis that the
researcher wants to support,
predicting that a significant
difference exists between the
groups being compared.
alternative hypothesis (H
a
),
or research hypothesis
(H
1
) The hypothesis that the
researcher wants to support,
predicting that a significant
difference exists between the
groups being compared.
one-tailed hypothesis
(directional hypothesis)
An alternative hypothesis in
which the researcher predicts
the direction of the expected
difference between the groups.
one-tailed hypothesis
(directional hypothesis)
An alternative hypothesis in
which the researcher predicts
the direction of the expected
difference between the groups.
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In other words, if the alternative hypothesis for a one-tailed test is
0
1
,
then the null hypothesis is
0
1
, and to reject H
0
, the children in academic
after-school programs have to have intelligence test scores higher than those
in the general population.
The alternative to a one-tailed or directional test is a two-tailed
hypothesis, or a nondirectional hypothesis—an alternative hypothesis in
which the researcher expects to find differences between the groups but is
unsure what the differences will be. In our example, the researcher would
predict a difference in IQ scores between children in academic after-school
programs and those in the general population, but the direction of the
difference would not be predicted. Those in academic programs would be
expected to have either higher or lower IQs but not the same IQs as the
general population of children. The statistical notation for a two-tailed
test is
H
0
:
0
1
, or
academic program
general population
H
a
:
0
1
, or
academic program
general population
In our example, a two-tailed hypothesis does not really make sense.
Assume that the researcher has selected a random sample of children
from academic after-school programs to compare their IQs with the IQs of
children in the general population (as noted previously, we know that the
mean IQ for the population is 100). If we collected data and found that the
mean intelligence level of the children in academic after-school programs
is “significantly” (a term that will be discussed shortly) higher than the
mean intelligence level for the population, we could reject the null hypoth-
esis. Remember that the null hypothesis states that no difference exists
between the sample and the population. Thus, the researcher concludes
that the null hypothesis—that there is no difference—is not supported.
When the null hypothesis is rejected, the alternative hypothesis—that
those in academic programs have higher IQ scores than those in the gen-
eral population—is supported. We can say that the evidence suggests
that the sample of children in academic after-school programs represents
a specific population that scores higher on the IQ test than the general
population.
If, on the other hand, the mean IQ score of the children in academic
after-school programs is not significantly different from the population
mean score, then the researcher has failed to reject the null hypothesis and,
by default, has failed to support the alternative hypothesis. In this case, the
alternative hypothesis—that the children in academic programs have higher
IQs than the general population—is not supported.
Type I and II Errors in Hypothesis Testing
Anytime we make a decision using statistics, four outcomes are pos-
sible (see Table 7.1). Two of the outcomes represent correct decisions,
whereas two represent errors. Let’s use our example to illustrate these
possibilities.
two-tailed hypothesis (non-
directional hypothesis)
An alternative hypothesis in
which the researcher predicts
that the groups being compared
differ but does not predict the
direction of the difference.
two-tailed hypothesis (non-
directional hypothesis)
An alternative hypothesis in
which the researcher predicts
that the groups being compared
differ but does not predict the
direction of the difference.
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If we reject the null hypothesis (that there is no IQ difference between
groups), we may be correct in our decision, or we may be incorrect. If our
decision to reject H
0
is correct, that means there truly is a difference in IQ
between children in academic after-school programs and the general popu-
lation of children. However, our decision could be incorrect. The result may
have been due to chance. Even though we observed a significant difference
in IQs between the children in our study and the general population, the
result might have been a fluke—maybe the children in our sample just
happened to guess correctly on a lot of the questions. In this case, we have
made what is known as a Type I error—we rejected H
0
, when in reality, we
should have failed to reject it (it is true that there really is no IQ difference
between the sample and the population). Type I errors can be thought of
as false alarms—we said there was a difference, but in reality, there is no
difference.
What if our decision is to not reject H
0
, meaning we conclude that
there is no difference in IQs between the children in academic after-
school programs and children in the general population? This decision
could be correct, meaning that in reality, there is no IQ difference between
the sample and the population. However, it could also be incorrect. In
this case, we would be making a Type II error—saying there is no dif-
ference between groups when, in reality, there is a difference. Somehow
we have missed the difference that really exists and have failed to reject
the null hypothesis when it is false. These possibilities are summarized
in Table 7.1.
Statistical Significance and Errors
Suppose we actually do the study on IQ levels and academic after-school
programs. In addition, suppose we find that there is a difference between
the IQ levels of children in academic after-school programs and children in
the general population (those in academic programs score higher). Last, sup-
pose this difference is statistically significant at the .05 (or the 5%) level (also
known as the .05 alpha level). To say that a result has statistical significance
at the .05 level means that a difference as large as or larger than what we
observed between the sample and the population could have occurred by
Type I error An error in
hypothesis testing in which the
null hypothesis is rejected when
it is true.
Type I error An error in
hypothesis testing in which the
null hypothesis is rejected when
it is true.
Type II error An error in
hypothesis testing in which
there is a failure to reject the
null hypothesis when it is false.
Type II error An error in
hypothesis testing in which
there is a failure to reject the
null hypothesis when it is false.
statistical significance
An observed difference between
two descriptive statistics (such
as means) that is unlikely to
have occurred by chance.
statistical significance
An observed difference between
two descriptive statistics (such
as means) that is unlikely to
have occurred by chance.
TABLE 7.1 The Four Possible Outcomes in Statistical Decision Making
THE TRUTH (UNKNOWN TO THE RESEARCHER)
THE RESEARCHER’S DECISION H
0
IS TRUE H
0
IS FALSE
Reject H
0
Type I error Correct decision
(say it is false)
Fail to reject H
0
Correct decision Type II error
(say it is true)
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chance only 5 times or less out of 100. In other words, the likelihood that this
result is due to chance is small. If the result is not due to chance, then it is
most likely due to a true or real difference between the groups. If our result
is statistically significant, we can reject the null hypothesis and conclude that
we have observed a significant difference in IQ scores between the sample
and the population.
Remember, however, that when we reject the null hypothesis, we could
be correct in our decision, or we could be making a Type I error. Maybe the
null hypothesis is true, and this is one of those 5 or less times out of 100
when the observed differences between the sample and the population did
occur by chance. This means that when we adopt the .05 level of significance
(the .05 alpha level), as often as 5 times out of 100, we could make a Type I
error. The .05 level, then, is the probability of making a Type I error (for this
reason, it is also referred to as a p value, which means probability value—the
probability of a Type I error). In the social and behavioral sciences, alpha is
typically set at .05 (as opposed to .01, .08, or anything else). This means that
researchers in these areas are willing to accept up to a 5% risk of making a
Type I error.
What if you want to reduce your risk of making a Type I error and
decide to use the .01 alpha level—reducing the risk of a Type I error to 1
out of 100 times? This seems simple enough: Simply reduce alpha to .01,
and you have reduced your chance of making a Type I error. By doing this,
however, you have now increased your chance of making a Type II error.
Do you see why? If I reduce my risk of making a false alarm—saying a
difference is there when it really is not—I increase my risk of missing a dif-
ference that really is there. When we reduce the alpha level, we are insist-
ing on more stringent conditions for accepting our research hypothesis,
making it more likely that we could miss a significant difference when it
is present. We will return to Type I and II errors later in this chapter when
we cover statistical power and discuss alternative ways of addressing this
problem.
Which type of error, Type I or Type II, do you think is considered more
serious by researchers? Most researchers consider a Type I error more
serious. They would rather miss a result (Type II error) than conclude
that there is a meaningful difference when there really is not (Type I
error). What about in other arenas, for example, in the courtroom? A jury
could make a correct decision in a case (find guilty when truly guilty or
find innocent when truly innocent). They could also make either a Type
I error (say guilty when innocent) or a Type II error (say innocent when
guilty). Which is more serious here? Most people believe that a Type I
error is worse in this situation also. How about in the medical profession?
Imagine a doctor attempting to determine whether or not a patient has
cancer. Here again, the doctor could make one of the two correct decisions
or one of the two types of errors. What would the Type I error be? This
would be saying that cancer is present when in fact it is not. What about
the Type II error? This would be saying that there is no cancer when in
fact there is. In this situation, most people would consider a Type II error
to be more serious.
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CRITICAL
THINKING
CHECK
7.1
1. A researcher hypothesizes that children in the South weigh less
(because they spend more time outside) than the national average.
Identify H
0
and H
a
. Is this a one- or two-tailed test?
2. A researcher collects data on children’s weights from a random
sample of children in the South and concludes that children in the
South weigh less than the national average. The researcher, however,
does not realize that the sample includes many children who are
small for their age and that in reality there is no difference in weight
between children in the South and the national average. What type
of error is the researcher making?
3. If a researcher decides to use the .10 level rather than the conventional
.05 level of significance, what type of error is more likely to be made?
Why? If the .01 level is used, what type of error is more likely? Why?
IN REVIEW Hypothesis Testing
CONCEPT DESCRIPTION EXAMPLE
Null Hypothesis The hypothesis stating that the independent H
0
:
0
1
(two-tailed)
variable has no effect and that there will be H
0
:
0
1
(one-tailed)
no difference between the two groups. H
0
:
0
1
(one-tailed)
Alternative Hypothesis The hypothesis stating that the independent H
a
:
0
1
(two-tailed)
or Research Hypothesis variable has an effect and that there will be H
a
:
0
1
(one-tailed)
a difference between the two groups. H
a
:
0
1
(one-tailed)
Two-Tailed or An alternative hypothesis stating that a The mean of the sample will be
Nondirectional Test difference is expected between the groups, different from or unequal to the
but there is no prediction as to which mean of the general population.
group will perform better or worse.
One-Tailed or An alternative hypothesis stating that a The mean of the sample will be
Directional Test difference is expected between the groups, greater than the mean of the
and it is expected to occur in a specific population, or the mean of the
direction. sample will be less than the mean
of the population.
Type I Error The error of rejecting H
0
when we should This error in hypothesis testing is
have failed to reject it. equivalent to a “false alarm,”
saying that there is a difference
when in reality there is no
difference between the groups.
Type II Error The error of failing to reject H
0
when we This error in hypothesis testing is
should have rejected it. equivalent to a “miss,” saying
that there is not a difference
between the groups when in
reality there is.
Statistical Significance When the probability of a Type I error The difference between the groups
is low (.05 or less). is so large that we conclude it is
due to something other than chance.
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Single-Sample Research and Inferential
Statistics
Now that you understand the concept of hypothesis testing, we can begin to
discuss how hypothesis testing can be applied to research. The simplest type
of study involves only one group and is known as a single-group design.
The single-group design lacks a comparison group—there is no control
group of any sort. We can, however, compare the performance of the group
(the sample) with the performance of the population (assuming that popula-
tion data are available).
Earlier in the chapter, we illustrated hypothesis testing using a single-
group design—comparing the IQ scores of children in academic after-school
programs (the sample) with the IQ scores of children in the general popula-
tion. The null and alternative hypotheses for this study were
H
0
:
0
1
, or
academic program
general population
H
a
:
0
1
, or
academic program
general population
To compare the performance of the sample with that of the population, we
need to know the population mean () and the population standard devia-
tion (). We know that for IQ tests, 100 and 15. We also need to
decide who will be in the sample. As noted in previous chapters, random
selection increases our chances of getting a representative sample of children
enrolled in academic after-school programs. How many children do we need
in the sample? You will see later in this chapter that the larger the sample,
the greater the power of the study. We will also see that one of the assump-
tions of the statistical procedure we will be using to test our hypothesis is a
sample size of 30 or more.
After we have chosen our sample, we need to collect the data. We have
discussed data collection in several earlier chapters. It is important to make
sure that the data are collected in a nonreactive manner as discussed in
Chapter 4. To collect IQ score data, we could either administer an intelli-
gence test to the children or look at their academic files to see whether they
have already taken such a test.
After the data are collected, we can begin to analyze them using inferential
statistics. As noted previously, inferential statistics involve the use of proce-
dures for drawing conclusions based on the scores collected in a research study
and going beyond them to make inferences about a population. In this chapter,
we will describe three inferential statistical tests. The first two, the z test and the
t test, are parametric tests—tests that require us to make certain assumptions
about estimates of population characteristics, or parameters. These assump-
tions typically involve knowing the mean () and standard deviation () of
the population and that the population distribution is normal. Parametric
tests are generally used with interval or ratio data. The third statistical test, the
chi-square (
2
) goodness-of-fit test is a nonparametric test—that is, a test that
does not involve the use of any population parameters. In other words, and
are not needed, and the underlying distribution does not have to be normal.
Nonparametric tests are most often used with ordinal or nominal data.
single-group design
A research study in which
there is only one group of
participants.
single-group design
A research study in which
there is only one group of
participants.
parametric test A statistical
test that involves making
assumptions about estimates
of population characteristics, or
parameters.
parametric test A statistical
test that involves making
assumptions about estimates
of population characteristics, or
parameters.
nonparametric test
A statistical test that does
not involve the use of any
population parameters; and
are not needed, and the
underlying distribution does not
have to be normal.
nonparametric test
A statistical test that does
not involve the use of any
population parameters; and
are not needed, and the
underlying distribution does not
have to be normal.
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