1. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

2. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

3. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

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

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

6. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

7. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.1 Hypothesis Testing on the Mean

8. 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.

9. 4-5 Inference on the Mean of a Population, Variance Unknown OPTIONS NOOVP NODATE NONUMBER; 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 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;

10. 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 : coeffs 적률 N 15 가중합 15 평균 0.83724 관측치 합 12.5586 표준 편차 0.0245571 분산 0.00060305 왜도 0.07243075 첨도 -1.128646 제곱합 10.523005 수정 제곱합 0.00844272 변동계수 2.93310158 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=0.82 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 default p-value for two- tailed test in SAS

11. 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 : coeffs 극 관측치 ------ 최소 ------ ------ 최대 ------ 값 관측치 값 관측치 0.7983 10 0.8532 7 0.8042 11 0.8580 6 0.8125 4 0.8660 15 0.8182 3 0.8730 12 0.8191 2 0.8750 5 줄기 잎 # 상자그림 87 35 2 | 86 6 1 | 85 38 2 +-----+ 84 18 2 | | 83 6 1 *--+--* 82 88 2 | | 81 289 3 +-----+ 80 4 1 | 79 8 1 | ----+----+----+----+ 값 : ( 줄기. 잎 )*10**-2 정규 확률도 0.875+ * +++* | *++++ | *+*++ | **++ 0.835+ ++*+ | ++++** | ++*+* * | +++* 0.795+ ++*+ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2

12. 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

13. 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.

14. 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

15. 4-5 Inference on the Mean of a Population, Variance Unknown Standardized Difference, d Figure 51. OC Curves for a Two—Sided t—Test (α = 0.05 ) (Source: Experimental Statistics, by M. G. Natrella, National Bureau of Standards Handbook 91, 1963)

16. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.3 Confidence Interval on the Mean

17. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.3 Confidence Interval on the Mean

18. 4-5 Inference on the Mean of a Population, Variance Unknown 4-5.4 Confidence Interval on the Mean

19. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

20. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

21. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

22. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

23. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

24. 4-6 Inference on the Variance of a Normal Population 4-6.1 Hypothesis Testing on the Variance of a Normal Population

25. 4-6 Inference on the Variance of a Normal Population 4-6.2 Confidence Interval on the Variance of a Normal Population

26. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion We will consider testing:

27. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion

28. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion

29. 4-7 Inference on Population Proportion 4-7.1 Hypothesis Testing on a Binomial Proportion

30. 4-7 Inference on Population Proportion 4-7.2 Type II Error and Choice of Sample Size

31. 4-7 Inference on Population Proportion 4-7.2 Type II Error and Choice of Sample Size

32. 4-7 Inference on Population Proportion 4-7.3 Confidence Interval on a Binomial Proportion

33. 4-7 Inference on Population Proportion 4-7.3 Confidence Interval on a Binomial Proportion

34. 4-7 Inference on Population Proportion 4-7.3 Confidence Interval on a Binomial Proportion Choice of Sample Size

35. 4-8 Other Interval Estimates for a Single Sample 4-8.1 Prediction Interval

36. 4-8 Other Interval Estimates for a Single Sample 4-8.2 Tolerance Intervals for a Normal Distribution

37. 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.