The Statistical Imagination
• Chapter 10.
Hypothesis Testing II:
Single-Sample Hypothesis
Tests: Establishing the
Representativeness of Samples
© 2008 McGraw-Hill Higher Education
Single-Sample
Hypothesis Tests
• Used to answer the question:
“Is a parameter equal to some chosen
target value?”
© 2008 McGraw-Hill Higher Education
Sources of Target Values
1. A known population parameter from a
comparison group
2. Known parameters from a past time period
3. A statistical ideal
4. A population is sampled, and the sample
statistics are compared to known
population parameters to determine
whether the sample is representative of the
population
© 2008 McGraw-Hill Higher Education
Small Single-Sample
Means Test
• Useful for testing a hypothesis that the
mean of a population is equal to a target
value
© 2008 McGraw-Hill Higher Education
When to Use a Small Single
Sample Means Test
1.There is only one variable
2. The level of measurement of the variable
is interval/ratio
3. There is a single sample and population
4. n < 121 cases
5. There is a target value of the variable to
which we may compare the mean of the
sample
© 2008 McGraw-Hill Higher Education
The Student’s t
Sampling Distribution
• The sampling distribution curve used
with especially small samples
(n < 121 cases) is called the
Student’s t-distribution
• The t-distribution is an approximately
normal distribution
© 2008 McGraw-Hill Higher Education
The t-distribution of Means
• For a sampling distribution of means, when
the sample size is 121 or fewer cases, the
probability curve begins to flatten into
a t-distribution
• See Figure 10-1 in the text
© 2008 McGraw-Hill Higher Education
Features of the t-distribution
• Standardized scores for the t-distribution
are called t-scores and are computed like
Z-scores
• The t-distribution, like the Z-distribution of
the normal curve, allows us to calculate
probabilities
• The t-distribution is symmetrical with mean,
median, and mode centered, but the curve
is flatter reflecting greater sampling error
© 2008 McGraw-Hill Higher Education
The t-distribution Table
• The t-distribution table (Appendix B,
Statistical Table C) is organized differently
from the normal curve table
• See Table 10-1 in the text
• The t-distribution table provides t-scores
for only the critical probabilities of .05, .01,
and .001; that is, this table provides
“critical t-scores”
© 2008 McGraw-Hill Higher Education
Degrees of Freedom
• Degrees of freedom ( df ) are the number
of opportunities in sampling to compensate
for limitations, distortions, and potential
weaknesses in statistical procedures
• Use of the t-distribution table requires the
calculation of degrees of freedom
• For single-sample means tests, df = n - 1
© 2008 McGraw-Hill Higher Education
More on Degrees of Freedom
• From repeated sampling we know that any single
sample is only an estimate. An estimate can be
distorted by limitations of the statistical procedures
used to obtain it
• E.g., the mean is influenced by outliers
• The larger the sample, the greater the opportunity
for an outlier to be neutralized in such a way as to
not distort a sample mean
• A small sample is especially vulnerable to outliers
© 2008 McGraw-Hill Higher Education
Z-Tests and t-Tests
• The small sample t-test applies when the
sample size, n < 121; we use the t-distribution
table
• However, regardless of sample size, all
single-sample means tests are typically
referred to as t-tests because the Z-test is a
special case of a t-test, a t-test where n >
121; the t-distribution table includes critical
scores for this situation under df = ∞
© 2008 McGraw-Hill Higher Education
Features of a
Single-Sample Means Test
• Follow the six steps of statistical inference
• Step 1: In general, state the H0 :
Parameter (sampled population) = known target parameter
• That is, the mean of a population
= a target value
• As in all hypotheses tests, the H0 and HA
refer to population parameters
© 2008 McGraw-Hill Higher Education
Features of a SingleSample Means Test (cont.)
• Step 2. t-distribution with df = n - 1
• The standard error is estimated by
dividing the sample standard
deviation by the square root of the
sample size, n
• Draw the t-distribution curve
• Obtain the critical t-score from the
t-table in Appendix B
© 2008 McGraw-Hill Higher Education
The Critical Test Score
for a t-Distribution
• Step 3: For a hypothesis test using the tdistribution curve, the critical test score is
the specific t-score that is large enough to
cause us to reject the H0
• After selecting the critical value from the tdistribution table, identify its location on the
probability curve in Step 2 and estimate
what kind of sample outcomes reach the
critical region
© 2008 McGraw-Hill Higher Education
Features of a Single-Sample
Means Test (Continued)
• Step 4. The effect is the difference
between the observed sample mean and
the hypothesized target value
• The test statistic is the effect divided by the
standard error
• The p-value is estimated from the tdistribution table
© 2008 McGraw-Hill Higher Education
Relationships Among
Mathematical Parts of the Test
• There are rules that assist in
understanding the relationships among
hypothesized parameters, observed
sample statistics, the test statistic, the pvalue, and the level of significance and its
critical test score
© 2008 McGraw-Hill Higher Education
Rule 1: The Test Effect and
the Critical Test Score
• The H0 is rejected when the test effect
is large enough that the test statistic
is greater than the critical test score
value
• If │tobserved │ > │ tα │, then p < α;
reject H0
© 2008 McGraw-Hill Higher Education
Rule 2: The Effect and the
Size of the p-Value
• The larger the test effect and the test
statistic, the smaller the p-value
• A large test effect implies a significant
difference between what was
hypothesized and what was observed
• A large test effect makes the H0
suspect
© 2008 McGraw-Hill Higher Education
Rule 3: Rejection and
Test Direction
• It is easier to reject with a one-tailed test
than with a two-tailed test
• Choosing to use a one-tailed test must be
done on the basis of theory or logic and is
done in Step 1 of the six steps
• Do not look to Step 4, the sample outcome,
to make a choice of test direction
© 2008 McGraw-Hill Higher Education
Rule 4: The Level of
Significance and Rejection
• The lower the level of significance, the
harder it is to reject the H0
• The H0 is rejected when the p-value is less
than or equal to α. The p-value must be
especially small to be smaller than a small
α
© 2008 McGraw-Hill Higher Education
Rule 5: Fail to Reject When Findings
are in the Wrong Direction
• When using a one-tailed test but the
observed sample statistic falls in the
opposite direction, immediately fail to
reject the H0
• Even if found statistically significant,
one cannot justify a wrong direction
finding
© 2008 McGraw-Hill Higher Education
Testing for Sample
Representativeness
• Single-sample hypothesis tests are
especially useful for determining whether a
sample is representative of the population
from which it was drawn
• To establish the representativeness of a
sample, data must be acquired on some
known parameters of the population to
provide the target value for the test
© 2008 McGraw-Hill Higher Education
Testing for Sample
Representativeness (cont.)
• Single-sample hypothesis tests
determine whether the test effect is
(a) due to random sampling error or
(b) due to a failure to sample and
represent some aspect of the
population adequately
© 2008 McGraw-Hill Higher Education
Testing for Sample
Representativeness (cont.)
• The H0 is stated as “the sample is
representative”
• If the difference between the known
parameter and the sample statistic is not
statistically significant but simply due to
sampling error, then we will not reject this
H0
© 2008 McGraw-Hill Higher Education
Testing for Sample
Representativeness (cont.)
• The HA is stated as “the sample is not
representative”
• If the difference between the known
parameter and the sample statistic is
significantly different, then we reject the H0
and conclude that the sampled population
and targeted population are not one and
the same – that there is sample bias
© 2008 McGraw-Hill Higher Education
Testing for Sample
Representativeness (cont.)
• Most tests of sample
representativeness use nominal/ordinal
variables, such as gender and race,
because known parameters are
available in government documents
(e.g., the U.S. Census)
• These parameters provide the target
values for such tests
© 2008 McGraw-Hill Higher Education
Large Single-Sample
Proportions Test
• The large single-sample proportions
test is used for testing sample
representativeness with
nominal/ordinal variables
© 2008 McGraw-Hill Higher Education
When to Use a Large SingleSample Proportions Test
1. There is only one variable
2. It is a nominal/ordinal variable with
P = p [of success category] and Q = p [of failure
category/categories]
3. There is a single sample and population
4. The sample size is sufficiently large that (psmaller) (n) >
5
5. There is a target value (known parameter) with which
to compare the sample proportion
© 2008 McGraw-Hill Higher Education
Features of a Large SingleSample Proportions Test
• Step 1. Stating the H0:
Parameter (sampled population) = a known target parameter
• That is: Pu = a known target value
© 2008 McGraw-Hill Higher Education
Features of a Large SingleSample Proportions Test (cont.)
• Step 2. The sampling distribution of
proportions is the approximately normal tdistribution with a standard error estimated
by taking the square root of Pu times Qu
divided by n
• In the t-distribution table, look to df = ∞;
this is essentially a normal curve test
© 2008 McGraw-Hill Higher Education
Features of a Large SingleSample Proportions Test (cont.)
• Step 4. The effect is the difference
between the observed sample proportion
(Ps) and the hypothesized target value (Pu)
• The test statistic is the effect divided by the
standard error
• The p-value is estimated from the tdistribution table
© 2008 McGraw-Hill Higher Education
Features of a Large SingleSample Proportions Test (cont.)
• Steps 5 and 6: When the H0 is rejected,
focus on the interpretation of the HA,
which is that the sample is NOT
representative
© 2008 McGraw-Hill Higher Education
Confidence Interval of the Mean
When n is Small (n < 121 cases)
• For a confidence interval of the mean
(Chapter 8), when the sample size is
121 or fewer cases, substitute the
appropriate t-score in place of a Z-score
• tα in place of Zα
© 2008 McGraw-Hill Higher Education
Statistical Follies: Sample Size
and Tests of Representativeness
• When testing for sample representativeness,
we wish to fail to reject the of H0 of no
significant difference
• With a small sample and its large standard
error, it is too easy to accomplish this, but
there is the chance of Type II error, in this
case, concluding the sample is representative
when it is not
© 2008 McGraw-Hill Higher Education
Statistical Follies: Sample Size and
Tests of Representativeness (cont.)
• To avoid a Type II error, the sample size
must be sufficiently large
• Statistical power is a test statistic’s
probability of not incurring a Type II
error for a given level of confidence
© 2008 McGraw-Hill Higher Education