2. 4-1 Statistical Inference
The field of statistical inference consists of those
methods used to make decisions or draw
conclusions about a population.
These methods utilize the information contained in
a random sample from the population in drawing
conclusions.

3. 4-1 Statistical Inference

4. 4-2 Point Estimation

5. 4-2 Point Estimation

6. 4-2 Point Estimation

7. 4-2 Point Estimation

8. 4-2 Point Estimation

9. 4-2 Point Estimation

10. 4-3 Hypothesis Testing 4-3.1 Statistical Hypotheses
We like to think of statistical hypothesis testing as the data analysis stage of a comparative experiment, in which the engineer is interested, for example, in comparing the mean of a population to a specified value (e.g. mean pull strength).

11. 4-3 Hypothesis Testing 4-3.1 Statistical Hypotheses
Two-sided Alternative Hypothesis
One-sided Alternative Hypotheses

12. 4-3 Hypothesis Testing 4-3.1 Statistical Hypotheses
Test of a Hypothesis
A procedure leading to a decision about a particular hypothesis
Hypothesis-testing procedures rely on using the information in a random sample from the population of interest.
If this information is consistent with the hypothesis, then we will conclude that the hypothesis is true; if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.

13. 4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses

14. 4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses

15. 4-3 Hypothesis Testing 4-3.2 Testing Statistical Hypotheses
Sometimes the type I error probability is called the significance level, or the -error, or the size of the test.

16. 4-3 Hypothesis Testing 4-3.2 Testing Statistical Hypotheses Reduce α
push critical regions further
Increasing sample size  increasing Z value

17. 4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses

18. 4-3 Hypothesis Testing 4-3.2 Testing Statistical Hypotheses
Increasing the sample size results in a decrease in β.
error when μ =52 and n=16.

19. 4-3 Hypothesis Testing 4-3.2 Testing Statistical Hypotheses
β increases as the true value of μ approaches the hypothesized value.
error when μ =50.5 and n=10.

20. 4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses

21. 4-3 Hypothesis Testing 4-3.2 Testing Statistical Hypotheses
The power is computed as 1 - b, and power can be interpreted as the probability of correctly rejecting a false null hypothesis. We often compare statistical tests by comparing their power properties.
For example, consider the propellant burning rate problem when
we are testing H 0 : m = 50 centimeters per second against H 1 : m not equal 50 centimeters per second . Suppose that the true value of the mean is m = 52. When n = 10, we found that b = , so the power of this test is 1 - b = = when m = 52.

22. 4-3 Hypothesis Testing 4-3.3 P-Values in Hypothesis Testing
P-value carries info about the weight of evidence against H0.
The smaller the p-value, the greater the evidence against H0.

23. 4-3 Hypothesis Testing 4-3.4 One-Sided and Two-Sided Hypotheses
Two-Sided Test:
One-Sided Tests:

24. 4-3 Hypothesis Testing
4-3.5 General Procedure for Hypothesis Testing

25. 4-3 Hypothesis Testing

26. 4-3 Hypothesis Testing OPTIONS NODATE NONUMBER; DATA EX415;
mu_h0=12; mu_h1=11.25; sigma=0.5; n=4; alpha=0.05;
xbar=11.7; sem=sigma/sqrt(n); /* sem = mean standard error */
Z=(xbar-mu_h0)/sem; z_beta= probit(alpha);
xbar_beta=mu_h0+z_beta*sem;
z_beta=(xbar_beta-mu_h1)/sem;
PVAL=PROBNORM(Z);
beta=1-probnorm(z_beta);
power=1-beta;
PROC PRINT;
VAR PVAL beta power;
TITLE '4-15 (page 168)';
RUN; QUIT;
4-15 (page 168)
OBS PVAL beta power

27. 4-4 Inference on the Mean of a Population, Variance Known
Assumptions

28. 4-4 Inference on the Mean of a Population, Variance Known
4-4.1 Hypothesis Testing on the Mean
We wish to test:
The test statistic is:

29. 4-4 Inference on the Mean of a Population, Variance Known
4-4.1 Hypothesis Testing on the Mean

30. 4-4 Inference on the Mean of a Population, Variance Known
4-4.1 Hypothesis Testing on the Mean
Reject H0 if the observed value of the test statistic z0 is either: or
Fail to reject H0 if

31. 4-4 Inference on the Mean of a Population, Variance Known
4-4.1 Hypothesis Testing on the Mean

32. 4-4 Inference on the Mean of a Population,
Variance Known

33. 4-4 Inference on the Mean of a Population, Variance Known
OPTIONS NODATE NONUMBER;
DATA EX43;
MU=50; SD=2; N=25; ALPHA=0.05; XBAR=51.3;
SEM=SD/SQRT(N); /*MEAN STANDARD ERROR*/
Z=(XBAR-MU)/SEM;
PVAL=2*(1-PROBNORM(Z));
PROC PRINT;
VAR Z PVAL ALPHA;
TITLE 'TEST OF MU=50 VS NOT=50';
RUN; QUIT;
TEST OF MU=50 VS NOT=50
OBS Z PVAL ALPHA

34. 4-4 Inference on the Mean of a Population, Variance Known
4-4.2 Type II Error and Choice of Sample Size
Finding The Probability of Type II Error 

35. 4-4 Inference on the Mean of a Population, Variance Known
4-4.2 Type II Error and Choice of Sample Size
Finding The Probability of Type II Error 

36. 4-4 Inference on the Mean of a Population, Variance Known
4-4.2 Type II Error and Choice of Sample Size
Sample Size Formulas

37. 4-4 Inference on the Mean of a Population, Variance Known
4-4.2 Type II Error and Choice of Sample Size
Sample Size Formulas

38. 4-4 Inference on the Mean of a Population, Variance Known
4-4.2 Type II Error and Choice of Sample Size

39. 4-4 Inference on the Mean of a Population, Variance Known
4-4.2 Type II Error and Choice of Sample Size

40. 4-4 Inference on the Mean of a Population, Variance Known
4-4.3 Large Sample Test
In general, if n  30, the sample variance s2 will be close to σ2 for most samples, and so s can be substituted for σ in the test procedures with little harmful effect.

41. 4-4 Inference on the Mean of a Population, Variance Known
4-4.4 Some Practical Comments on Hypothesis
Testing
The Seven-Step Procedure
Only three steps are really required:

42. 4-4 Inference on the Mean of a Population, Variance Known
4-4.4 Some Practical Comments on Hypothesis
Testing
Statistical versus Practical Significance

43. 4-4 Inference on the Mean of a Population, Variance Known
4-4.4 Some Practical Comments on Hypothesis
Testing
Statistical versus Practical Significance

44. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean
Two-sided confidence interval:
One-sided confidence intervals:
Confidence coefficient:

45. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean

46. 4-4 Inference on the Mean of a Population, Variance Known
4-4.6 Confidence Interval on the Mean

47. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean

48. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean
Relationship between Tests of Hypotheses and Confidence Intervals
If [l,u] is a 100(1 - a) percent confidence interval for the parameter, then the test of significance level a of the hypothesis
will lead to rejection of H0 if and only if the hypothesized value is not in the 100(1 - a) percent confidence interval [l, u].

49. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean
Confidence Level and Precision of Estimation
The length of the two-sided 95% confidence interval is
whereas the length of the two-sided 99% confidence
interval is

50. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean
Choice of Sample Size

51. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean
Choice of Sample Size

52. 4-4 Inference on the Mean of a Population, Variance Known
4-4.5 Confidence Interval on the Mean
One-Sided Confidence Bounds

53. 4-4 Inference on the Mean of a Population, Variance Known
4-4.6 General Method for Deriving a Confidence Interval

54. 4-4 Inference on the Mean of a Population, Variance Known
Ex4-32
in pp.185

55. 4-4 Inference on the Mean of a Population, Variance Known
Ex4-37
in pp.185

56. 4-4 Inference on the Mean of a Population, Variance Known
Ex4-37
in pp.185

57. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean

58. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean

59. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean

60. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean
Calculating the P-value

61. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean

62. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean
Fixed Significance Level Approach

63. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.1 Hypothesis Testing on the Mean

64. 4-5 Inference on the Mean of a Population,
Variance Unknown

65. 4-5 Inference on the Mean of a Population,
Variance Unknown

66. 4-5 Inference on the Mean of a Population,
Variance Unknown

67. 4-5 Inference on the Mean of a Population, Variance Unknown
OPTIONS NODATE NONUMBER;
DATA ex47;
xbar= ; sd= ;
mu=0.82; n=15;alpha=0.05; df=n-1;
t0=(xbar-mu)/(sd/sqrt(n));
pval = 1- PROBT(t0, df); /* PROBT function returns the probability from a t distribution
less than or equal to x */
cv=tinv(1-alpha, df); /* critical value */
/* TINV function returns a quantile from the t distribution */
PROC PRINT;
var t0 pval cv;
title 'Test of mu=0.82 vs mu>0.82';
title2 'Ex. 4-7 in pp193';
RUN; QUIT;

68. 4-5 Inference on the Mean of a Population, Variance Unknown
OPTIONS NOOVP NODATE NONUMBER;
DATA ex47;
input coeffs
cards;
proc univariate data=ex47 mu0=0.82 plot normal;
var coeffs;
title 'using proc uinivariate for t-test for one population mean';
title2 “Example 4-7 in page 191”;
RUN; QUIT;

69. 4-5 Inference on the Mean of a Population,
Variance Unknown

70. 4-5 Inference on the Mean of a Population,
Variance Unknown

71. 4-5 Inference on the Mean of a Population,
Variance Unknown

72. 4-5 Inference on the Mean of a Population, Variance Unknown
OPTIONS NOOVP NODATE NONUMBER LS=80;
DATA ex47;
input coeffs
cards;
ods graphics on;
proc ttest data=ex47 h0=0.82 sides=u;
/* h0는 귀무가설, sides는 양측일 경우 2, lower 단측 검정일경우 L, upper 단측 검정일 경우 u*/
var coeffs;
title 'using t-test for one population mean';
title “Example 4-7 in page 191”;
RUN; ods graphics off;
QUIT;

73. 4-5 Inference on the Mean of a Population,
Variance Unknown

74. 4-5 Inference on the Mean of a Population,
Variance Unknown

75. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.2 Type II Error and Choice of Sample Size
Fortunately, this unpleasant task has already been done, and the results are summarized in a series of graphs in Appendix A Charts Va, Vb, Vc, and Vd that plot for the t-test against a parameter d for various sample sizes n.

76. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.2 Type II Error and Choice of Sample Size
These graphics are called operating characteristic (or OC) curves. Curves are provided for two-sided alternatives on Charts Va and Vb. The abscissa scale factor d on these charts is defined as

77. 4-5 Inference on the Mean of a Population, Variance Unknown
Chart V (c) in pp. 496

78. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.3 Confidence Interval on the Mean

79. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.3 Confidence Interval on the Mean

80. 4-5 Inference on the Mean of a Population, Variance Unknown
4-5.4 Confidence Interval on the Mean

81. 4-5 Inference on the Mean of a Population,
Variance Unknown

82. 4-5 Inference on the Mean of a Population, Variance Unknown
OPTIONS NOOVP NODATE NONUMBER LS=80;
ods pdf file='ols_out.pdf';
DATA ex60;
input diam
cards;
ods graphics on;
proc univariate plot normal mu0=8.2;
var diam;
title 'univariate for t-test with H0: mu=8.2';
proc ttest data=ex60 h0=8.2;
title 't-test for one population mean';
title “Example 4-60 in page 198”;
RUN; ods graphics off;
QUIT; ods pdf close;

83. 4-5 Inference on the Mean of a Population,
Variance Unknown

84. 4-5 Inference on the Mean of a Population,
Variance Unknown

85. 4-6 Inference on the Variance of a Normal Population
4-6.1 Hypothesis Testing on the Variance of a
Normal Population

86. 4-6 Inference on the Variance of a Normal Population
4-6.1 Hypothesis Testing on the Variance of a
Normal Population

87. 4-6 Inference on the Variance of a Normal Population
4-6.1 Hypothesis Testing on the Variance of a
Normal Population

88. 4-6 Inference on the Variance of a Normal Population
4-6.1 Hypothesis Testing on the Variance of a
Normal Population

89. 4-6 Inference on the Variance of a Normal Population
4-6.1 Hypothesis Testing on the Variance of a
Normal Population

90. 4-6 Inference on the Variance of a Normal Population
4-6.1 Hypothesis Testing on the Variance of a
Normal Population

91. 4-6 Inference on the Variance of a
Normal Population

92. 4-6 Inference on the Variance of a Normal Population
4-6.2 Confidence Interval on the Variance of a
Normal Population

93. 4-6 Inference on the Variance of a
Normal Population

94. 4-6 Inference on the Variance of a Normal Population
OPTIONS NODATE NONUMBER;
DATA ex410;
p_var=0.01; s_var=0.0153;
n=20;alpha=0.05; df=n-1;
chisq=(n-1)*s_var/p_var;
p_value=1-probchi(chisq, df);
cv1=cinv(1-alpha, df); /* critical value */
cv2=cinv(alpha, df);
lcl=sqrt((n-1)*s_var/cv1); /* ucl= upper confidence limit */
PROC PRINT;
var chisq p_value cv1 cv2 lcl;
title 'Test of H0: var=0.01 vs H0: var>0.01';
title2 'Ex in pp202';
RUN; QUIT;

95. 4-6 Inference on the Variance of a
Normal Population

96. 4-8 Other Interval Estimates for a Single Sample
4-8.1 Prediction Interval

97. 4-8 Other Interval Estimates for a Single Sample
4-8.2 Tolerance Intervals for a Normal Distribution