4. 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 sample from the population in drawing conclusions.

5. 4-1 Statistical Inference

6. 4-2 Point Estimation

10. 4-3 Hypothesis Testing 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). 4-3.1 Statistical Hypotheses

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 1.push critical regions further 2.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 error when μ =52 and n=16. Increasing the sample size results in a decrease in β.

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

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 - , 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 :  = 50 centimeters per second against H 1 :  not equal 50 centimeters per second. Suppose that the true value of the mean is  = 52. When n = 10, we found that  = 0.2643, so the power of this test is 1 -  = 1 - 0.2643 = 0.7357 when  = 52.

22. 4-3 Hypothesis Testing 4-3.3 P-Values in Hypothesis Testing P-value carries info about the weight of evidence against H 0. The smaller the p-value, the greater the evidence against H 0.

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. 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 1 0.11507 0.087685 0.91231 4-3 Hypothesis Testing

26. 4-4 Inference on the Mean of a Population, Variance Known Assumptions

27. 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 :

28. 4-4 Inference on the Mean of a Population, Variance Known 4-4.1 Hypothesis Testing on the Mean

29. 4-4 Inference on the Mean of a Population, Variance Known 4-4.1 Hypothesis Testing on the Mean Reject H 0 if the observed value of the test statistic z 0 is either: or Fail to reject H 0 if

30. 4-4 Inference on the Mean of a Population, Variance Known 4-4.1 Hypothesis Testing on the Mean

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

32. 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 1 3.25.001154050 0.05

33. 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 

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 Sample Size Formulas

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

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 4-4.3 Large Sample Test In general, if n  30, the sample variance s 2 will be close to σ 2 for most samples, and so s can be substituted for σ in the test procedures with little harmful effect.

40. 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:

41. 4-4 Inference on the Mean of a Population, Variance Known 4-4.4 Some Practical Comments on Hypothesis Testing Statistical versus Practical Significance

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 4-4.5 Confidence Interval on the Mean Two-sided confidence interval: One-sided confidence intervals: Confidence coefficient:

44. 4-4 Inference on the Mean of a Population, Variance Known 4-4.5 Confidence Interval on the Mean

45. 4-4 Inference on the Mean of a Population, Variance Known 4-4.6 Confidence Interval on the Mean

46. 4-4 Inference on the Mean of a Population, Variance Known 4-4.5 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 Relationship between Tests of Hypotheses and Confidence Intervals If [l,u] is a 100(1 -  ) percent confidence interval for the parameter, then the test of significance level  of the hypothesis will lead to rejection of H 0 if and only if the hypothesized value is not in the 100(1 -  ) percent confidence interval [l, u].

48. 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

49. 4-4 Inference on the Mean of a Population, Variance Known 4-4.5 Confidence Interval on the Mean Choice of Sample Size

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

55. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

56. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

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

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 Fixed Significance Level Approach

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

61. 4-5 Inference on the Mean of a Population, Variance Unknown Fig 4-20 Normal probability plot of the coefficient of restitution data from Example 4-7.

62. 4-5 Inference on the Mean of a Population, Variance Unknown OPTIONS NODATE NONUMBER; DATA ex47; xbar=0.83725; sd=0.02456; 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 */ cr=tinv(1-alpha, df); /* critical region */ /* TINV function returns a quantile from the t distribution */ PROC PRINT; var t0 pval cr; title 'Test of mu=0.82 vs mu>0.82'; title2 'Ex. 4-7 in pp193'; RUN; QUIT; Test of mu=0.82 vs mu>0.82 Ex. 4-7 in pp193 OBS t0 pval cr 1 2.72023.008292926 1.76131

63. 4-5 Inference on the Mean of a Population, Variance Unknown OPTIONS NOOVP NODATE NONUMBER; DATA ex47; input coeffs @@; coeffs1=coeffs-0.82; cards; 0.8411 0.8191 0.8182 0.8125 0.8750 0.8580 0.8532 0.8483 0.8276 0.7983 0.8042 0.8730 0.8282 0.8359 0.8660 proc univariate data=ex47 plot normal; var coeffs1; title 'using proc uinivariate for t-test for one population mean'; title2 “Example 4-7 in page 191”; RUN; QUIT;

64. 4-5 Inference on the Mean of a Population, Variance Unknown using proc uinivariate for t-test for one population mean Example 4-7 in page 191 UNIVARIATE 프로시저 변수 : coeffs1 적률 N 15 가중합 15 평균 0.01724 관측치 합 0.2586 표준 편차 0.0245571 분산 0.00060305 왜도 0.07243075 첨도 -1.128646 제곱합 0.01290098 수정 제곱합 0.00844272 변동계수 142.442573 평균의 표준 오차 0.00634062 기본 통계 측도 위치측도 변이측도 평균 0.837240 표준 편차 0.02456 중위수 0.835900 분산 0.0006031 최빈값. 범위 0.07670 사분위 범위 0.03980 위치모수 검정 : Mu0=0 검정 -- 통계량 --- -------p 값 ------- 스튜던트의 t t 2.718979 Pr > |t| 0.0166 부호 M 2.5 Pr >= |M| 0.3018 부호 순위 S 39 Pr >= |S| 0.0256 정규성 검정 검정 ---- 통계량 ---- -------p 값 ------- Shapiro-Wilk W 0.960869 Pr < W 0.7075 Kolmogorov-Smirnov D 0.110275 Pr > D >0.1500 Cramer-von Mises W-Sq 0.026454 Pr > W-Sq >0.2500 Anderson-Darling A-Sq 0.193435 Pr > A-Sq >0.2500

65. 4-5 Inference on the Mean of a Population, Variance Unknown using proc uinivariate for t-test for one population mean Example 4-7 in page 191 UNIVARIATE 프로시저 변수 : coeffs1 극 관측치 ------ 최소 ------ ------ 최대 ------ 값 관측치 값 관측치 -0.0217 10 0.0332 7 -0.0158 11 0.0380 6 -0.0075 4 0.0460 15 -0.0018 3 0.0530 12 -0.0009 2 0.0550 5 줄기 잎 # 상자그림 5 35 2 | 4 6 1 | 3 38 2 +-----+ 2 18 2 | | 1 6 1 *--+--* 0 88 2 | | -0 821 3 +-----+ -1 6 1 | -2 2 1 | ----+----+----+----+ 값 : ( 줄기. 잎 )*10**-2 정규 확률도 0.055+ * +++* | *++++ | *+*++ | **++ 0.015+ ++*+ | ++++** | ++*+* * | +++* -0.025+ ++*+ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2

66. 4-5 Inference on the Mean of a Population, Variance Unknown OPTIONS NOOVP NODATE NONUMBER LS=80; DATA ex47; input coeffs @@; cards; 0.8411 0.8191 0.8182 0.8125 0.8750 0.8580 0.8532 0.8483 0.8276 0.7983 0.8042 0.8730 0.8282 0.8359 0.8660 proc ttest data=ex47 h0=0.82 sides=u; /* h0 는 귀무가설, sides 는 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; QUIT; using t-test for one population mean The TTEST Procedure Variable: coeffs N Mean Std Dev Std Err Minimum Maximum 15 0.8372 0.0246 0.00634 0.7983 0.8750 Mean 95% CL Mean Std Dev 95% CL Std Dev 0.8372 0.8261 Infty 0.0246 0.0180 0.0387 DF t Value Pr > t 14 2.72 0.0083

67. 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  for various sample sizes n.

68. 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 d efi ned as

69. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.2 Type II Error and Choice of Sample Size Chart V (c) in pp. 496

70. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.3 Confidence Interval on the Mean

71. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.3 Confidence Interval on the Mean

72. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.4 Confidence Interval on the Mean

73. 4-5 Inference on the Mean of a Population, Variance Unknown OPTIONS NOOVP NODATE NONUMBER LS=80; DATA ex60; input diam @@; diam1=diam-8.2; cards; 8.23 8.31 8.42 8.29 8.19 8.24 8.19 8.29 8.30 8.14 8.32 8.40 proc univariate plot normal; var diam; title 'univariate for t-test with H0: mu=8.2'; proc univariate; var diam1; title 'univariate for t-test with H0: mu=0'; proc ttest data=ex60 h0=8.2; var diam; title 't-test for one population mean'; title “Example 4-60 in page 198”; RUN; QUIT;

74. 4-5 Inference on the Mean of a Population, Variance Unknown univariate for t-test with H0: mu=8.2 UNIVARIATE 프로시저 변수 : diam 적률 N 12 가중합 12 평균 8.27666667 관측치 합 99.32 표준 편차 0.08359353 분산 0.00698788 왜도 0.17762405 첨도 -0.3971019 제곱합 822.1154 수정 제곱합 0.07686667 변동계수 1.00999033 평균의 표준 오차 0.02413137 기본 통계 측도 위치측도 변이측도 평균 8.276667 표준 편차 0.08359 중위수 8.290000 분산 0.00699 최빈값 8.190000 범위 0.28000 사분위 범위 0.10500 Note: 표시된 최빈값은 2 개의 최빈값 ( 개수 : 2) 중에 가장 작습니다. univariate for t-test with H0: mu=8.2 UNIVARIATE 프로시저 변수 : diam 적률 위치모수 검정 : Mu0=0 검정 -- 통계량 --- -------p 값 ------- 스튜던트의 t t 342.9836 Pr > |t| <.0001 부호 M 6 Pr >= |M| 0.0005 부호 순위 S 39 Pr >= |S| 0.0005 정규성 검정 검정 ---- 통계량 ---- -------p 값 ------- Shapiro-Wilk W 0.961555 Pr < W 0.8058 Kolmogorov-Smirnov D 0.146697 Pr > D >0.1500 Cramer-von Mises W-Sq 0.03945 Pr > W-Sq >0.2500 Anderson-Darling A-Sq 0.249801 Pr > A-Sq >0.2500

75. 4-5 Inference on the Mean of a Population, Variance Unknown univariate for t-test with H0: mu=8.2 UNIVARIATE 프로시저 변수 : diam 극 관측치 ----- 최소 ----- ----- 최대 ----- 값 관측치 값 관측치 8.14 10 8.30 9 8.19 7 8.31 2 8.19 5 8.32 11 8.23 1 8.40 12 8.24 6 8.42 3 줄기 잎 # 상자그림 84 02 2 | 83 | 83 012 3 +-----+ 82 99 2 *--+--* 82 34 2 +-----+ 81 99 2 | 81 4 1 | ----+----+----+----+ 값 : ( 줄기. 잎 )*10**-1 정규 확률도 8.425+ * +*++++ | ++++++ | *+*++* 8.275+ ++*+*+ | ++++*+* 8.125+ +++++* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2

76. 4-5 Inference on the Mean of a Population, Variance Unknown univariate for t-test with H0: mu=0 UNIVARIATE 프로시저 변수 : diam1 적률 N 12 가중합 12 평균 0.07666667 관측치 합 0.92 표준 편차 0.08359353 분산 0.00698788 왜도 0.17762405 첨도 -0.3971019 제곱합 0.1474 수정 제곱합 0.07686667 변동계수 109.035043 평균의 표준 오차 0.02413137 기본 통계 측도 위치측도 변이측도 평균 0.07667 표준 편차 0.08359 중위수 0.09000 분산 0.00699 최빈값 -0.01000 범위 0.28000 사분위 범위 0.10500 Note: 표시된 최빈값은 2 개의 최빈값 ( 개수 : 2) 중에 가장 작습니다. 위치모수 검정 : Mu0=0 검정 -- 통계량 --- -------p 값 ------- 스튜던트의 t t 3.177053 Pr > |t| 0.0088 부호 M 3 Pr >= |M| 0.1460 부호 순위 S 31 Pr >= |S| 0.0122

77. 4-5 Inference on the Mean of a Population, Variance Unknown “Example 4-60 in page 198” The TTEST Procedure Variable: diam N Mean Std Dev Std Err Minimum Maximum 12 8.2767 0.0836 0.0241 8.1400 8.4200 Mean 95% CL Mean Std Dev 95% CL Std Dev 8.2767 8.2236 8.3298 0.0836 0.0592 0.1419 DF t Value Pr > |t| 11 3.18 0.0088

78. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

79. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

80. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

81. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

82. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

83. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

84. 4-6 Inference on the Variance of a Normal Population 4-6.2 Confidence Interval on the Variance of a Normal Population

85. 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. 4-10 in pp202'; RUN; QUIT; Test of H0: var=0.01 vs H0: var>0.01 Ex. 4-10 in pp202 OBS chisq p_value cv1 cv2 lcl 1 29.07 0.064892 30.1435 10.1170 0.098203

86. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion We will consider testing:

87. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion

88. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion

89. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion

90. 4-7 Inference on Population Proportion 4-7.2 Type II Error and Choice of Sample Size

91. 4-7 Inference on Population Proportion 4-7.2 Type II Error and Choice of Sample Size

92. 4-7 Inference on Population Proportion 4-7.3 Confidence Interval on a Binomial Proportion

93. 4-7 Inference on Population Proportion 4-7.3 Confidence Interval on a Binomial Proportion

94. 4-7 Inference on Population Proportion 4-7.3 Confidence Interval on a Binomial Proportion Choice of Sample Size

95. 4-8 Other Interval Estimates for a Single Sample 4-8.1 Prediction Interval

96. 4-8 Other Interval Estimates for a Single Sample 4-8.2 Tolerance Intervals for a Normal Distribution

97. 4-10 Testing for Goodness of Fit So far, we have assumed the population or probability distribution for a particular problem is known. There are many instances where the underlying distribution is not known, and we wish to test a particular distribution. Use a goodness-of-fit test procedure based on the chi- square distribution.