Chapter 21
Univariate Statistical Analysis
AT-A-GLANCE
I. Introduction
II. Hypothesis Testing
A. The hypothesis-testing procedure
Process
Significance levels and p-values
B. An example of hypothesis testing
C. Type I and Type II errors
Type I error
Type II error
III. Choosing the Appropriate Statistical Technique
A. Type of question to be answered
B. Number of variables
C. Level of scale of measurement
D. Parametric versus nonparametric hypothesis tests
IV. The t-Distribution
A. Calculating a confidence interval estimate using the t-distribution
One and two-tailed t-tests
B. Univariate hypothesis test using the t-distribution
V. The Chi-Square Test for Goodness of Fit
VI. Hypothesis Test of a Proportion
VII. Additional Applications of Hypothesis Testing
LEARNING OUTCOMES
1. Implement the hypothesis-testing procedure
2. Use p-values to assess statistical significance
3. Test a hypothesis about an observed mean compared to some standard
4. Know the difference between Type I and Type II errors
5. Know when a univariate 2 test is appropriate and how to conduct one
CHAPTER VIGNETTE: Well, Are They Satisfied or Not?
Ed Bond is a research analyst for a company, and Rob Baer, the COO, and Kathy Hahn, the CEO, asked
his take on employee satisfaction for one the company’s plants. Ed replied that they had put together an
index of three questions that asked about job satisfaction, and the average satisfaction for that plant is 3.9.
Kathy and Rob didn’t understand what that meant, and Ed realized that he can’t just speak about scores—
he’s there to help them understand.
SURVEY THIS!
Hypothesis testing is often a critical part of what business researchers do for the organization, and it is
particularly important to understand how data you gather compare to benchmarks set by your work group.
Here’s a short exercise that will help students understand the importance of this analysis:
1. Select two variables for the survey that could serve as a possible benchmark.
2. Using a frequencies distribution of both variables, identify the mean and standard deviation of
both.
3. Develop a hypothesis statement for both variables.
4. Conduct a hypothesis test for both, setting your benchmark value to the scale midpoint.
5. Notice and comment on the significance of these tests for both variables. What do the results
tell you?
RESEARCH SNAPSHOTS
The “Freshman 7.8”
There is a common belief that college freshmen gain 15 pounds in their first year, commonly
referred to as the “Freshman 15.” Researchers at Purdue University conducted a study using 907
freshmen and found the average weight gain being 7.8 pounds. As a test, we asked a freshman class
for their own weight gain. While this is a subjective assessment, it does allow for a hypothesis test:
Do students gain 7.8 pounds in their first year? The test results found an average weight gain of
5.63 pounds, and the univariate statistical test suggests the answer is no (p < 0.0001). The
“Freshman 15” should lose a few pounds.
The Law and Type I and Type II Errors
While most attorneys and judges do not use statistical terminology such as Type I and Type II
errors, they do follow this logic. For example, the null hypothesis is that an innocent individual is
sent to prison. A Type I error is equivalent to sending an innocent person to prison, while a Type
II error would occur if a guilty party were set free. Our society places such a high value on
avoiding Type I errors that Type II errors are more likely to occur.
Living in a Statistical Web
Having trouble learning statistical concepts? Several web sources are described:
STATLIB (http://lib.stat.cmu.edu) – a system for distributing statistical software,
datasets, and information electronically.
STAT-HELP (http://www.stat-help.com) – provides helps with statistics via the
Internet and contains spreadsheets for performing many basic calculations.
SURFSTAT.AUSTRALIA(http://surfstat.anu.edu.au/surfstat-home/surfstat-main.html
) – an online text in introductory statistics from the University of Newcastle and the
Australian government.
RICE VIRTUAL LAB IN STATISTICS(http://onlinestatbook.com/rvls.html) –
provides hypertext materials.
STATCRUNCH (http://www.statcrunch.com) – a statistical software package via the
World Wide Web.
GRAPHPAD (http://www.graphpad.com/quickcalcs/Statratio1.cfm) – a p-value
calculator.
Interested in Retirement? It Often Depends on Your Age
Chi-square tests are used often in business research. Consider a business that sponsors a program
to educate employees on retirement issues. They need to plan, and one question is whether or not
an equal number of younger versus older employees will come to the sessions. They decide to
observe the relative frequencies of younger versus older employees based on the number of
sign-ups they receive in the first week, with a cut-off set at 200. The χ2 value is significant, so the
researchers conclude that the older workers are much more likely to attend the retirement
seminar, and they can design the seminar to meet the needs associated with this group.
OUTLINE
I. INTRODUCTION
Statistical analysis can be divided into several groups:
Univariate statistical analysis – tests hypotheses involving only one variable.
Bivariate statistical analysis – tests hypotheses involving two variables.
Multivariate statistical analysis – tests hypotheses and models involving multiple (3 or
more) variables or sets of variables.
II. HYPOTHESIS TESTING
Descriptive research and causal research designs often climax with hypotheses tests.
Hypotheses are defined as formal statements of explanations stated in a testable form, which
means they should be stated in concrete fashion so that the method of empirical testing seems
almost obvious.
Type of hypotheses tested commonly in business research:
1. Relational hypotheses – examining how changes in one variable vary with changes in
another are usually tested by assessing covariance in some way (i.e., regression analysis).
2. Hypotheses about differences between groups – examine how some variable varies from
one group to another (very common in causal designs).
3. Hypotheses about differences from some standard – examines how some variable differs
from some preconceived standard.
Empirical testing typically involves inferential statistics, meaning that inference will be
drawn about some population based on observations of a sample representing that population.
The Hypothesis-Testing Procedure
Process
Hypotheses are tested by comparing the researcher’s educated guess with empirical
reality.
The process can be described as follows:
1. First, the hypothesis is derived from the research objectives and should be stated
as specifically as possible.
2. Next, a sample is obtained and the relevant variable is measured.
3. The sample value is compared to the value either stated or implied in the
hypothesis.
If the value is consistent with the hypothesis, the hypothesis is
supported.
If it is not consistent with the hypothesis, the hypothesis is not
supported.
Univariate hypotheses are typified by tests comparing some observed sample mean
against a benchmark value.
When the observed mean is close to the benchmark, we would not have sufficient
confidence that a second set of data from a new sample taken from the same
population would produce a finding that was very different from the benchmark.
In contrast, when the mean turns out well above the benchmark, then we could
more easily trust that another sample would not produce a mean equal to or less
than the benchmark.
In statistics classes, students are exposed to hypothesis testing as a contrast between a
null and an alternative hypothesis.
A “null” hypothesis can be thought of as the expectation of findings as if no
hypothesis existed (i.e., “no” or “null” hypothesis).
The state implied by the null hypothesis is the opposite of the state represented
by the actual hypothesis.
The researcher’s hypothesis is generally stated in the form of an “alternative”
hypothesis.
The idea of a null hypothesis can be confusing, so it will be avoided when at all
possible.
A statistical test’s significance level or p-level becomes a key indicator of whether or
not a hypothesis can be supported.
Significance Levels and P-Values
A significance level is a critical probability associated with a statistical hypothesis
test that indicates how likely an inference supporting a difference between an
observed value and some statistical expectation is true.
P-value stands for probability-value and is essentially another name for an observed
or computed significance level.
The probability in a p-value is that the statistical expectation (null) for a given test is
true.
Traditionally, researchers have specified an acceptable significance level for a test
prior to analysis, and acceptable levels are 0.1, 0.05, or 0.01.
If the p-value resulting from a statistical test is less than the pre-specified level, then
a hypothesis about differences is supported.
Exhibit 21.2 illustrates an important property of p-values – as the observed value gets
further from the benchmark, the p-value gets smaller, meaning that the chance of the
mean actually equaling the benchmark is smaller.
In discussing confidence intervals, statisticians use the term confidence level, or
confidence coefficient, to refer to the level of probability associated with an interval
estimate.
However, when discussing hypothesis testing, the terminology is changed to a
significance level, α (the Greek letter alpha).
An Example of Hypothesis Testing
The example illustrates the conventional statistical approach to testing a univariate
hypothesis with an interval or ratio variable.
The Pizza-In restaurant is concerned about its store image, one aspect of which is the
friendliness of the service.
A sample of 225 customers is asked to indicate their perception of the service on a
5-point scale, where 1 indicates “very unfriendly” and 5 indicates “very friendly” service.
Suppose Pizza-In believes the service has to be different from 3.0 before they can make a
decision about expansion.
The researcher formulates the null hypothesis that the population mean is equal to 3.0:
H0 :
m
= 3.0
The alternative hypothesis is that the mean does not equal 3.0:
H1 :
m
¹
3.0
More practically, the researcher is likely to write the substantive hypothesis (as it would
be stated in a research report or proposal) as:
H1: Customer perceptions of friendly service are significantly greater than three.
The substantive hypothesis matches the “alternative” phrasing, and in practical terms,
is the only hypothesis formally stated.
Next, the researcher must decide on a significance level, which corresponds to a region of
rejection on a normal sampling distribution shown in Exhibit 21.1.
The shaded area shows the region of rejection when
a
= .025 in each tail of the curve.
The values within the unshaded area are called acceptable at the 95 percent confidence
level (or 5 percent significance level, or 0.05 alpha level), and if we find that our sample
mean lies within this region of acceptance, we conclude that the means are not different
from the expected value
More precisely, we fail to reject the null hypothesis.
In other words, the range of acceptance:
1. identifies those acceptable values that reflect a difference from the hypothesized
mean in the null hypothesis and
2. shows the range within which any difference is so minuscule that we would conclude
that this difference was due to random sampling error rather than to a false null
hypothesis.
The Pizza-In restaurant hired research consultants who collected a sample of 225
interviews.
The mean friendliness score on the 5-point scale was 3.78.
If
s
is known, then this is used in the analysis; however, this is rarely true and was not
true in this case.
The sample standard deviation was S = 1.5.
The researchers have set the significance level at the .05 level, which means that the
researcher wishes to draw a conclusion that will be erroneous 5 times in 100 (.05) or
fewer.
From the tables of standardized normal distribution, the researcher finds that the Z score
of 1.96 represented a probability of .025 that a sample mean will lie above 1.96 standard
errors from
m
.
Likewise .025 of all sample means will fall below -1.96 standard errors from
m
.
Adding these two “tails” together, we get .05.
The values that lie exactly on the boundary of the region of rejection are called the
critical values of
m
.
Now we must transform these critical Z-values to the sampling distribution of the mean
for this study. The critical values are:
)/( nSZorZS
x
255/5.1(96.10.3
)
)1(.96.10.3
196.00.3
Lower limit = 2.804
Upper limit = 3.196
Based on the survey,
X
= 3.78.
Thus, since the sample mean is greater than the critical value, 3.196, the researchers say
that the sample result is statistically significant beyond the .05 level.
Therefore, the results indicate that customers believe the service is friendly. It is unlikely
(less than five in 100) that this result would occur because of random sampling error.
An alternative way to test the hypothesis is to formulate the decision rule in terms of the
Z-statistic, and in this example, the value is 7.8 with a p-value of .000001, which is less
than the acceptable level and the hypothesis is supported.
Statistical packages usually return a p-value for a given test.
Type I and Type II Errors
Because we cannot make any statement about a sample with complete certainty, there is
always the chance that an error will be made.
When using a census (i.e., every unit in a population is measured), conclusions are certain
(but researchers rarely use a census).
The researcher runs the risk of committing two types of errors (Exhibit 21.4 summarizes
the state of affairs in the population and the nature of Type I and Type II errors).
Type I Error
A Type I error occurs when a condition that is true in the population is rejected
based on statistical observations.
When a researcher sets an acceptable significance level a-priori (
), he or she is
determining the tolerance for type I error.
When testing for relationships, a type I error occurs when the researcher concludes a
relationship exists when in fact one does not.
Type II Error
If the alternative condition is in fact true but we conclude that we should not reject
the null hypothesis, we make a Type II error
It is the probability of failing to reject a false null hypothesis.
This incorrect decision is called beta (
).
In practical terms, a type II error means that we fail to reach the conclusion that some
difference between an observed mean and a benchmark exists when in fact the
difference is very real.
Without increasing the sample size the researcher cannot simultaneously reduce Type I
and Type II errors because there is an inverse relationship between the two.
Thus, reducing the probability of a Type II error increases the probability of a Type I
error.
In business problems, Type I errors are generally more serious then Type II errors so
more emphasis is placed on determining the significance level, α, than in determining β.
III. CHOOSING THE APPROPRIATE STATISTICAL TECHNIQUE
Numerous statistical techniques are available to assist the researcher in interpreting data.
Making the correct choice can be determined by considering:
1. The type of questions to be answered.
2. The number of variables involved.
3. The level of scale measurement.
Hypotheses are tested by using a correct click-through sequence in a statistical software
package, which are highly reliable.
Type of Question to Be Answered
The researcher should consider the method of statistical analysis before choosing the
research design and before determining the type of data to collect.
Number of Variables
Univariate, bivariate and multivariate statistical procedures exist and are appropriate
based on the number of variables involved in an analysis.
Level of Scale Measurement
Testing a hypothesis about a mean is appropriate for interval scaled or ratio scaled data.
Where data are measured on an ordinal scale, the median may be used as the average or a
percentile may be used as a measure of dispersion.
Nominal and ordinal data are often analyzed using frequencies or cross-tabulation.
Parametric versus Nonparametric Hypothesis Tests
The terms parametric statistics and nonparametric statistics refer to the two major
groupings of statistical procedures.
The major distinction lies in the underlying assumptions about the data.
Parametric statistics involve numbers with known, continuous distributions.
When the data are interval or ratio scales and the sample size is larger, parametric
procedures are appropriate.
Based on the assumption that the data in the study are drawn from a population with
a normal (bell-shaped) distribution and/or normal sampling distribution.
Nonparametric statistics are appropriate when the numbers do not conform to a known
distribution.
Making the assumption that the population distribution or sampling distribution is
normal generally is inappropriate when data are either ordinal or nominal.
Thus, nonparametric statistics are referred to as distribution free.
Exhibit 21.5 illustrates how statistical techniques vary according to scale properties and
the type of question being asked.
IV. THE t-DISTRIBUTION
A univariate t-test is appropriate for testing hypotheses involving some observed mean
against some specified value.
The t-distribution, like the standardized normal curve, is a symmetrical, bell-shaped
distribution with a mean of zero and a unit standard deviation.
When sample size (n) is larger than 30, the t-distribution and Z-distribution are almost
identical.
Therefore, while the t-test is strictly appropriate for tests involving small sample sizes with
unknown standard deviations, researchers commonly apply the t-test for comparisons
involving the mean of an interval or ratio measure.
The shape of the t-distribution is influenced by its degrees of freedom (df).
The degrees of freedom are determined by the number of distinct calculations that are
possible given a set of information.
In the case of a Univariate t-test, the degrees of freedom are equal to the sample size (n)
minus one.
Calculation of t closely resembles the calculation of the Z-value. To calculate t, use the
formula:
t = X – m
S
X
with n – 1 degrees of freedom.
If the population standard deviation
is known, then the Z-test is most appropriate.
1
8
1
8
When
is unknown (i.e., the situation in most business research studies), and the sample
size is greater than 30, the Z-test also can be used.
When
is unknown and the sample size is small, the t-test is most appropriate.
Calculating a Confidence Interval Estimate Using the t-Distribution
Procedure for calculating the confidence interval:
1. Calculate
X
from the sample.
2. Since is unknown, estimate the population standard deviation by finding S, the
sample standard deviation.
3. Estimate the standard error of the mean.
4. Determine the t-values associated with the desired confidence level – go to Table A.3
in the appendix (e.g., look under the .05 column for two-tailed tests at the row in
which degrees of freedom (d.f.) equal the appropriate value (n-1).
5. Calculate the confidence interval.
Example: Suppose a business organization is interested in finding out how long newly
hired MBA graduates remain on their first job. On the basis of a small sample of
employees with MBAs, the researcher wishes to estimate the population mean with 95
confidence (see data in the textbook). To find the confidence estimates of the population
mean for this small sample:
m = X ± tc . 1 .
S
X
or
m = X ± tc . 1 . S n
where
m
= population mean
X
= sample mean
t
c . 1 .
= critical value of t at a specified confidence level
S
X
= standard error of the mean
S = sample standard deviation
n = sample size
In the example about the MBA students, we know
X
= 3.89, S = 2.81, and n = 18.
To calculate the confidence interval, we must go to Table A.3 in the Appendix and
look under 17 degrees of freedom (n – 1, 18 – 1 = 17) for the t value at the 95 percent
confidence level. In this case t = 2.12. Thus,
Upper limit = 3.89 + 2.12(2.12/ ) = 2.49
Lower limit = 3.89 — 2.12(2.66/ ) = 5.28
It may be concluded with 95 percent confidence that the population mean for the
number of years spent on the first job by MBAs is between 2.49 and 5.28.
One and Two-Tailed t-Tests
Univariate tests can be one or two-tailed.
One-tailed test – appropriate when a research hypothesis implies that an observed mean
can only be greater than or less than a hypothesized value.
Thus, only one of the “tails” of the bell-shaped normal curve is relevant.
Practically, a one-tailed test can be determined from a two-tailed test result by taking
½ of the observed p-value.
Two-tailed tests – one that tests for differences from the population mean that are either
greater or less.
Thus, the extreme values of the normal curve (or tails) on both the right and the left
are considered.
When a research question does not specify whether a difference should be greater
than or less than, a two-tailed test is most appropriate.
When not sure whether a one or two-tailed test is appropriate, opt for the less
conservative two-tailed test.
Most computer software will assume a two-tailed test unless otherwise specified.
Univariate Hypothesis Test Using the t-Distribution
The step-by-step procedure for a t-test is conceptually similar to that for hypothesis
testing with the Z-distribution:
1. Calculate a sample mean and standard deviation.
2. Compute the standard error.
3. Find the t-value associated with the desired level of confidence level or statistical
significance (e.g., if a 95 percent confidence level is desired, the significance level
is .05).
4. Find the critical values for the t-test by locating the upper and lower limits of the
confidence interval. The result defines the regions of rejection.
The researcher makes the statistical decision by determining whether the sample mean
falls between the critical limits.
V. THE CHI-SQUARE TEST FOR GOODNESS OF FIT
The chi-square (2) test is one of the most basic tests for statistical significance and is
particularly appropriate for testing hypotheses about frequencies arranged in a frequency or
contingency table.
Used for univariate tests involving nominal or ordinal variables.
The 2 test is associated with goodness-of-fit (GOF).
GOF can be thought of as how well some matrix (table) of numbers matches or fits another
matrix of the same size, and most often, the test is between a table of observed frequency
counts and another table of expected values (central tendency) for those counts.
In statistical terms, a 2 test determines whether the difference between an observed frequency
distribution and the corresponding expected frequency distribution is due to sampling variation.
Computing it is fairly straightforward and easy:
1. Gather data and tally the observed frequencies for the categorical variable.
2. Compute the expected values for each value of the categorical variable.
3. Calculate the 2 value, using the observed frequencies from the sample and the expected
frequencies.
4. Find the degrees of freedom for the test.
5. Make the statistical decision by comparing the p-value associated with the calculated 2
against the predetermined significance level (acceptable type I error rate).
The
2
distribution is not a single probability curve but a family of curves.
These curves vary slightly with the degrees of freedom.
Degrees of freedom: d.f. = k – 1, where k = the number of cells associated with column or row
data.
The chi-square test is further discussed in Chapter 22, as it is also frequently used to analyze
contingency tables.
VI. HYPOTHESIS TEST OF A PROPORTION
The population proportion (
) can be estimated on the basis of an observed sample
proportion (p).
Conducing a hypothesis test of a proportion is conceptually similar to hypothesis testing
when the mean is the characteristic of interest.
However, the formulation of the standard error of the proportion is mathematically different.
Even though a population proportion is unknown, a large sample allows the use of the Z-test.
Using the following formula, we can calculate the observed value of Z, given a certain sample
proportion:
Z
obs
= p – p
S
p
where p = sample proportion
p
= hypothesized population proportion
Sp= estimate of the standard error of the proportion
The formula for Sp is
n
pq
S
p
or
n
pp
S
p
1
where
Sp estimate of the standard error of the proportion
p proportion of successes
q 1 − p, proportion of failures
VII. ADDITIONAL APPLICATIONS OF HYPOTHESIS TESTING
Other hypothesis tests for population parameters estimated from sample statistics exist but are
not mentioned here.
Many of these other tests are conceptually no different in their methods of hypothesis testing.
However, the formulas for conducting statistical tests are mathematically different.