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

3

• 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

13

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

14

• 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

15

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

16

• Note

17

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

18

• 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

19

• 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.

20

• 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.

21

• 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)?
  

22

• 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

24

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

25

• 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
  .

26

• 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
  
• .

28

• 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?
• .

29

• 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.
• .

30

• 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.

31

• 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

33

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

34

• 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

35

• 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

36

• 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

37

• 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

38

• 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

39

• 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

41

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

42

• 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

43

• 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

44

• 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

45

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

46

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

47

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

48

• 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
  ?

49

• 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

51

• 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.

52

• 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

54

• 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

56

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

57

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

58

• 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

59

• 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

60

• 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

61

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

62

• 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

63

• 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
  

64

• 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?