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

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

4-5.1 Hypothesis Testing on the Mean







Calculating the P-value








Ex. 4-7 – mean coefficient of restitution n = 15

i) HypothesisH0: μ = 0.82 H1: μ > 0.82

ii) Test Statistict = 2.72

iii) Fixed significance level approachCritical Region (CR) 또는 Region of Rejection (ROR): t0 > t0.05= 1.761P-value approacht0= 2.72 는 t0.01;14= 2.624 값과 t0.005;14= 2.977 사이에 있으므로 0.005<p-value<0.01

iv) ConclusionReject H0 . In other words, the mean coefficient of restitution exceeds 0.82.

Fig 4-20 Normal probability plot of the coefficient of restitution data from Example 4-7.


OPTIONS NOOVP NODATE NONUMBER;DATA ex47;input coeffs @@;cards;

0.8411 0.8191 0.8182 0.8125 0.87500.8580 0.8532 0.8483 0.8276 0.79830.8042 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;


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


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


OPTIONS NOOVP NODATE NONUMBER LS=80;DATA ex47;input coeffs @@;cards;

0.8411 0.8191 0.8182 0.8125 0.87500.8580 0.8532 0.8483 0.8276 0.79830.8042 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


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.



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


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)


4-5.3 Confidence Interval on the Mean





4-6 Inference on the Variance of a Normal Population

4-6.1 Hypothesis Testing on the Variance of a Normal Population












4-6.2 Confidence Interval on the Variance of a Normal Population

4-7 Inference on Population Proportion

4-7.1 Hypothesis Testing on a Binomial Proportion

We will consider testing:












4-7.3 Confidence Interval on a Binomial Proportion


4-7.3 Confidence Interval on a Binomial Proportion


4-7.3 Confidence Interval on a Binomial ProportionChoice of Sample Size

4-8 Other Interval Estimates for a Single Sample

4-8.1 Prediction Interval

4-8 Other Interval Estimates for a Single Sample

4-8.2 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.

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