1. Tests of Hypotheses for a Single Sample
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
is the null hypothesis and is a two- sided alternative hypothesis

3. Probability of 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 error False positive
Correct outcome True positive
Fail to reject null hypothesis
Correct outcome True negative
Type II error False negative
From Wikipedia,

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
: > 1.5 Mpa (We want) Or
: < 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
: > 1.5 Mpa (We want) Or
: < 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
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

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 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 patient’s 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

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,

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: , …
and
Example 9-7 Golf Club Design Sample Size
If the mean coefficient of restitution exceeds 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 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 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

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 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

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 upper percentile of the standard normal distribution. Then

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

47. Type II error and choice of sample size
Lower-tailed alternative
Let be the 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 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

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

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

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

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

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

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?