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Chapter 10
Inference from Small Samples
General Objective:
This chapter supplements the large-sample techniques by
presenting small-sample tests and confidence intervals for
population means and variances. Unlike their large-sample
counterparts, these small-sample techniques require the
samples populations to be normal or approximately so.
©1998 Brooks/Cole Publishing/ITP
Specific Topics
1. Student’s t distribution
2. Small-sample inferences concerning a population mean
3. Small sample inferences concerning the difference in two
means: Independent random samples
4. Paired-difference test: dependent samples
5. Inferences concerning a population variance
6. Comparing two populations variances
7. Small-sample assumptions
©1998 Brooks/Cole Publishing/ITP
10.1 Introduction

When the sample size is small, the estimation and testing
procedures involve these familiar parameters:
- A single population mean, m
- The difference between two population means, ( m 1 - m 2 )
- A single population variance, s 2
- The comparison of two population variances, s 12 and s 22
©1998 Brooks/Cole Publishing/ITP
10.2 Student’s t Distribution

In discussing the sampling distribution of x in Chapter 7,
we made these points:
- When the original sampled population is normal, x and
z  x - m  s n  both have normal distributions, for any
sample size.
- When the original sampled population is not normal,
z  x - m  s n  all have approximately normal distributions.
©1998 Brooks/Cole Publishing/ITP



To find the sampling distribution of the statistic x - m  s n  :
- Use an empirical approach. Draw repeated samples and
compute the statistic for each sample. The relative frequency
distribution that you construct using these values will
approximate the shape and location of the sampling
distribution.
- Use a mathematical approach to derive the actual density
function or curve that describes the sampling distribution.
x-m
Student’s t distribution: t 
s n
Student's t has the following characteristics:
- Mound-shaped and symmetric about t  0, just like z.
- More variable than z, with “ heavier tails.”
- Shape depends on the sample size n. For larger n, it
approximates the z.
©1998 Brooks/Cole Publishing/ITP

Figure 10.1 shows the standard normal z and the t distribution
with 5 degrees of freedom:
©1998 Brooks/Cole Publishing/ITP

The divisor (n - 1) in the formula for the sample variance s 2
is called the number of degrees of freedom (d f ) associated
with s 2. It determines the shape of the t distribution.

Table 10.1 gives the format of the Student’s t table from
Table 4 in Appendix I.

Figure 10.2 shows tabulated values of Student’s t.

Examples 10. 1 and 10.2 deal with the t distribution.
©1998 Brooks/Cole Publishing/ITP
Table 10.1 Format of the Student’s t table
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Figure 10.2 Tabluated values of Student’s t
©1998 Brooks/Cole Publishing/ITP
Example 10.1
For a t distribution with 5 degrees of freedom, the value of t that
has area .05 to its right is found in row 5 in the column marked
t.050. For this particular t distribution, the area to the right of
t  2.015 is .05; only 5% of all values of the t statistic will
exceed this value.
©1998 Brooks/Cole Publishing/ITP

When n  30 (df  29), the critical value of t for p .05 is 1.699,
which is quite close to z  1.645 for the same p-value.
Assumptions Behind Student’s t Distribution
- The sample must be randomly chosen.
- The population from which you are sampling must be
normally distributed.

The t statistic is robust, meaning that the distribution of the
statistic does not change significantly when the normality
assumptions are violated.
©1998 Brooks/Cole Publishing/ITP
10.3 Small-Sample Inferences
Concerning a Population Mean
Small sample inference can involve either estimation or
hypothesis testing.
Small Sample Hypothesis Test for m :
1. Null Hypothesis: H 0 : m  m 0
2. Alternative Hypothesis:
One-Tailed Test
Two-Tailed Test
Ha : m > m0
Ha : m  m0
(or H a : m < m 0 )

3. Test Statistic:
t
x - m0
s
n
©1998 Brooks/Cole Publishing/ITP
4. Rejection Region: Reject H 0 when
One-Tailed Test
Two-Tailed Test
t > ta
t > ta/2 or t < -ta/2
(or t < -ta when the
alternative hypothesis
is H a : m < m 0 )
or when the p-value < a

Assumption: The sample is randomly selected from a normally
distributed population.
©1998 Brooks/Cole Publishing/ITP
Small-Sample (1 - a )100% Confidence Interval for m :
s
2
n
where s n is the estimated standard error of x , often
referred to as the standard error of the mean (SEOM).
x  ta

Example 10.3 is a further application of the t distribution.
Example 10.3
A new process for producing synthetic diamonds can be
operated at a profitable level only if the average weight of the
diamonds is greater than .5 karat. To evaluate the profitability of
the process, six diamonds are generated, with recorded weights
.46, .61, .52, .48, .57, and .54 karat. Do the six measurements
present sufficient evidence to indicate that the average weight of
the diamonds produced by the process is in excess of .5 karat?
©1998 Brooks/Cole Publishing/ITP
Solution
The population of diamond weights produced by this new
process has mean m , the value in question. The hypotheses to
be tested are
H 0 : m  .5 versus H a : m > .5
and the test statistic is a t-statistic with (n - 1)  (6 - 1)  5
degrees of freedom. You can use your calculator to verify that
the mean and standard deviation for the six diamond weights
are .53 and .0559, respectively. The calculated value of the test
statistic is then
x- m
.53 - .5
t

 1.32
s / n .0559 / 6
_
0
As with the large-sample tests, the test statistic provides
evidence for either rejecting or accepting H 0 depending on how
far from the center of the t distribution it lies.
©1998 Brooks/Cole Publishing/ITP
If you choose a 5% level of significance (a  .05 ), the righttailed rejection region is found using the critical values of t from
Table 4 in Appendix I. With d f  n - 1  5, you can reject H 0 if
t > t.05  2.015, as shown in Figure 10.4. Since the calculated
value of the test statistic, 1.32, does not fall into the rejection
region, you cannot reject H 0. The data do not present sufficient
evidence to indicate that the mean diamond weight exceeds
.5 karat.
©1998 Brooks/Cole Publishing/ITP
Figure 10.4 Rejection region from Example 10.3
©1998 Brooks/Cole Publishing/ITP

There are two ways to conduct a test of a hypothesis:
- The critical value approach
- The p-value approach

Example 10.4 further applies the t distribution in a two-tailed
test.
©1998 Brooks/Cole Publishing/ITP
Figure 10.5 Calculating the p-value for Example 10.4
©1998 Brooks/Cole Publishing/ITP

Most statistical computing packages contain programs that will
implement the Student’s t test or construct a confidence interval
for m when the data are properly entered.
Figure 10.6 Minitab output for Example 10.4
©1998 Brooks/Cole Publishing/ITP

The value of using the computer output to evaluate statistical
results:
- The exact p-value eliminates the need for tables and
critical values.
- All of the numerical calculations are done for you.
©1998 Brooks/Cole Publishing/ITP
10.4 Small-Sample Inferences
for the Difference Between Two
Population Means:
Independent Random Samples
Test of a Hypothesis Concerning the Difference Between
Two Means: Independent Random Samples
1. Null hypothesis: H 0 : ( m 1 - m 2)  D 0 , where D 0 is some
specified difference that you wish to test. For many tests,
you will hypothesize that there is no difference between m 1
and m 2; that is, D 0  0.
©1998 Brooks/Cole Publishing/ITP
2. Alternative hypothesis:
One-Tailed Test
H a : ( m 1 - m 2) > D 0
[or H a : ( m 1 - m 2) < D 0 ]
Two-Tailed Test
H a : ( m 1 - m 2)  D 0
3. Test statistic:

x 1 - x 2  - D0
t
1
1

n  n 
 1
2
s2
where s 2 
 n1 - 1s12   n2 - 1s22
n1  n2 - 2
©1998 Brooks/Cole Publishing/ITP
4. Rejection region: Reject H 0 when
One-Tailed Test
Two-Tailed Test
t > ta
t > ta/2 or zt < -ta/2
[or t < -ta when the
alternative hypothesis
is H a : ( m 1 - m 2) < D 0 ]
or when the p-value is a


The critical values are based on (n1  n2 - 2) d f. The tabulated
values can be found in Table 4 in Appendix I.
Assumptions: The samples are randomly and independently
selected from normally distributed populations. The variances of
2
the populations s 12 and s 21
are equal.
©1998 Brooks/Cole Publishing/ITP
Small-Sample (1 - a )100% Confidence Interval for ( m 1 - m 2)
Based on Independent Random Samples:
x1 - x 2   ta 2
1
1

n  n 
 1
2
s2
where s 2 is the pooled estimate of s 2 .

Example 10.5 provides a one-tailed t test for the difference
between two population means. Example 10.6 determines the
p-value for the previous example. Example 10.7 determines a
confidence limit for the previous example.
©1998 Brooks/Cole Publishing/ITP
Example 10.5
An assembly operation in a manufacturing plant requires
approximately a 1-month training period for a new employee to
reach maximum efficiency in assembling a device. A new
method of training was suggested, and a test was conducted to
compare the new method with the standard procedure. Two
groups of nine new employees were trained for a period of
3 weeks, one group using the new method and the other
following the standard training procedure. The length of time
(in minutes) required for each employee to assemble the device
was recorded at the end of the 3-week period. These measurements appear in Table 10.2. Do the data present sufficient
evidence to indicate that the mean time to assemble at the
end of a 3-week training period is less for the new training
procedure?
©1998 Brooks/Cole Publishing/ITP
Table 10.2 Assembly times after two training procedures
Standard
Procedure
32
37
35
28
41
44
35
31
34
New
Procedure
35
31
29
25
34
40
27
32
31
©1998 Brooks/Cole Publishing/ITP
Figure 10.8 Rejection region for Example 10.5
©1998 Brooks/Cole Publishing/ITP

The two-sample procedure that uses a pooled estimate of the
common variance s 2 relies on four important assumptions:
- The samples must be randomly selected.
- The samples must be independent.
- The populations from which you sample must be normal.
- The population variances should be equal or nearly equal.

You should use an approximate t distribution if
Larger s 2
>3
2
Smaller s
The resulting test statistic is
 x 1 - x 2  - D0
s12 s 22

n1 n2
©1998 Brooks/Cole Publishing/ITP

When the sample sizes are small, critical values for this statistic
are found in Table 4 of Appendix I, using degrees of freedom
approximated by the formula:
df 
 s12 s22 
  
n

 1 n2 
s12 n1 2  s22
n1 - 1

2
2
n2
n2 - 1
Figure 10.9 gives the Minitab output for Example 10.5.
©1998 Brooks/Cole Publishing/ITP
10.5 Small-Sample Inferences for
the Difference Between Two Means:
A Paired-Different Test

Table 10.3 shows the average wear on two types of tires. Figure
10.10 shows the Minitab output using a t test for independent
samples for the tire data. Table 10.4 shows differences in tire
wear using the data of Table 10.3.
Table 10.3
Automobile
1
2
3
4
5
Tire A
10.6
9.8
12.3
9.7
8.8
x 1  10.24
s1  1.1316
Tire B
10.2
9.4
11.8
9.1
8.3
x 2  9.76
s2  1.328
©1998 Brooks/Cole Publishing/ITP
Figure 10.10
Table 10.4
Automobile
1
2
3
4
5
A
B
10.6
9.8
12.3
9.7
8.8
10.2
9.4
11.8
9.1
8.3
d=A-B
.4
.4
.5
.6
.5
d  .48
©1998 Brooks/Cole Publishing/ITP

The paired-difference or matched pairs design allows us to
eliminate the variability by looking at only the difference
measurements.
Paired Difference Test of Hypothesis for ( m 1 - m 2)  m d :
Dependent Samples
1. Null hypothesis: H 0 : md  0
2. Alternative hypothesis:
One-Tailed Test
H a : md > 0
(or H a : md < 0)
Two-Tailed Test
H a : md  0
©1998 Brooks/Cole Publishing/ITP
3. Test statistic:
t
d -0
d

sd n sd n
where n  Number of paired differences
d  Mean of the sample differences
sd  Standard deviation of the sample differences
4. Rejection region: Reject H 0 when
One-Tailed Test
Two-Tailed Test
t > ta
t > ta/2 or t < -ta/2
(or t < -ta when the
alternative hypothesis
is H a : md < 0 )
or when p-value < a
©1998 Brooks/Cole Publishing/ITP

The critical values are based on (n - 1) d f. These tabulated
values are given in Table 4 in Appendix I.
A (1- a )100% Small-Sample Confidence Interval for
( m 1 - m 2)  m d , Based on a Paired-Difference Experiment:
 sd 

d  ta 2 
 n

Assumption: The experiment is designed as a paired-difference
test so that the n differences represent a random sample from a
normal population.

Example 10.8 shows a t test for the difference in the mean for
two small samples. Example 10.9 computes a 95% confidence
interval from the data in the previous example.
©1998 Brooks/Cole Publishing/ITP
Example 10.8
Do the data in Table 10.3 provide sufficient evidence to indicate
a difference in the mean wear for tire types A and B ? Test
using a  .05.
Solution
You can verify using your calculator that the average and
standard deviation of the five difference measurements are
d  .48 and sd  .0837
Then
and
H 0 : md  0 and H a : md  0
t
d -0
.48

 12.8
sd n .0837 5
The critical value of t for a two-tailed statistical test, a  .05 and
4 df, is 2.776. Certainly, the observed value of t 12.8 is
extremely large and highly significant. Hence, you can conclude
that there is a difference in the mean wear for tire types A and B.
©1998 Brooks/Cole Publishing/ITP

Blocking: Comparing the different procedures within groups of
relatively similar experimental units called blocks. In this way,
the “noise” caused by the large variability does not mask the
true differences between the procedures.
©1998 Brooks/Cole Publishing/ITP
10.6 Inferences Concerning
a Population Variance
Definition: The standardized statistic
c2

n - 1s 2

s2
is called a chi-square variable and has a sampling distribution
called the chi-square probability distribution.

Figure 10.12 shows a c2 distribution. Table 10.5 shows
the format of the c2 table from Table 5 in Appendix I.
Example 10.10 requires the use of the table.
©1998 Brooks/Cole Publishing/ITP
Figure 10.12
©1998 Brooks/Cole Publishing/ITP
Table 10.5
©1998 Brooks/Cole Publishing/ITP
Example 10.10
Check you ability to use Table 5 in Appendix I by verifying the
following statements:
1. The probability that c2, based on n  16 measurements
(d f  15), exceeds 24.9958 is .05.
2. For a sample of n  6 measurements, 95% of the area under
the c2 distribution lies to the right of 1.145476.
These values are shaded in Table 10.5.
©1998 Brooks/Cole Publishing/ITP
Test of Hypothesis Concerning a Population Variance
1. Null hypothesis: H 0 : s 2  s 02
2. Alternative hypothesis:
One-Tailed Test
H a : s 2 > s 02
(or H a : s 2 < s 02 )
3. Test statistic:
c2
Two-Tailed Test
H a : s 2  s 02

n - 1 s 2

s 02
4. Rejection region: Reject H 0 when
One-Tailed Test
Two-Tailed Test
c 2 > ca2 2 or c 2 < c 21-a 2 
c 2 > c a2
[or c 2 < c 21-a  when the
alternative hypothesis
is H a : s 2 < s 02 ]
or when p-value < a
©1998 Brooks/Cole Publishing/ITP


The critical values of c2 are based on (n - 1) d f. These
tabulated values are given in Table 5 in Appendix I.
The unnumbered figure on page 419 shows the one- and
two-tailed rejection regions for the c2 distribution.
©1998 Brooks/Cole Publishing/ITP
A (1- a )100% Confidence Interval for s 2 :
n - 1s 2 < s 2 < n - 1s 2
2
2
ca


2
c 1-a 2 
Assumption: The sample is randomly selected from a normal
population.
Examples 10.11 and 10.12 are applications of the c2
distribution to the determination of equality of variances.
©1998 Brooks/Cole Publishing/ITP
Example 10.11
A cement manufacturer claims that concrete prepared from his
product has a relatively stable compressive strength and that
the strength measured in kilograms per square centimeter
(km/cm 2 ) lies within a range of 40 km/cm 2. A sample of
n  10 measurements produced a mean and variance equal to,
respectively,
x  312 and s 2  195
Do these data present sufficient evidence to reject the
manufacturers claim?
Solution
In Section 2.5, you learned that the range of a set of
measurements should be approximately four standard
deviations.
©1998 Brooks/Cole Publishing/ITP
The manufacturer’s claim that the range of the strength
measurements is within 40 km/cm 2 must mean that the standard
deviation of the measurements is roughly 10 km/cm 2 or less. To
test his claim, the appropriate hypotheses are
H 0 : s 2  10 2  100
versus
H a : s 2 > 100
If the sample variance is much larger than the hypothesized
value of 100, then the test statistic

n - 1s 2

1755
 17.55
2
100
s0
will be unusually large, favoring rejection of H 0 and acceptance
of H a. There are two ways to use the test statistic to make a
decision for this test:
c2

- The critical value approach
- The p-value approach
©1998 Brooks/Cole Publishing/ITP
Figure 10.13 Rejection region and p-value (shaded)
for Example 10.11
©1998 Brooks/Cole Publishing/ITP
10.7 Comparing Two Populations
Variances


When independent random samples are drawn from two normal
populations with equal variances — that is, s 12  s 22 — then
s12 s22 has a probability distribution in repeated sampling that
is known to statisticians as an F distribution.
Figure 10.14 shows an F distribution with d f 1  10 and d f 2  10.
Example 10.13 requires the use of Table 6 in Appendix I.
Assumptions for s12 s22 to Have an F Distribution:
- Random and independent samples are drawn from each of
two normal populations.
- The variability of the measurements in the two populations is
the same and can be measured by a common variance, s 2;
that is, s 12  s 22  s 2 .
©1998 Brooks/Cole Publishing/ITP
Figure 10.14
©1998 Brooks/Cole Publishing/ITP
Example 10.13
Check your ability to use Table 6 in Appendix I by verifying the
following statements:
1. The value of F with area .05 to its right for d f 1  6 and
d f 2  9 is 3.37.
2. The value of F with area .05 to its right for d f 1  5 and
d f 2  10 is 3.33.
3. The value of F with area .05 to its right for d f 1  6 and
d f 2  9 is is 5.80.
These values are shaded in Table 10.6.
©1998 Brooks/Cole Publishing/ITP
Test of Hypothesis Concerning the Equality of
Two Population Variances.
1. Null hypothesis: H 0 : s 2  s 2
1
2
2. Alternative hypothesis:
One-Tailed Test
H a : s 12 > s 22
Two-Tailed Test
H a : s 12  s 22
3. Test statistic:
One-Tailed Test
F  s12 s22
Two-Tailed Test
F  s12 s22
(or H a : s 12 < s 22 )
where s12 is the larger sample variance
©1998 Brooks/Cole Publishing/ITP
4. Rejection region: Reject H 0 when
One-Tailed Test
Two-Tailed Test
F > Fa
F > Fa / 2
or when p-value < a

Assumptions: The samples are randomly and independently
selected from normally distributed populations.
©1998 Brooks/Cole Publishing/ITP
Confidence Interval for s 12 s 22 :
 s12 
s 12  s12 
1
1
 
 
<
<
2
 s2  F
 2
 2  df1, df2  s 2  s2  Fdf2 , df1 

where d f 1  (n 1 - 1) and d f 2  (n 2 - 1).
Assumption: The samples are randomly and independently
selected from normally distributed populations.

Example 10.14 applies the F test, as do Examples 10.15
and 10.16.

The F test for the difference in two population variances
completes the battery of tests in this chapter for making
inferences about population parameters under these conditions.
- The sample sizes are small.
- The sample or samples are drawn from normal populations.
©1998 Brooks/Cole Publishing/ITP
Example 10.14
An experimenter is concerned that the variability of responses
using two different experimental procedures may not be the
same. Before conducting his research, he conducts a prestudy
with random samples of 10 and 8 responses and gets
s12  7.14 and s22  3.21, respectively. Do the sample variances
present sufficient evidence to indicate that the population
variances are unequal?
Solution
Assume that the populations have probability distributions that
are reasonably mound-shaped and hence satisfy, for all
practical purposes, the assumptions that the populations are
normal. You wish to test these hypotheses:
H 0 : s 12  s 22
versus
Ha : s 12  s 22
©1998 Brooks/Cole Publishing/ITP
Using Table 6 in Appendix I for a / 2  .025, you can reject H 0
when F > 4.82 with a  .05. The calculated value of the test
statistic is
s12 7.14
F 2 
 2.22
s2 3.21
Because the test statistic does not fall into the rejection region,
you can not reject H 0 : s 12  s 22 .
Thus, there is insufficient evidence to indicate a difference in the
population variances.
©1998 Brooks/Cole Publishing/ITP

How do I decide which test to use?
- Are you interested in testing means? If the design involves:
a. One random sample, use the one-sample t statistic.
b. Two independent random samples: Are the population
variances equal?
i. If equal, use the two-sample t statistic with pooled s 2.
ii. If unequal, use the unpooled t with estimated df.
c. Two paired samples with random pairs, use a
one-sample t for analyzing differences.
- Are you interested in testing variances? If the design involves:
a. One random sample, use the c 2 test for a single
variance.
b. Two independent random samples, use the F test to
compare two variances.
©1998 Brooks/Cole Publishing/ITP
10.8 Revisiting the
Small-Sample Assumptions
Assumptions:
1.For all tests and confidence intervals described in this chapter,
it is assumed that samples are randomly selected from
normally distributed populations.
2. When two samples are selected, it is assumed that they are
selected in an independent manner except in the case of the
paired-difference experiment.
3.For tests or confidence intervals concerning the difference
between two population means m 1 and m 2, it is assumed
that s 12  s 22 .
©1998 Brooks/Cole Publishing/ITP
Key Concepts and Formulas
I. Experimental Designs for Small Samples
1. Single random sample: The sampled population must be
normal.
2. Two independent random samples: Both sampled populations
must be normal.
a. Populations have a common variance s 2.
b. Populations have different variances s 12 and s 22 .
3. Paired-difference or matched-pairs design: The samples are
not independent.
©1998 Brooks/Cole Publishing/ITP
II. Statistical Tests of Significance
1. Based on the t, F, and c 2 distributions
2. Use the same procedure as in Chapter 9
3. Rejection region— critical values and significance levels:
based on the t, F, and c 2 distributions with the appropriate
degrees of freedom
4. Tests of population parameters: a single mean, the difference
between two means, a single variance, and the ratio of two
variances
III. Small Sample Test Statistics
To test one of the population parameters when the sample sizes
are small, use the following test statistics:
©1998 Brooks/Cole Publishing/ITP
©1998 Brooks/Cole Publishing/ITP