158
Chapter 7
Testing Hypotheses
I
n the past, the increase in the price of gasoline could be attributed to major national or global
event, such as the Lebanon and Israeli war or Hurricane Katrina. However, in 2005, the price for a
gallon of regular gasoline reached $3.00 and remained high for a long time afterward. The impact
of unpredictable fuel prices is still felt across the nation, but the burden is greater among distinct social
economic groups and geographic areas.
Lower-income Americans spend eight times more of their disposable income on gasoline than
wealthier Americans do.
1
For example, in Wilcox, Alabama, individuals spend 12.72% of their income
to fuel one vehicle, while in Hunterdon Co., New Jersey, people spend 1.52%. Nationally, Americans
spend 3.8% of their income fueling one vehicle.
The first state to reach the $3.00-per-gallon milestone was California in 2005. California’s drivers
were especially hit hard by the rising price of gas, due in part to their reliance on automobiles, especially
for work commuters. Analysts predicted that gas prices would continue to rise nationally. Declines
in consumer spending and confidence in the economy have been attributed in part to the high (and
rising) cost of gasoline.
In 2010, gasoline prices have remained higher for states along the West Coast, particularly in Alaska
and California. Let’s say we drew a random sample of California gas stations (N = 100) and calculated
the mean price for a gallon of regular gas. Based on consumer information,
2
we also know that nation-
ally the mean price of a gallon was $2.86, with a standard deviation of 0.17 for the same week. We can
thus compare the mean price of gas in California with the mean price of all gas stations in April 2010.
By comparing these means, we are asking whether it is reasonable to consider our random sample
of California gas as representative of the population of gas stations in the United States. Actually, we
Understanding the assumptions of statistical hypothesis testing
Defining and applying the components in hypothesis testing: the research and null
hypotheses, sampling distribution, and test statistic
Understanding what it means to reject or fail to reject a null hypothesis
Applying hypothesis testing to two sample cases, with means or proportions
Chapter Learning Objectives
Testing Hypotheses— 159
expect to find that the average price of gas from a sample of California gas stations will be unrepresen-
tative of the population of gas stations because we assume higher gas prices in the state.
The mean price for our sample is $3.11. This figure is higher than $2.86, the mean price per gallon
across the nation. But is the observed gap of 25 cents ($3.11 - $2.86) large enough to convince us that
the sample of California gas stations is not representative of the population?
There is no easy answer to this question. The sample mean of $3.11 is higher than the population mean,
but it is an estimate based on a single sample. Thus, it could mean one of two things: (1) The average price
of gas in California is indeed higher than the national average or (2) the average price of gas in California is
about the same as the national average, and this sample happens to show a particularly high mean.
How can we decide which of these explanations makes more sense? Because most estimates are based
on single samples and different samples may result in different estimates, sampling results cannot be used
directly to make statements about a population. We need a procedure that allows us to evaluate hypotheses
about population parameters based on sample statistics. In Chapter 6, we saw that population parameters
can be estimated from sample statistics. In this chapter, we will learn how to use sample statistics to make
decisions about population parameters. This procedure is called statistical hypothesis testing.
Statistical hypothesis testing
A procedure that allows us to evaluate hypotheses about
population parameters based on sample statistics.
- ASSUMPTIONS OF STATISTICAL HYPOTHESIS TESTING
Statistical hypothesis testing requires several assumptions. These assumptions include considerations
of the level of measurement of the variable, the method of sampling, the shape of the population distri-
bution, and the sample size. The specific assumptions may vary, depending on the test or the conditions
of testing. However, without exception, all statistical tests assume random sampling. Tests of hypoth-
eses about means also assume interval-ratio level of measurement and require that the population
under consideration be normally distributed or that the sample size be larger than 50.
Based on our data, we can test the hypothesis that the average price of gas in California is higher
than the average national price of gas. The test we are considering meets these conditions:
1. The sample of California gas stations was randomly selected.
2. The variable price per gallon is measured at the interval-ratio level.
3. We cannot assume that the population is normally distributed. However, because our sample
size is sufficiently large (N > 50), we know, based on the central limit theorem, that the sampling
distribution of the mean will be approximately normal.
- STATING THE RESEARCH AND NULL HYPOTHESES
Hypotheses are usually defined in terms of interrelations between variables and are often based on
a substantive theory. Earlier, we defined hypotheses as tentative answers to research questions. They
160— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
are tentative because we can find evidence for them only after being empirically tested. The testing of
hypotheses is an important step in this evidence-gathering process.
The Research Hypothesis (H
1
)
Our first step is to formally express the hypothesis in a way that makes it amenable to a statistical
test. The substantive hypothesis is called the research hypothesis and is symbolized as H
1
. Research
hypotheses are always expressed in terms of population parameters because we are interested in
making statements about population parameters based on our sample statistics.
Research hypothesis (H
1
) A statement reflecting the substantive hypothesis. It is always
expressed in terms of population parameters, but its specific form varies from test to test.
In our research hypothesis (H
1
), we state that the average price of gas in California is higher than
the average price of gas nationally. Symbolically, we use m
Y
to represent the population mean; our
hypothesis can be expressed as
H
1
: m
Y
> $2.86
In general, the research hypothesis (H
1
) specifies that the population parameter is one of the following:
1. Not equal to some specified value: m
Y
≠ some specified value
2. Greater than some specified value: m
Y
> some specified value
3. Less than some specified value: m
Y
< some specified value
The Null Hypothesis (H
0
)
Is it possible that in the population there is no real difference between the mean price of gas in
California and the mean price of gas in the nation and that the observed difference of 0.25 is actually
due to the fact that this particular sample happened to contain California gas stations with higher
prices? Since statistical inference is based on probability theory, it is not possible to prove or disprove
the research hypothesis directly. We can, at best, estimate the likelihood that it is true or false.
To assess this likelihood, statisticians set up a hypothesis that is counter to the research hypothesis.
The null hypothesis, symbolized as H
0
, contradicts the research hypothesis and usually states that
there is no difference between the population mean and some specified value. It is also referred to
as the hypothesis of “no difference.” Our null hypothesis can be stated symbolically as
H
0
: m
Y
= $2.86
Rather than directly testing the substantive hypothesis (H
1
) that there is a difference between the
mean price of gas in California and the mean price nationally, we test the null hypothesis (H
0
) that there
Testing Hypotheses— 161
is no difference in prices. In hypothesis testing, we hope to reject the null hypothesis to provide support
for the research hypothesis. Rejection of the null hypothesis will strengthen our belief in the research
hypothesis and increase our confidence in the importance and utility of the broader theory from which
the research hypothesis was derived.
Null hypothesis (H
0
) A statement of “no difference” that contradicts the research
hypothesis and is always expressed in terms of population parameters.
More About Research Hypotheses: One- and Two-Tailed Tests
In a one-tailed test, the research hypothesis is directional; that is, it specifies that a population
mean is either less than (<) or greater than (>) some specified value. We can express our research
hypothesis as either
H
1
: m
Y
< some specified value
or
H
1
: m
Y
> some specified value
The research hypothesis we’ve stated for the average price of a gallon of regular gas in California is
a one-tailed test.
When a one-tailed test specifies that the population mean is greater than some specified value, we
call it a right-tailed test because we will evaluate the outcome at the right tail of the sampling distribu-
tion. If the research hypothesis specifies that the population mean is less than some specified value, it is
called a left-tailed test because the outcome will be evaluated at the left tail of the sampling distribu-
tion. Our example is a right-tailed test because the research hypothesis states that the mean gas prices
in California are higher than $2.86. (Refer to Figure 7.1 on page 163.)
Sometimes, we have some theoretical basis to believe that there is a difference between groups,
but we cannot anticipate the direction of that difference. For example, we may have reason to believe
that the average price of California gas is different from that of the general population, but we may not
have enough research or support to predict whether it is higher or lower. When we have no theoretical
reason for specifying a direction in the research hypothesis, we conduct a two-tailed test. The research
hypothesis specifies that the population mean is not equal to some specified value. For example, we can
express the research hypothesis about the mean price of gas as
H
1
: m
Y
≠ $2.86
With both one- and two-tailed tests, our null hypothesis of no difference remains the same. It can
be expressed as
H
0
: m
Y
= some specified value
162— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
One-tailed test A type of hypothesis test that involves a directional research hypothesis.
It specifies that the values of one group are either larger or smaller than some specified
population value.
Right-tailed test A one-tailed test in which the sample outcome is hypothesized to be at
the right tail of the sampling distribution.
Left-tailed test
A one-tailed test in which the sample outcome is hypothesized to be at
the left tail of the sampling distribution.
Two-tailed test A type of hypothesis test that involves a nondirectional research
hypothesis. We are equally interested in whether the values are less than or greater than
one another. The sample outcome may be located at both the lower and the higher ends of
the sampling distribution.
- DETERMINING WHAT IS
SUFFICIENTLY IMPROBABLE:
PROBABILITY VALUES AND ALPHA
Now let’s put all our information together. We’re assuming that our null hypothesis (m
Y
= $2.86) is true,
and we want to determine whether our sample evidence casts doubt on that assumption, suggesting
that there is evidence for research hypothesis, m
Y
> $2.86. What are the chances that we would have
randomly selected a sample of California gas stations such that the average price per gallon is higher
than $2.86, the average for the nation? We can determine the chances or probability because of what we
know about the sampling distribution and its properties. We know, based on the central limit theorem,
that if our sample size is larger than 50, the sampling distribution of the mean is approximately normal,
with a mean
l
Y
¼ l
Y
and a standard deviation (standard error) of
r
Y
¼
r
Y
ffiffiffiffi
N
p
We are going to assume that the null hypothesis is true and then see if our sample evidence casts
doubt on that assumption. We have a population mean m
Y
= $2.86 and a standard deviation s
Y
=
0.17. Our sample size is N = 100, and the sample mean is $3.11. We can assume that the distribution
of means of all possible samples of size N = 100 drawn from this distribution would be approximately
normal, with a mean of $2.86 and a standard deviation of
r
Y
¼
r
Y
ffiffiffiffi
N
p
¼
0:17
ffiffiffiffiffiffiffi
100
p
¼ 0:017
This sampling distribution is shown in Figure 7.1. Also shown in Figure 7.1 is the mean gas price we
observed for our sample of California gas stations.
Testing Hypotheses— 163
Because this distribution of sample means is normal, we can use Appendix A to determine the prob-
ability of drawing a sample mean of $3.11 or higher from this population. We will translate our sample
mean into a Z score so that we can determine its location relative to the population mean. In Chapter 5,
we learned how to translate a raw score into a Z score by using Formula 5.1:
Z
¼
Y
Y
S
Y
Because we are dealing with a sampling distribution in which our raw score is
Y
, the mean, and the
standard deviation (standard error) is
r
Y
=
ffiffiffiffi
N
p
, we need to modify the formula somewhat:
Z ¼
Y l
Y
r
Y
=
ffiffiffiffi
N
p
(7.1)
Converting the sample mean to a Z-score equivalent is called computing the test statistic. The Z value
we obtain is called the Z statistic (obtained). The obtained Z gives us the number of standard devia-
tions (standard errors) that our sample is from the hypothesized value (m
Y
or m
Y
–
), assuming the null
hypothesis is true. For our example, the obtained Z is
Z ¼
3:11 2:86
0:17=
ffiffiffiffiffiffiffi
100
p
¼ 14:70
Z statistic (obtained) The test statistic computed by converting a sample statistic (such
as the mean) to a Z score. The formula for obtaining Z varies from test to test.
Y
–
= $3.11
µ
Y
–
= $2.86
Figure 7.1 Sampling Distribution of Sample Means Assuming H
0
Is True for a Sample N = 100
164— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
Before we determine the probability of our obtained Z statistic, let’s determine whether it is consis-
tent with our research hypothesis. Recall that we defined our research hypothesis as a right-tailed test
(m
Y
> $2.86), predicting that the difference would be assessed on the right tail of the sampling distribu-
tion. The positive value of our obtained Z statistic confirms that we will be evaluating the difference on
the right tail. (If we had a negative obtained Z, it would mean the difference would have to be evaluated
at the left tail of the distribution, contrary to our research hypothesis.)
To determine the probability of observing a Z value of 14.70, assuming that the null hypothesis is
true, look up the value in Appendix A to find the area to the right of (above) the Z of 14.70. Our calcu-
lated Z value is not listed in Appendix A, so we’ll need to rely on the last Z value reported in the table,
4.00. Recall from Chapter 5, where we calculated Z scores and their probability, that the Z values are
located in Column A. The P value is the probability to the right of the obtained Z, or the “area beyond
Z” in Column C. This area includes the proportion of all sample means that are $3.11 or higher. The
proportion is less than 0.0001 (Figure 7.2). This value is the probability of getting a result as extreme as
the sample result if the null hypothesis is true; it is symbolized as P. Thus, for our example, P ≤ .0001.
0
.0001 of the area,
or P ≤ .0001
Z ≥ 14.70
Figure 7.2 The Probability (P) Associated With Z ≥ 14.70
A P value can be defined as the actual probability associated with the obtained value of Z. It is
a measure of how unusual or rare our obtained statistic is compared with what is stated in our null
hypothesis. The smaller the P value, the more evidence we have that the null hypothesis should be
rejected in favor of the research hypothesis.
P value
The probability associated with the obtained value of Z.
Testing Hypotheses— 165
Researchers usually define in advance what a sufficiently improbable Z value is by specifying a
cutoff point below which P must fall to reject the null hypothesis. This cutoff point, called alpha and
denoted by the Greek letter a, is customarily set at the .05, .01, or .001 level. Let’s say that we decide
to reject the null hypothesis if P ≤ .05. The value .05 is referred to as alpha (a); it defines for us what
result is sufficiently improbable to allow us to take the risk and reject the null hypothesis. An alpha (a)
of .05 means that even if the obtained Z statistic is due to sampling error, so that the null hypothesis is
true, we would allow a 5% risk of rejecting it. Alpha values of .01 and .001 are more cautionary levels of
risk. The difference between P and alpha is that P is the actual probability associated with the obtained
value of Z, whereas alpha is the level of probability determined in advance at which the null hypothesis
is rejected. The null hypothesis is rejected when P ≤ a.
alpha (a)
The level of probability at which the null hypothesis is rejected. It is custom-
ary to set alpha at the .05, .01, or .001 level.
We have already determined that our obtained Z has a probability value less than .0001. Since our
observed P is less than .05 (P = .0001 < a = .05), we reject the null hypothesis. The value of .0001
means that fewer than 1 out of 10,000 samples drawn from this population are likely to have a mean
that is 14.70 Z scores above the hypothesized mean of $2.86. Another way to say it is as follows: There
is only 1 chance out of 10,000 (or .0001%) that we would draw a random sample with a Z ≥ 14.70 if the
mean price of California gas were equal to the national mean price.
Based on the P value, we can also make a statement regarding the “significance” of the results. If
the P value is equal to or less than our alpha level, our obtained Z statistic is considered statistically
significant—that is to say, it is very unlikely to have occurred by random chance or sampling error. We
can state that the difference between the average price of gas in California and nationally is significantly
different at the .05 level, or we can specify the actual level of significance by saying that the level of
significance is less than .0001.
Recall that our hypothesis was a one-tailed test (m
Y
> $2.86). In a two-tailed test, sample outcomes
may be located at both the higher and the lower ends of the sampling distribution. Thus, the null
hypothesis will be rejected if our sample outcome falls either at the left or right tail of the sampling
distribution. For instance, a .05 alpha or P level means that H
0
will be rejected if our sample outcome
falls among either the lowest or the highest 5% of the sampling distribution.
Suppose we had expressed our research hypothesis about the mean price of gas as
H
1
: m
Y
≠ $2.86
The null hypothesis to be directly tested still takes the form H
0
: m
Y
= $2.86 and our obtained Z is
calculated using the same formula (7.1) as was used with a one-tailed test. To find P for a two-tailed
test, look up the area in Column C of Appendix A that corresponds to your obtained Z (as we did ear-
lier) and then multiply it by 2 to obtain the two-tailed probability. Thus, the two-tailed P value for Z =
14.70 is .0001 × 2 = .0002. This probability is less than our stated alpha (.05), and thus, we reject the
null hypothesis.
166— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
- THE FIVE STEPS IN HYPOTHESIS TESTING: A SUMMARY
Regardless of the particular application or problem, statistical hypothesis testing can be organized into
five basic steps. Let’s summarize these steps:
1. Making assumptions
2. Stating the research and null hypotheses and selecting alpha
3. Selecting the sampling distribution and specifying the test statistic
4. Computing the test statistic
5. Making a decision and interpreting the results
Making Assumptions. Statistical hypothesis testing involves making several assumptions regarding the
level of measurement of the variable, the method of sampling, the shape of the population distribution,
and the sample size. In our example, we made the following assumptions:
1. A random sample was used.
2. The variable price per gallon is measured on an interval-ratio level of measurement.
3. Because N > 50, the assumption of normal population is not required.
Stating the Research and Null Hypotheses and Selecting Alpha. The substantive hypothesis is called the
research hypothesis and is symbolized as H
1
. Research hypotheses are always expressed in terms of
population parameters because we are interested in making statements about population parameters
based on sample statistics. Our research hypothesis was
H
1
: m
Y
> $2.86
The null hypothesis, symbolized as H
0
, contradicts the research hypothesis in a statement of no dif-
ference between the population mean and our hypothesized value. For our example, the null hypothesis
was stated symbolically as
H
0
: m
Y
= $2.86
We set alpha at .05, meaning that we would reject the null hypothesis if the probability of our
obtained Z was less than or equal to .05.
Selecting the Sampling Distribution and Specifying the Test Statistic. The normal distribution and the Z
statistic are used to test the null hypothesis.
Computing the Test Statistic. Based on Formula 7.1, our Z statistic is 14.70.
Making a Decision and Interpreting the Results. We confirm that our obtained Z is on the right tail of
the distribution, consistent with our research hypothesis. Based on our obtained Z statistic of 14.70, we
Testing Hypotheses— 167
determine that its P value is less than .0001, less than our .05 alpha levels. We have evidence to reject
the null hypothesis of no difference between the mean price of California gas and the mean price of
gas nationally. We thus conclude that the price of California gas is, on average, significantly higher than
the national average.
- ERRORS IN HYPOTHESIS TESTING
We should emphasize that because our conclusion is based on sample data, we will never really know
if the null hypothesis is true or false. In fact, as we have seen, there is a 0.01% chance that the null
hypothesis is true and that we are making an error by rejecting it.
The null hypothesis can be either true or false, and in either case, it can be rejected or not rejected.
If the null hypothesis is true and we reject it nonetheless, we are making an incorrect decision. This
type of error is called a Type I error. Conversely, if the null hypothesis is false but we fail to reject it,
this incorrect decision is a Type II error.
Type I error The probability associated with rejecting a null hypothesis when it is true.
Type II error The probability associated with failing to reject a null hypothesis when it
is false.
In Table 7.1, we show the relationship between the two types of errors and the decisions we make
regarding the null hypothesis. The probability of a Type I error—rejecting a true hypothesis—is equal
to the chosen alpha level. For example, when we set alpha at the .05 level, we know that the probability
that the null hypothesis is in fact true is .05 (or 5%).
Table 7.1
Type I and Type II Errors
True State of Affairs
Decision Made
H
0
Is Tr ue H
0
Is False
Reject H
0
Type I error (a)
Correct decision
Do not reject H
0
Correct decision Type II error
We can control the risk of rejecting a true hypothesis by manipulating alpha. For example, by setting
alpha at .01, we are reducing the risk of making a Type I error to 1%. Unfortunately, however, Type I and
Type II errors are inversely related; thus, by reducing alpha and lowering the risk of making a Type I
error, we are increasing the risk of making a Type II error (Table 7.1).
As long as we base our decisions on sample statistics and not population parameters, we have to
accept a degree of uncertainty as part of the process of statistical inference.
168— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
✓ Learning
Check
The implications of research findings are not created equal. For example, researchers might hypoth-
esize that eating spinach increases the strength of weight lifters. Little harm will be done if the null
hypothesis that eating spinach has no effect on the strength of weight lifters is rejected in error. The
researchers would most likely be willing to risk a high probability of a Type I error, and all weight
lifters would eat spinach. However, when the implications of research have important consequences,
the balancing act between Type I and Type II errors becomes more important. Can you think of
some examples where researchers would want to minimize Type I errors? When might they want to
minimize Type II errors?
The t Statistic and Estimating the Standard Error
The Z statistic we have calculated (Formula 7.1) to test the hypothesis involving a sample of
California gas stations assumes that the population standard deviation s
Y
is known. The value of s
Y
is
required to calculate the standard error
r
Y
ffiffiffiffi
N
p
In most situations, s
Y
will not be known, and we will need to estimate it using the sample standard
deviation S
Y
. We then use the t statistic instead of the Z statistic to test the null hypothesis. The formula
for computing the t statistic is
t ¼
Y l
Y
S
Y
=
ffiffiffiffi
N
p
(7.2)
The t value we calculate is called the t statistic (obtained). The obtained t represents the number
of standard deviation units (or standard error units) that our sample mean is from the hypothesized
value of m
Y
, assuming that the null hypothesis is true.
t statistic (obtained)
The test statistic computed to test the null hypothesis about a
population mean when the population standard deviation is unknown and is estimated
using the sample standard deviation.
The t Distribution and Degrees of Freedom
To understand the t statistic, we should first be familiar with its distribution. The t distribution
is actually a family of curves, each determined by its degrees of freedom. The concept of degrees of
freedom is used in calculating several statistics, including the t statistic. The degrees of freedom (df)
represent the number of scores that are free to vary in calculating each statistic.
Testing Hypotheses— 169
t distribution A family of curves, each determined by its degrees of freedom (df). It is
used when the population standard deviation is unknown and the standard error is esti-
mated from the sample standard deviation.
Degrees of freedom (df
) The number of scores that are free to vary in calculating a
statistic.
To calculate the degrees of freedom, we must know the sample size and whether there are any
restrictions in calculating that statistic. The number of restrictions is then subtracted from the sample
size to determine the degrees of freedom. When calculating the t statistic for a one-sample test, we
start with the sample size N and lose 1 degree of freedom for the population standard deviation we
estimate.
3
Note that the degrees of freedom will increase as the sample size increases. In the case of a
single-sample mean, the df is calculated as follows:
df = N – 1 (7.3)
Comparing the t and Z Statistics
Notice the similarities between the formulas for the t and Z statistics. The only apparent difference
is in the denominator. The denominator of Z is the standard error based on the population standard
deviation s
Y
. For the denominator of t, we replace
r
Y
=
ffiffiffiffi
N
p
with
S
Y
=
ffiffiffiffi
N
p
, the estimated standard error
based on the sample standard deviation.
However, there is another important difference between the Z and t statistics
: Because it is esti-
mated from sample data, the denominator of the t statistic is subject to sampling error. The sampling
distribution of the test statistic is not normal, and the standard normal distribution cannot be used to
determine probabilities associated with it.
In Figure 7.3, we present the t distribution for several dfs. Like the standard normal distribution, the
t distribution is bell shaped. The t statistic, similar to the Z statistic, can have positive and negative val-
ues. A positive t statistic corresponds to the right tail of the distribution; a negative value corresponds to
the left tail. Note that when the df is small, the t distribution is much flatter than the normal curve. But
as the degrees of freedom increase, the shape of the t distribution gets closer to the normal distribution,
until the two are almost identical when df is greater than 120.
Appendix B summarizes the t distribution. We have reproduced a small part of this appendix in
Table 7.2. Note that the t table differs from the normal (Z) table in several ways. First, the column on the
left side of the table shows the degrees of freedom. The t statistic will vary depending on the degrees
of freedom, which must first be computed (df = N - 1). Second, the probabilities or alpha, denoted as
significance levels, are arrayed across the top of the table in two rows, the first for a one-tailed and the
second for a two-tailed test. Finally, the values of t, listed as the entries of this table, are a function of
(1) the degrees of freedom, (2) the level of significance (or probability), and (3) whether the test is a
one- or a two-tailed test.
170— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
To illustrate the use of this table, let’s determine the probability of observing a t value of 2.021 with 40
degrees of freedom and a two-tailed test. Locating the proper row (df = 40) and column (two-tailed test),
we find the t statistic of 2.021 corresponding to the .05 level of significance. Restated, we can say that the
probability of obtaining a t statistic of 2.021 is .05, or that there are less than 5 chances out of 100 that we
would have drawn a random sample with an obtained t of 2.021 if the null hypothesis were correct.
Statistics in Practice: The Earnings of White Women
We drew a 2002 General Social Survey (GSS) sample (N = 371) of white females who worked full-
time. We found their mean earnings to be $28,889, with a standard deviation S
Y
= $21,071. Based on the
Current Population Survey,
4
we also know that the 2002 mean earnings nationally for all women was
m
Y
= $24,146. However, we do not know the value of the population standard deviation. We want to
determine whether the sample of white women was representative of the population of all full-time
women workers in 2002. Although we suspect that white American women experienced a relative
advantage in earnings, we are not sure enough to predict that their earnings were indeed higher than
the earnings of all women nationally. Therefore, the statistical test is two-tailed.
Let’s apply the five-step model to test the hypothesis that the average earnings of white women dif-
fered from the average earnings of all women working full-time in the United States in 2002.
Making Assumptions. Our assumptions are as follows:
1. A random sample is selected.
2. Because N > 50, the assumption of normal population is not required.
3. The level of measurement of the variable income is interval ratio.
df = ∞ (the normal curve) df = 20
df = 5
df = 1
Figure 7.3 The Normal Distribution and t Distributions for 1, 5, 20, and ∞ Degrees of Freedom
Testing Hypotheses— 171
Stating the Research and the Null Hypotheses and Selecting Alpha. The research hypothesis is
H
1
: m
Y
≠ $24,146
and the null hypothesis is
H
0
: m
Y
= $24,146
We’ll set alpha at .05, meaning that we will reject the null hypothesis if the probability of our
obtained statistic is less than or equal to .05.
Selecting the Sampling Distribution and Specifying the Test Statistic. We use the t distribution and the
t statistic to test the null hypothesis.
Computing the Test Statistic. We first calculate the df associated with our test:
df = (N - 1) = (371 - 1) = 370
Table 7.2
Values of the t Distribution
Source: Abridged from R. A. Fisher and F. Yates, Statistical Tables for Biological,
Agricultural and Medical Research, Table 111. Copyright R. A. Fisher and F. Yates,
1963. Published by Pearson Education Limited.
Level of Significance for One-Tailed Test
.10 .05
.025 .01
.005 .0005
Level of Significance for Two-Tailed Test
df
.20 .10
.05 .02 .01 .001
1 3.078 6.314 12.706 31.821 63.657 636.619
2 1.886 2.920 4.303 6.965 9.925 31.598
3 1.638 2.353 3.182 4.541 5.841 12.941
4 1.533 2.132 2.776 3.747 4.604 8.610
5 1.476 2.015 2.571 3.365 4.032 6.859
10 1.372 1.812 2.228 2.764 3.169 4.587
15 1.341 1.753 2.131 2.602 2.947 4.073
20 1.325 1.725 2.086 2.528 2.845 3.850
25 1.316 1.708 2.060 2.485 2.787 3.725
30 1.310 1.697 2.042 2.457 2.750 3.646
40 1.303 1.684 2.021 2.423 2.704 3.551
60 1.296 1.671 2.000 2.390 2.660 3.460
80 1.289 1.658 1.980 2.358 2.617 3.373
∞
1.282 1.645
1.960 2.326 2.576 3.291
172— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
We need to calculate the obtained t statistic by using Formula 7.2:
t ¼
Y l
Y
S
Y
=
ffiffiffiffi
N
p
¼
28;889 24;146
21;071
=
ffiffiffiffiffiffiffi
371
p
¼ 4:33
Making a Decision and Interpreting the Results. Given our research hypothesis, we will conduct a two-
tailed test. To determine the probability of observing a t value of 4.33 with 370 degrees of freedom, let’s
refer to Table 7.2. From the first column, we can see that 370 degrees of freedom is not listed, so we’ll
have to use the last row, df = ∞, to locate our obtained t statistic.
Though our obtained t statistic of 4.33 is not listed in the last row of t statistics, in fact, it is greater
than the last value listed in the row, 3.291. The t statistic of 3.291 corresponds to the .001 level of sig-
nificance for two-tailed tests. Restated, we can say that our obtained t statistic of 4.33 is greater than
3.291 or the probability of obtaining a t statistic of 4.33 is less than .001 (P < .001). This P value is below
our .05 alpha level. The probability of obtaining the difference of $4,743 ($28,889 - $24,146) between
the income of white women and the national average for all women, if the null hypothesis were true, is
extremely low. We have sufficient evidence to reject the null hypothesis and conclude that the average
earnings of white women in 2002 were significantly different from the average earnings of all women.
The difference of $4,743 is significant at the .05 level. We can also say that the level of significance is
less than .001.
- TESTING HYPOTHESES ABOUT TWO SAMPLES
In practice, social scientists are often more interested in situations involving two (sample) parameters
than those involving one. For example, we may be interested in finding out whether the average years
of education for one racial/ethnic group is the same, lower, or higher than another group.
U.S. data on educational attainment reveal that Asians and Pacific Islanders have more years
of education than any other racial/ethnic groups; this includes the percentage of those earning a
high school degree or higher or a college degree or higher. Though years of education have steadily
increased for blacks and Hispanics since 1990, their numbers remain behind Asians and Pacific
Islanders and whites.
Using data from the 2008 GSS, we examine the difference in black and Hispanic educational attain-
ment. From the GSS sample, black respondents reported an average of 12.80 years of education and
Hispanics an average of 10.63 years as shown in Table 7.3. These sample averages could mean either
(1) the average number of years of education for blacks is higher than the average for Hispanics or
(2) the average for blacks is actually about the same as for Hispanics, but our sample just happens to
indicate a higher average for blacks. What we are applying here is a bivariate analysis (for more infor-
mation, refer to Chapter 8), a method to detect and describe the relationship between two variables—
race/ethnicity and educational attainment.
The statistical procedures discussed in the following sections allow us to test whether the differ-
ences that we observe between two samples are large enough for us to conclude that the populations
from which these samples are drawn are different as well. We present tests for the significance of the
differences between two groups. Primarily, we consider differences between sample means and differ-
ences between sample proportions.
Testing Hypotheses— 173
The Assumption of Independent Samples
With a two-sample case, we assume that the samples are independent of each other. The choice of sample
members from one population has no effect on the choice of sample members from the second population.
In our comparison of blacks and Hispanics, we are assuming that the selection of blacks is independent of the
selection of Hispanics. (The requirement of independence is also satisfied by selecting one sample randomly,
then dividing the sample into appropriate subgroups. For example, we could randomly select a sample and
then divide it into groups based on gender, religion, income, or any other attribute that we are interested in.)
Stating the Research and Null Hypotheses
With two-sample tests, we compare two population parameters.
Our research hypothesis (H
1
) is that the average years of education for blacks is not equal to the
average years of education for Hispanic respondents. We are stating a hypothesis about the relationship
between race/ethnicity and education in the general population by comparing the mean educational
attainment of blacks with the mean educational attainment of Hispanics. Symbolically, we use m to
represent the population mean; the subscript 1 refers to our first sample (blacks) and subscript 2 to our
second sample (Hispanics). Our research hypothesis can then be expressed as
H
1
: m
1
≠ m
2
Because H
1
specifies that the mean education for blacks is not equal to the mean education for
Hispanics, it is a nondirectional hypothesis. Thus, our test will be a two-tailed test. Alternatively, if
there were sufficient basis for deciding which population mean score is larger (or smaller), the research
hypothesis for our test would be a one-tailed test:
H
1
: m
1
< m
2
or H
1
: m
1
> m
2
In either case, the null hypothesis states that there are no differences between the two population
means:
H
0
: m
1
= m
2
Table 7.3 Years of Education for Black and Hispanic Men and
Women, GSS 2008
Blacks (Sample 1) Hispanics (Sample 2)
Mean 12.80 10.63
Standard deviation 2.55 3.76
Vari ance 6.50 14.14
N
279 82
174— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
✓ Learning
Check
We are interested in finding evidence to reject the null hypothesis of no difference so that we have
sufficient support for our research hypothesis.
For the following research situations, state your research and null hypotheses:
• There is a difference between the mean statistics grades of social science majors and the mean
statistics grades of business majors.
• The average number of children in two-parent black families is lower than the average number
of children in two-parent nonblack families.
• Grade point averages are higher among girls who participate in organized sports than among
girls who do not.
- THE SAMPLING DISTRIBUTION OF THE
DIFFERENCE BETWEEN MEANS
The sampling distribution allows us to compare our sample results with all possible sample outcomes
and estimate the likelihood of their occurrence. Tests about differences between two sample means are
based on the sampling distribution of the difference between means. The sampling distribution
of the difference between two sample means is a theoretical probability distribution that would be
obtained by calculating all the possible mean differences
Y
1
Y
2
ðÞ
by drawing all possible indepen-
dent random samples of size N
1
and N
2
from two populations.
Sampling distribution of the difference between means A theoretical probability
distribution that would be obtained by calculating all the possible mean differences
Y
1
Y
2
ðÞ
that would be obtained by drawing all the possible independent random
samples of size N
1
and N
2
from two populations where N
1
and N
2
are both greater than 50.
The properties of the sampling distribution of the difference between two sample means are
determined by a corollary to the central limit theorem. This theorem assumes that our samples are
independently drawn from normal populations, but that with sufficient sample size (N
1
> 50, N
2
> 50)
the sampling distribution of the difference between means will be approximately normal, even if the
original populations are not normal. This sampling distribution has a mean
l
Y
1
l
Y
2
and a standard
deviation (standard error)
r
Y
1
Y
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r
2
Y
1
N
1
þ
r
2
Y
2
N
2
s
(7.4)
which is based on the variances in each of the two populations (
r
2
Y
1
and
r
2
Y
2
).
Testing Hypotheses— 175
Estimating the Standard Error
Formula 7.4 assumes that the population variances are known and that we can calculate the standard
error
r
Y
1
Y
2
(the standard deviation of the sampling distribution). However, in most situations, the only
data we have are based on sample data, and we do not know the true value of the population variances,
r
2
Y
1
and
r
2
Y
2
. Thus, we need to estimate the standard error from the sample variances,
S
2
Y
1
and
S
2
Y
2
. The
estimated standard error of the difference between means is symbolized as
S
Y
1
Y
2
(instead of
r
Y
1
Y
2
).
Calculating the Estimated Standard Error
When we can assume that the two population variances are equal, we combine information from the
two sample variances to calculate the estimated standard error.
S
Y
1
Y
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðN
1
1ÞS
2
Y
1
þðN
2
1ÞS
2
Y
2
ð
N
1
þ
N
2
Þ
2
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
1
þ N
2
N
1
N
2
s
(7.5)
where
S
Y
1
Y
2
is the estimated standard error of the difference between means, and
S
2
Y
1
and
S
2
Y
2
are the
variances of the two samples. As a rule of thumb, when either sample variance is more than twice as large
as the other, we can no longer assume that the two population variances are equal and would need to use
Formula 7.8.
The t Statistic
As with single sample means, we use the t distribution and the t statistic whenever we estimate
the standard error for a difference between means test. The t value we calculate is the obtained t. It
represents the number of standard deviation units (or standard error units) that our mean difference
Y
1
Y
2
ðÞ
is from the hypothesized value of m
1
– m
2
, assuming that the null hypothesis is true.
The formula for computing the t statistic for a difference between means test is
t ¼
Y
1
Y
2
S
Y
1
Y
2
(7.6)
where
S
Y
1
Y
2
is the estimated standard error.
Calculating the Degrees of Freedom
for a Difference Between Means Test
To use the t distribution for testing the difference between two sample means, we need to calculate the
degrees of freedom. As we saw earlier, the degrees of freedom (df) represent the number of scores that
are free to vary in calculating each statistic. When calculating the t statistic for the two-sample test, we
lose 2 degrees of freedom, one for every population variance we estimate. When population variances are
assumed to be equal or if the size of both samples is greater than 50, the df is calculated as follows:
df
¼
ð
N
1
þ
N
2
Þ
2
(7.7)
176— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
When we cannot assume that the population variances are equal and when the size of one or both
samples is equal to or less than 50, we use Formula 7.9 to calculate the degrees of freedom.
- POPULATION VARIANCES ARE ASSUMED TO BE UNEQUAL
If the variances of the two samples (
S
2
Y
1
and
S
2
Y
2
) are very different (one variance is twice as large as the
other), the formula for the estimated standard error becomes
S
Y
1
Y
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
2
Y
1
N
1
þ
S
2
Y
2
N
2
s
(7.8)
When the population variances are unequal and the size of one or both samples is equal to or less
than 50, we use another formula to calculate the degrees of freedom associated with the t statistic:
5
df ¼
ðS
2
Y
1
=N
1
þ S
2
Y
2
=N
2
Þ
2
ð
S
2
Y
1
=N
1
Þ
2
=
ð
N
1
1
Þþð
S
2
Y
2
=N
2
Þ
2
=
ð
N
2
1
Þ
(7.9)
- THE FIVE STEPS IN HYPOTHESIS TESTING
ABOUT DIFFERENCE BETWEEN MEANS: A SUMMARY
As with single-sample tests, statistical hypothesis testing involving two sample means can be organized
into five basic steps. Let’s summarize these steps:
1. Making assumptions
2. Stating the research and null hypotheses and selecting alpha
3. Selecting the sampling distribution and specifying the test statistic
4. Computing the test statistic
5. Making a decision and interpreting the results
Making Assumptions. In our example, we made the following assumptions:
1. Independent random samples are used.
2. The variable years of education is measured at an interval-ratio level of measurement.
3. Because N
1
> 50 and N
2
> 50, the assumption of normal population is not required.
4. The population variances are assumed to be equal.
Stating the Research and Null Hypotheses and Selecting Alpha. Our research hypothesis is that the mean
education of blacks is different from the mean education of Hispanics, indicating a two-tailed test.
Symbolically, the research hypothesis is expressed as
H
1
: m
1
≠ m
2
with m
1
representing the mean education of blacks and m
2
the mean education of Hispanics.
Testing Hypotheses— 177
The null hypothesis states that there are no differences between the two population means, or
H
0
: m
1
= m
2
We are interested in finding evidence to reject the null hypothesis of no difference so that we have
sufficient support for our research hypothesis. We will reject the null hypothesis if the probability of t
(obtained) is less than or equal to .05 (our alpha value).
Selecting the Sampling Distribution and Specifying the Test Statistic. The t distribution and the t statistic
are used to test the significance of the difference between the two sample means.
Computing the Test Statistic. To test the null hypothesis about the differences between the mean educa-
tion of blacks and Hispanics, we need to translate the ratio of the observed differences to its standard
error into a t statistic (based on data presented in Table 7.3). The obtained t statistic is calculated using
Formula 7.6:
t ¼
Y
1
Y
2
S
Y
1
Y
2
where
S
Y
1
Y
2
is the estimated standard error of the sampling distribution. Because the population
variances are assumed to be equal, df is (N
1
+ N
2
) - 2 = (279 + 82) - 2 = 359 and we can combine
information from the two sample variances to estimate the standard error (Formula 7.5):
S
Y
1
Y
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð279 1Þ2:55
2
þð82 1Þ3:76
2
ð
279
þ
82
Þ
2
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
279 þ 82
279
ð
82
Þ
s
¼ 2:87ðÞ0:13ðÞ¼ 0:37
We substitute this value into the denominator for the t statistic (Formula 7.6):
t
¼
12:80 10:63
0
:
37
¼
5:86
Making a Decision and Interpreting the Results. We confirm that our obtained t is on the right tail of
the distribution. Since our obtained t statistic of 5.86 is greater than t = 3.291 (df = ∞, two-tailed; see
Appendix B), we can state that its probability is less than .001. This is less than our .05 alpha level,
and we can reject the null hypothesis of no difference between the educational attainment of blacks
and Hispanics. We conclude that black men and women, on average, have significantly higher years of
education than Hispanic men and women do.
- TESTING THE SIGNIFICANCE OF THE DIFFERENCE
BETWEEN TWO SAMPLE PROPORTIONS
Numerous variables in the social sciences are measured at a nominal or an ordinal level. These vari-
ables are often described in terms of proportions. For example, we might be interested in comparing
the proportion of those who support immigrant policy reform among Hispanics and non-Hispanics
or the proportion of men and women who supported the Democratic candidate during the last
178— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
presidential election. In this section, we present statistical inference techniques to test for significant
differences between two sample proportions.
Hypothesis testing with two sample proportions follows the same structure as the statistical tests
presented earlier: The assumptions of the test are stated, the research and null hypotheses are formu-
lated, the sampling distribution and the test statistic are specified, the test statistic is calculated, and a
decision is made whether or not to reject the null hypothesis.
Statistics in Practice: Political Party Affiliation
and Confidence in the Executive Branch
Pollsters often collect data to measure public opinions on current social and political issues.
Differences are often categorized by income, race/ethnicity, gender, region, or political groups. For
example, do Republicans and Democrats have the same confidence in the executive branch of govern-
ment? We can use data from the 2008 GSS to test the null hypothesis that the proportion of Republicans
and Democrats who have a “great deal” of confidence in the executive branch of the government is
equal. The proportion of Republicans who reported a great deal of confidence was 0.21 (p
1
); the pro-
portion of Democrats with the same response was lower at 0.06 (p
2
). A total of 141 Republicans (N
1
)
and 267 Democratic respondents (N
2
) answered this question.
Making Assumptions. Our assumptions are as follows:
1. Independent random samples of N
1
> 50 and N
2
> 50 are used.
2. The level of measurement of the variable is nominal.
Stating the Research and Null Hypotheses and Selecting Alpha. We propose a two-tailed test that the
population proportions for Republicans and Democrats are not equal.
H
1
: p
1
≠ p
2
H
0
: p
1
= p
2
We decide to set alpha at .05.
Selecting the Sampling Distribution and Specifying the Test Statistic. The population distributions of dichot-
omies are not normal. However, based on the central limit theorem, we know that the sampling distribu-
tion of the difference between sample proportions is normally distributed when the sample size is large
(when N
1
> 50 and N
2
> 50), with mean
l
p
1
p
2
and the estimated standard error
S
p
1
p
2
. Therefore, we
can use the normal distribution as the sampling distribution and we can calculate Z as the test statistic.
6
The formula for computing the Z statistic for a difference between proportions test is
Z ¼
p
1
p
2
S
p
1
p
2
(7.10)
where p
1
and p
2
are the sample proportions for Republicans and Democrats, and
S
p
1
p
2
is the estimated
standard error of the sampling distribution of the difference between sample proportions.
Testing Hypotheses— 179
The estimated standard error is calculated using the following formula:
S
p
1
p
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
1
ð1 p
1
Þ
N
1
þ
p
2
ð1 p
2
Þ
N
2
r
(7.11)
Calculating the Test Statistic. We calculate the standard error using Formula 7.11:
S
p
1
p
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:21ð1 0:21Þ
141
þ
0:06ð1 0:06Þ
267
r
¼
:037
¼
0:04
Substituting this value into the denominator of Formula 7.10, we get
Z
¼
0:21 0:06
0
:
0
4
¼
3:75
Making a Decision and Interpreting the Results. Our obtained Z of 3.75 indicates that the difference
between the two proportions will be evaluated at the right tail of the Z distribution. To determine the
probability of observing a Z value of 3.75 if the null hypothesis is true, look up the value in Appendix A
(Column C) to find the area to the right of (above) the obtained Z.
The P value corresponding to a Z score of 3.75 is .0001. However, for a two-tailed test, we’ll have to
multiply P by 2 (.0001 × 2 = .0002). The probability of 3.75 for a two-tailed test is less than our alpha
level of .05 (.0002 < .05).
Thus, we reject the null hypothesis of no difference and conclude that there is a significant political
party difference in confidence in the executive branch of the government. Republicans are more likely
than Democrats to report a great deal of confidence in the executive branch.
- IS THERE A SIGNIFICANT DIFFERENCE?
The news media made note of a 2010 Centers for Disease Control (CDC) study that examined the dif-
ference in length of marriage between couples in 2005 who first cohabited before marriage and couples
who did not cohabit before marriage. Several news services released stories noting the “troubles” asso-
ciated with living together. As reported, the percentage of marriages surviving to the 10th anniversary,
among those who cohabited before marriage, was lower than those who did not cohabit before their
first marriage. A closer look at the report reveals important (overlooked) details.
CDC researchers Goodwin, Mosher, and Chandra (2010) reported that previous cohabitation expe-
rience was significantly associated with marriage survival probabilities for men. On the other hand,
though the probability that a woman’s marriage would last at least 10 years was lower for those who
cohabited before marriage (60%) than for women who did not (66%), the researchers wrote, “however,
in the 2002 data, the difference was not significant at the 5% level” (p. 7).
7
Throughout this chapter, we’ve assessed the difference between two means and two proportions,
attempting to determine whether the difference between them is due to real effects in the population or
due to sampling error. A significant difference is one that confirms that effects of the independent variable,
such as cohabiting before marriage, are real. As in the case of marriage survival rate, cohabitation before
marriage makes a significant difference in marital outcomes for men, but not for women in the CDC
180— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
sample. Take caution in accepting comparative statements that fail to mention significance. There may be a
difference, but you have to ask, is it a significant difference?
- READING THE RESEARCH LITERATURE: REPORTING
THE RESULTS OF STATISTICAL HYPOTHESIS TESTING
Robert Emmet Jones and Shirley A. Rainey (2006) examined the relationship between race, environ-
mental attitudes, and perceptions about environmental health and justice.
8
Researchers have docu-
mented how people of color and the poor are more likely than whites and more affluent groups to live
in areas with poor environmental quality and protection, exposing them to greater health risks. Yet little
is known about how this disproportional exposure and risk are perceived by those affected. Jones and
Rainey studied black and white residents from the Red River community in Tennessee, collecting data
from interviews and a mail survey during 2001 to 2003.
They created a series of index scales measuring residents’ attitudes pertaining to environmental
problems and issues. The Environmental Concern (EC) Index measures public concern for specific envi-
ronmental problems in the neighborhood. It includes questions on drinking water quality, landfills, loss
of trees, lead paint and poisoning, the condition of green areas, and stream and river conditions. EC-II
measures public concern (very unconcerned to very concerned) for the overall environmental quality
in the neighborhood. EC-III measures the seriousness (not serious at all to very serious) of environ-
mental problems in the neighborhood. Higher scores on all EC indicators indicate greater concern for
environmental problems in their neighborhood. The Environmental Health (EH) Index measures public
perceptions of certain physical side effects, such as headaches, nervous disorders, significant weight loss
or gain, skin rashes, and breathing problems. The EH Index measures the likelihood (very unlikely to
very likely) that the person believes that he or she or a household member experienced health problems
due to exposure to environmental contaminants in his or her neighborhood. Higher EH scores reflect a
greater likelihood that respondents believe that they have experienced health problems from exposure
to environmental contaminants. Finally, the Environmental Justice (EJ) Index measures public percep-
tions about environmental justice, measuring the extent to which they agreed (or disagreed) that public
officials had informed residents about environmental problems, enforced environmental laws, or held
meetings to address residents’ concerns. A higher mean EJ score indicates a greater likelihood that
respondents think public officials failed to deal with environmental problems in their neighborhood.
Index score comparisons between black and white respondents are presented in Table 7.4.
Let’s examine the table carefully. Each row represents a single index measurement, reporting means
and standard deviations separately for black and white residents. Obtained t-test statistics are reported
in the second to last column. The probability of each t test is reported in the last column (P < .001),
indicating a significant difference in responses between the two groups. All index score comparisons
are significant at the .001 level.
While not referring to specific differences in index scores or to t-test results, Jones and Rainey use
data from this table to summarize the differences between black and white residents on the three envi-
ronmental index measurements:
The results presented [in Table 1] suggest that as a group, Blacks are significantly more concerned
than Whites about local environmental conditions (EC Index). . . . The results . . . also indicate that
Testing Hypotheses— 181
Table 7.4 Environmental Concern (EC), Environmental Health (EH), and Environmental Justice (EJ)
Indicator Group Mean Standard Deviation
t
Significance (one-tailed)
EC Index Blacks 56.2 13.7 6.2
<0.001
Whites 42.6 15.5
EC-II Blacks
4.4 1.0 5.6
<0.001
Whites
3.5 1.3
EC-III Blacks 3.4 1.1 6.7
<0.001
Whites
2.3 1.0
EH Index Blacks 23.0 10.5 5.1
<0.001
Whites 16.0
7.3
EJ Index Blacks 31.0 7.3 3.8
<0.001
Whites 27.2
6.3
Source: Robert E. Jones and Shirley A. Rainey, “Examining Linkages Between Race, Environmental Concern, Health and Justice
in a Highly Polluted Community of Color,” Journal of Black Studies 36, no. 4 (2006): 473–496.
Note: N = 78 blacks, 113 whites.
as a group, Blacks believe they have suffered more health problems from exposure to poor environ-
mental conditions in their neighborhood than Whites (EH Index). . . . [T]here is greater likelihood
that Blacks feel local public agencies and officials failed to deal with environmental problems in
their neighborhood in a fair, just, and effective manner (EJ Index). (p. 485)
MAIN POINTS
•• Statistical hypothesis testing is a decision-
making process that enables us to determine
whether a particular sample result falls within
a range that can occur by an acceptable level
of chance. The process of statistical hypothesis
testing consists of five steps: (1) making assump-
tions, (2) stating the research and null hypoth-
eses and selecting alpha, (3) selecting a sampling
distribution and a test statistic, (4) computing
the test statistic, and (5) making a decision and
interpreting the results.
•• Statistical hypothesis testing may involve a
comparison between a sample mean and a popula-
tion mean or a comparison between two sample
means. If we know the population variance(s) when
testing for differences between means, we can use
the Z statistic and the normal distribution. However,
in practice, we are unlikely to have this information.
•• When testing for differences between means
when the population variance(s) are unknown,
we use the t statistic and the t distribution.
•• Tests involving differences between pro-
portions follow the same procedure as tests for
differences between means when population vari-
ances are known. The test statistic is Z, and the
sampling distribution is approximated by the
normal distribution.
182— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
KEY TERMS
alpha (a)
degrees of freedom (df)
left-tailed test
null hypothesis (H
0
)
one-tailed test
P value
research hypothesis (H
1
)
right-tailed test
sampling distribution of the
difference between means
statistical hypothesis testing
t distribution
t statistic (obtained)
two-tailed test
Type I error
Type II error
Z statistic (obtained)
ON YOUR OWN
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study questions, web quizzes, web resources, flashcards, codebooks and datasets, web exer-
cises, appendices, and links to social science journal articles reflecting the statistics used in
this chapter.
CHAPTER EXERCISES
1. It is known that, nationally, doctors working for health maintenance organizations (HMOs) average 13.5 years
of experience in their specialties, with a standard deviation of 7.6 years. The executive director of an HMO in
a western state is interested in determining whether or not its doctors have less experience than the national
average. A random sample of 150 doctors from HMOs shows a mean of only 10.9 years of experience.
a. State the research and the null hypotheses to test whether or not doctors in this HMO have less experi-
ence than the national average.
b. Using an alpha level of .01, make this test.
2. Consider the problem facing security personnel at a military facility in the Southwest. Their job is to
detect infiltrators (spies trying to break in). The facility has an alarm system to assist the security officers.
However, sometimes the alarm doesn’t work properly, and sometimes the officers don’t notice a real alarm.
In general, the security personnel must decide between these two alternatives at any given time:
H
0
: Everything is fine; no one is attempting an illegal entry.
H
1
: There are problems; someone is trying to break into the facility.
Based on this information, fill in the blanks in these statements:
a. A “missed alarm” is a Type ___ error, and its probability of occurrence is denoted as ___.
b. A “false alarm” is a Type ___ error.
3. For each of the following situations determine whether a one- or a two-tailed test is appropriate. Also, state
the research and the null hypotheses.
a. You are interested in finding out if the average household income of residents in your state is differ-
ent from the national average household. According to the U.S. Census, for 2010, the national average
household income is $50,303.
b. You believe that students in small liberal arts colleges attend more parties per month than students
nationwide. It is known that, nationally, undergraduate students attend an average of 3.2 parties per
month. The average number of parties per month will be calculated from a random sample of students
from small liberal arts colleges.
c. A sociologist believes that the average income of elderly women is lower than the average income of
elderly men.
Exercises
Testing Hypotheses— 183
d. Is there a difference in the amount of study time on-campus and off-campus students devote to their
schoolwork during an average week? You prepare a survey to determine the average number of study
hours for each group of students.
e. Reading scores for a group of third graders enrolled in an accelerated reading program are predicted
to be higher than the scores for nonenrolled third graders.
f. Stress (measured on an ordinal scale) is predicted to be lower for adults who own dogs (or other pets)
than for non–pet owners.
4. a. For each situation in Exercise 3, describe the Type I and Type II errors that could occur.
b. What are the general implications of making a Type I error? Of making a Type II error?
c. When would you want to minimize Type I error? Type II error?
5. One way to check on how representative a survey is of the population from which it was drawn is to com-
pare various characteristics of the sample with the population characteristics. A typical variable used for
this purpose is age. The 2008 GSS of the American adult population found a mean age of 47.71 and
a standard deviation of 17.35 for its sample of 2,013 adults. Assume that we know from census data
that the mean age of all American adults is 37.7. Use this information to answer these questions.
a. State the research and the null hypotheses for a two-tailed test.
b. Calculate the t statistic and test the null hypothesis at the .001 significance level. What did you find?
c. What is your decision about the null hypothesis? What does this tell us about how representative the
sample is of the American adult population?
6. Using data on average school grade for first-time cigarette use from MTF 2008, use the t test to conduct
a one-tailed test of the null hypothesis, assuming that the average grade is higher for males than for
females. Set alpha at .05. What can you conclude? Would your conclusions have been different if you had
used a two-tailed test?
7. In this exercise, we will examine the attitudes of liberals and conservatives toward affirmative action
policies in the workplace. Data from the 2008 GSS reveal that 10% of conservatives (N = 424) and 28%
of liberals (N = 336) indicate that they “strongly support” or “support” affirmative action policies for
African Americans in the workplace.
a. What is the appropriate test statistic? Why?
b. Test the null hypothesis with a one-tailed test (conservatives are less likely to support affirmative
action policies than liberals); a = .05. What do you conclude about the difference in attitudes between
conservatives and liberals?
c. If you conducted a two-tailed test with a = .05, would your decision have been different?
Males Females
Grade first tried cigarettes Mean = 4.90 Mean = 4.76
S
Y
= 1.74 S
Y
= 1.73
N = 150 N = 169
Source: Monitoring the Future, 2008.
Note: Only valid responses are included in the table. Students who indicated that they
have never used the drug are not included.
Exercises
184— ESSENTIALS OF SOCIAL STATISTICS FOR A DIVERSE SOCIETY
8. Let’s continue our analysis of liberals and conservatives, taking a look this time at differences in their
educational attainment. We obtain the following information from the 2008 GSS—the average edu-
cational attainment for liberals is 13.90 years (S
Y
= 3.27) and the average educational attainment for
conservatives is 13.55 years (S
Y
= 2.82). Data are based on 187 liberals and 227 conservative responses.
a. Test the research hypothesis that there is a difference in level of education between liberals and con-
servatives; set alpha at .01.
b. Would your decision have been different if alpha were set at .05?
9. During the 2008 Democratic presidential campaign, gender was considered more of an issue for
Hillary Clinton’s campaign than for her opponent, Barack Obama. During the campaign, pollsters
consistently reported how Clinton’s supporters were mostly (older) women, with men less likely to
support her candidacy. In surveys conducted during December 2007 and January 2008, the Pew
Research Center reported that among 240 men, 41% indicated that Senator Clinton was their first-
choice candidate. Among 381 women, 49% reported the same. Do these differences reflect a gender
gap among Clinton supporters?
a. If you wanted to test the research hypothesis that the proportion of male voters identifying Senator
Clinton as their first-choice candidate is less than female voters, would you conduct a one- or a two-
tailed test?
b. Test the null hypothesis at the .05 alpha level. What do you conclude?
c. If alpha were changed to .01, would your decision remain the same?
10. Data from the MTF 2008 survey reveal that 75.7% (493 out of 651) of males and 70.4% (501 out of
712) of females reported trying alcohol. You wonder whether there is any difference between males
and females in the population trying alcohol (this variable does not measure regular use, only if the
student had ever tried alcohol). Use a test of the difference between proportions when answering
these questions.
a. What is the research hypothesis? Should you conduct a one- or a two-tailed test? Why?
b. Test your hypothesis at the .05 level. What do you conclude?
11. Marcelline Fusilier, Subhash Durlabhji, Alain Cucchi, and Michael Collins (2005) examined Internet
usage among college students in the United States and in India.
9
Usage was divided into two categories,
for personal use and course-related use. We compare average hours between the U.S. and Indian students
in the following table.
Course Work Hours Personal Hours
U.S. students
Y
= 1.76
Y
= 2.08
N = 149 S
Y
= 1.52 S
Y
= 1.91
Indian students
Y
= 0.73
Y
= 0.87
N = 306 S
Y
= 0.79 S
Y
= 0.78
Source: Marcelline Fusilier, Subhash Durlabhji, Alain Cucchi, and Michael Collins, “A four country
investigation of factors facilitating student Internet use,” CyberPsychology and Behavior 8(5):
454–464. Copyright © Mary Ann Liebert, Inc. Publishers.
Internet Use Means and Standard Deviations for U.S. and Indian Students Sample
Exercises
Testing Hypotheses— 185
a. Determine whether U.S. students have significantly higher Internet use for course work than the
Indian students. Test at the .05 alpha level.
b. Test whether there is a significant difference in Internet use for personal use between the U.S. and
Indian students. Test at the .01 alpha level.
12. Do men and women have different beliefs on the ideal number of children in a family? Based on the following
GSS 2008 data and obtained t statistic, what would you conclude? (Hint: Assume a two-tailed test; a = .05.)
Men Women
Mean ideal number of children 3.06 3.22
Standard deviation 1.92 1.99
N
604 678
Obtained t statistic -1.450
Ver y Happy Not Too Happy
Mean ideal number of children 3.49 2.73
Standard deviation 2.24 1.35
N
228 90
Obtained t statistic 2.98
13. Does access to cocaine vary by sex? Data from the MTF 2008 survey indicate that among male students
(N = 670) 18% say that it would be “very easy” to obtain cocaine compared with 14% of female students
(N = 723). Is there a significant difference in proportion of easy cocaine access between the two student
groups? Set alpha at .01. What can you conclude?
14. We recalculated our comparison of ideal number of children, this time only for women, separating them
into two groups, those who indicated that they were “very happy” or “not too happy” in general. Results,
based on the GSS 2008, are presented below.
Based on a two-tailed test, a = .05. What do you conclude?
Exercises
