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

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

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

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

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

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

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

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.

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;

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 관측치 합 표준 편차 분산 왜도 첨도 제곱합 수정 제곱합 변동계수 평균의 표준 오차 기본 통계 측도 위치측도 변이측도 평균 표준 편차 중위수 분산 최빈값. 범위 사분위 범위 위치모수 검정 : Mu0=0.82 검정 -- 통계량 p 값 스튜던트의 t t Pr > |t| 부호 M 2.5 Pr >= |M| 부호 순위 S 39 Pr >= |S| 정규성 검정 검정 ---- 통계량 p 값 Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq > Anderson-Darling A-Sq Pr > A-Sq > default p-value for two- tailed test in SAS

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 줄기 잎 # 상자그림 | | | | *--+--* | | | | 값 : ( 줄기. 잎 )*10**-2 정규 확률도 * +++* | *++++ | *+*++ | ** *+ | ++++** | ++*+* * | +++* *

4-5 Inference on the Mean of a Population, Variance Unknown OPTIONS NOOVP NODATE NONUMBER LS=80; DATA ex47; input coeffs cards; 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 Mean 95% CL Mean Std Dev 95% CL Std Dev Infty DF t Value Pr > t

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

4-7 Inference on Population Proportion Hypothesis Testing on a Binomial Proportion

4-7 Inference on Population Proportion Hypothesis Testing on a Binomial Proportion

4-7 Inference on Population Proportion Hypothesis Testing on a Binomial Proportion

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

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

4-7 Inference on Population Proportion Confidence Interval on a Binomial Proportion

4-7 Inference on Population Proportion Confidence Interval on a Binomial Proportion

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

4-8 Other Interval Estimates for a Single Sample Prediction Interval

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

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.