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Title: Statistics

1
Statistics

Tests of Hypotheses for a Single Sample

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
2
Hypothesis Testing

Statistical hypothesis
A statistical hypothesis is a statement about the
parameters of one or more populations.
For example,
centimeters per second
centimeters per second
is the null hypothesis and is a
two-sided alternative hypothesis

Type I error
Rejecting the null hypothesis when it is
true is defined as a type I error
Type II error
Failing to reject the null hypothesis when it is
false is defined as a type II error
Probability of type I error
P(type I error) P(reject when
is true)
Probability of type II error
P(type II error) P(fail to reject
when is false)

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
4
Null hypothesis (H0) is true Null hypothesis (H0) is false
Reject null hypothesis Type I errorFalse positive Correct outcomeTrue positive
Fail to reject null hypothesis Correct outcomeTrue negative Type II errorFalse negative
From Wikipedia, http//www.wikipedia.org.
5

Properties
The size of the critical region and can be
reduced by appropriate selection of the critical
values
Type I and type II errors are related. Decrease
one will increase the other
An increase in sample size reduces
increases as the true value of the
parameter approaches the value hypothesized in
the null hypothesis
0.05
Widely used

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
6

Power
The probability of correctly rejecting a false
null hypothesis
Sensitivity the ability to detect differences

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
7

Formulating one-sided hypothesis
1.5 MPa
gt 1.5 Mpa (We want)
Or
1.5 MPa
lt 1.5 Mpa (We want)

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
8

Formulating one-sided hypothesis
1.5 MPa
gt 1.5 Mpa (We want)
Or
1.5 MPa
lt 1.5 Mpa (We want)
P-value
The P-value is the smallest level of significance
that would lead to rejection of the null
hypothesis with the given data

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
9

General procedure for hypothesis tests
Specify the test statistic to be used (such as
)
Specify the location of the critical region
(two-tailed, upper-tailed, or lower-tailed)
Specify the criteria for rejection (typically,
the value of , or the P-value at which
rejection should occur)
Practical significance
Be careful when interpreting the results from
hypothesis testing when the sample size is large,
because any small departure from the hypothesized
value will probably be detected, even when
the difference is of little or no practical
significance

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
10

Example 9-1 Propellant Burning Rate
Suppose that if the burning rate is less than 50
centimeters per second, we wish to show this with
a strong conclusion.
centimeters per second
centimeters per second
Since the rejection of is always a strong
conclusion, this statement of the hypotheses will
produce outcome if is rejected.

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
11

Exercise 9-27
A random sample of 500 registered voters in
Phoenix is asked if they favor the use of
oxygenated fuels year-round to reduce air
pollution. If more than 400 voters respond
positively, we will conclude that more than 60
of the voters favor the use of these fuels.
(a) Find the probability of type I error if
exactly 60 of the voters favor the use of these
fuels.
(b) What is the type II error probability if
75 of the voters favor this action?
Hint use the normal approximation to the
binomial.

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
12
Tests on the Mean of a Normal Distribution,
Variance Known

Hypothesis tests on the mean
Hypotheses, two-sided alternative
Test statistic
P-value
Reject if or

Hypotheses, upper-tailed alternative
P-value
Reject if
Hypotheses, lower-tailed alternative
P-value
Reject if

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, two-sided alternative
Suppose the true value of the mean under is
Test statistic
Under

Type II error and choice of sample size
Sample size formulas
If
Let be the 100 upper percentile of the
standard normal distribution. Then

Note

Sample size for a two-sided test on the mean,
variance known
Sample size for a one-sided test on the mean,
variance known

Operating characteristic (OC) curves
Curves plotting against a parameter for
various sample size
See Appendix VII
For a given and , find .
For a given and , find
Large-sample test
If , the sample standard deviation
can be substituted for in the test
procedures with little effect

Example 9-2 Propellant Burning Rate
, , ,
,
Specifications require that the mean burning rate
must be 50 centimeters per second. What
conclusions should be drawn?
Example 9-3 Propellant Burning Rate Type II Error
Suppose that the true burning rate is 49
centimeters per second. What is for the
two-sided test with , ,
and ?
Example 9-4 Propellant Burning Rate Type II Error
from OC Curve
Suppose the true mean burning rate is
centimeters per second.

Example 9-4 Propellant Burning Rate Sample Size
from OC Curve
Design the test so that if the true mean burning
rate differs from 50 centimeters per second by as
much an 1 centimeter per second, the test will
detect this with a high probability 0.90.

Exercise 9-47
Medical researchers have developed a new
artificial heart constructed primarily of
titanium and plastic. The heart will last and
operate almost indefinitely once it is implanted
in the patients body, but the battery pack needs
to be recharged about every four hours. A random
sample of 50 battery packs is selected and
subjected to a life test. The average life of
these batteries is 4.05 hours. Assume that
battery life is normally distributed with
standard deviation
hour.
(a) Is there evidence to support the claim that
mean battery life exceeds 4 hours? Use
.
(b) What is the P-value for the test in part (a)?

Exercise 9-47
(c) Compute the power of the test if the true
mean battery life is 4.05 hours.
(d) What sample size would be required to detect
a true mean battery life of 4.5 hours if we
wanted the power of the test to be at least 0.9?
(e) Explain how the question in part (a) could be
answered by constructing a one-sided confidence
bound on the mean life.

23
Tests on the Mean of a Normal Distribution,
Variance Unknown

Hypothesis tests on the mean
Hypotheses, two-sided alternative
Test statistic
P-value
Reject if or

Hypotheses, upper-tailed alternative
P-value
Reject if
Hypotheses, lower-tailed alternative
P-value
Reject if

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, two-sided alternative
Suppose the true value of the mean under is
Test statistic
Under
is of the noncentral distribution with
degrees of freedom and noncentrality parameter
.

PDF of noncentral distribution

From Wikipedia, http//www.wikipedia.org.
27

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, two-sided alternative
where denotes the noncentral random
variable
Operating characteristic (OC) curves
Curves plotting against a parameter for
various sample size
See Appendix VII
Note that depends on the unknown parameter
.

Example 9-6 Golf Club Design
It is of interest to determine if there is
evidence (with ) to support a claim
that the mean coefficient of restitution exceeds
0.82.
Data 0.8411,
and
Example 9-7 Golf Club Design Sample Size
If the mean coefficient of restitution exceeds
0.82 by as much as 0.02, is the sample size
adequately to ensure that
will be rejected with probability at least 0.8?
.

Exercise 9-59
A 1992 article in the Journal of the American
Medical Association (A Critical Appraisal of
98.6 Degrees F, the Upper Limit of the Normal
Body Temperature, and Other Legacies of Carl
Reinhold August Wunderlich) reported body
temperature, gender, and heart rate for a number
of subjects. The body temperatures for 25 female
subjects follow 97.8,
(a) Test the hypothesis versus
using . Find the
P-value.
(b) Check the assumption that female body
temperature is normally distributed.
(c) Compute the power of the test if the true
mean female body temperature is as low as 98.0.
.

Exercise 9-59
(d) What sample size would be required to detect
a true mean female body temperature as low as
98.2 if we wanted the power of the test to be at
least 0.9?
(e) Explain how the question in part (a) could be
answered by constructing a two-sided confidence
interval on the mean female body temperature.

Exercise 9-59
Normality plot

32
Tests on the Variance and Standard Deviation of a
Normal Distribution

Hypothesis tests on the variance
Hypotheses, two-sided alternative
Test statistic
P-value
Reject if or

Hypotheses, upper-tailed alternative
P-value
Reject if
Hypotheses, lower-tailed alternative
P-value
Reject if

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, two-sided alternative
Suppose the true value of the variance under
is

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, upper-tailed alternative
Suppose the true value of the variance under
is

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, lower-tailed alternative
Suppose the true value of the variance under
is

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, two-sided alternative
Operating characteristic (OC) curves
Curves plotting against a parameter for
various sample size
See Appendix VII

Example 9-8 Automated Filling
, , .
Is there evidence in the sample data to suggest
that the manufacture has a problem with
underfilled or overfilled bottles? (
)
Example 9-8 Automated Filling Sample Size
,
Find

Exercise 9-83
Recall the sugar content of the syrup in canned
peaches from Exercise 8-46. Suppose that the
variance is thought to be
(milligrams)2. Recall that a random sample of
cans yields a sample standard deviation of
milligrams.
(a) Test the hypothesis versus
using . Find the
P-value for this test.
(b) Suppose that the actual standard deviation is
twice as large as the hypothesized value. What is
the probability that this difference will be
detected by the test described in part (a)?
(c) Suppose that the true variance is
. How large a sample would be required to
detect this difference with probability at least
0.90?

40
Tests on a Population Proportion

Large-sample tests on a proportion
Hypotheses, two-sided alternative
Test statistic
P-value
Reject if or

Hypotheses, upper-tailed alternative
P-value
Reject if
Hypotheses, lower-tailed alternative
P-value
Reject if

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, two-sided alternative
Suppose the true value of the proportion under
is

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, upper-tailed alternative
Suppose the true value of the proportion under
is

Type II error and choice of sample size
Finding the probability of type II error
Hypotheses, lower-tailed alternative
Suppose the true value of the proportion under
is

Type II error and choice of sample size
Two-sided alternative
Let be the 100 upper percentile of the
standard normal distribution. Then

Type II error and choice of sample size
Upper-tailed alternative
Let be the 100 upper percentile of the
standard normal distribution. Then

Type II error and choice of sample size
Lower-tailed alternative
Let be the 100 upper percentile of the
standard normal distribution. Then

Example 9-10 Automobile Engine Controller
, ,
The semiconductor manufacturer takes a random
sample of 200 devices and finds that four of them
are defective. Can the manufacturer demonstrate
process capability for the customer? (
)
Example 9-11 Automobile Engine Controller Type II
Error
Suppose that its process fallout is really
. What is the -error for a test of process
capability that uses and
?

Exercise 9-95
In a random sample of 85 automobile engine
crankshaft bearings, 10 have a surface finish
roughness that exceeds the specifications. Does
this data present strong evidence that the
proportion of crankshaft bearings exhibiting
excess surface roughness exceeds 0.10?
(a) State and test the appropriate hypotheses
using
.
(b) If it is really the situation that
, how likely is it that the test procedure in
part (a) will not reject the null hypotheses?
(c) If , how large would the sample
size have to be for us to have a probability of
correctly rejecting the null hypothesis of 0.9?
, ,

50
Testing for Goodness of Fit

Test the hypothesis that a particular
distribution will be satisfactory as a population
model
Based on the chi-square distribution
observations, is the number of
parameters of the hypothesized distribution
estimated by sample statistics
the observed frequency in the th class
interval
the expected frequency in the th class
interval
Test statistic
P-value
Reject the hypothesis if

Example 9-12 Printed Circuit Board Defects,
Poisson Distribution
Number of defects 0, observed frequency 32
Number of defects 1, observed frequency 15
Number of defects 2, observed frequency 9
Number of defects 3, observed frequency 4
Example 9-13 Power Supply Distribution,
Continuous Distribution
, ,
A manufacturer engineer is testing a power supply
used in a notebook computer and, using
, wishes to determine whether output voltage is
adequately described by a normal distribution.

Exercise 9-101
The number of cars passing eastbound through the
intersection of Mill and University Avenues has
been tabulated by a group of civil engineering
students. They have obtained the data in the
adjacent table
(a) Does the assumption of a Poisson distribution
seem appropriate as a probability model for this
process? Use .
(b) Calculate the P-value for this test.
Data (40, 14), (41, 24),

53
Contingency Table Tests

Test the hypothesis that two methods of
classification are statistically independent
Based on the chi-square distribution
observations, contingency table
the observed frequency for level of
the first classification and level for the
second classification
,
,
Test statistic
P-value
Reject the hypothesis if

Example 9-13 Health Insurance Plan Preference
A company has to choose among three health
insurance plans. Management wishes to know
whether the preference for plans is independent
of job classification and wants to use
.
, data
Exercise 9-107
A study is being made of the failure of an
electronic component. There are four types of
failures possible and two mounting positions for
the device
Would you conclude that the type of failure is
independent of the mounting position? Use
. Find the P-value for this test.

A B C D
1 20 48 20 7
2 4 17 6 12
55
Nonparametric Procedures

The sign test
Test hypotheses about the median of a
continuous distribution
the observed number of plus signs (
)
Hypotheses, two-sided alternative
P-value
if
or
if
Reject if

Hypotheses, upper-tailed alternative
P-value
Reject if
Hypotheses, lower-tailed alternative
P-value
Reject if

Appendix Table VIII ( )
Hypotheses, two-sided alternative
Reject if
Hypotheses, upper-tailed alternative
Reject if
Hypotheses, lower-tailed alternative
Reject if

Ties in the sign test
Values of exactly equal to should be
set aside and the sign test applied to the
remaining data
Normal approximation for sign test statistic
Reject if for
or if for
or if for

Type II error for the sign test
Finding the probability of type II error
Not only a particular value of , say,
, must be used but also the form of the
underlying distribution will affect the
calculations

Wilcoxon signed-rank test
Appendix Table IX ( )
Rank the absolute differences in
ascending order, and then give the ranks the
signs of their corresponding differences
the sum of the positive ranks
the absolute value of the sum of negative
ranks
Hypotheses, two-sided alternative
Reject if

Wilcoxon signed-rank test
Appendix Table IX ( )
Hypotheses, upper-tailed alternative
Reject if
Hypotheses, lower-tailed alternative
Reject if

Ties in the Wilcoxon signed-rank test
If several observations have the same absolute
magnitude, they are assigned the average of the
ranks that they would receive if they differed
slightly from one another
Normal approximation for Wiocoxon signen-rank
test statistic
Reject if for
or if for
or if for

Example 9-15 Propellant Shear Strength Sign Test
We would like to test the hypothesis that the
median shear strength is 13790 kN/m2, using
Example 9-16 Propellant Shear Strength Wilcoxon
Signed-Rank Test
We would like to test the hypothesis that the
median shear strength is 13790 kN/m2, using

Exercise 9-117
A primer paint can be used on aluminum panels.
The drying time of the primer is an important
consideration in the manufacturing process.
Twenty panels are selected and the drying times
are as follows 1.6,
Is there evidence that the mean drying time of
the primer exceeds 1.5 hr?

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