Chapter 13: Tests of Hypothesis Section 13.1 Introduction

Transcription

1 Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed i Sectio 1.6. Example: Sample of size from a ormal populatio N (, ). H H : agaist :. The LR test leads to the critical regio X c, for some costat c. / X U der : (, / ),. / H X N P c P Z c X If P Type I Error, the Rejectio regio: z. / / i.e., R eject H : =, if X z, or X z, / / i. e., ( X z, X z ), the1(1- )% C I for. / / The alterative hypothesis H :, or equivaletly, or Two-sided alterative: True mea could be less tha (or more tha) the mea uder the ull. Therefore, the Critical Regio is two-tailed. Reject the ull if the sample mea is away, from the populatio mea uder the ull hypothesis, i either directio!

2 How far away is too far away: determied by the desired P T ype I Error What if the alterative hypothesis is oe-sided? H [Also called, size of the test procedure.] :, or H : The, as discussed i Sec. 1.5, the MP critical regio is oe tailed: Reject H : i favor of H : ; if X z, or Reject H : i favor of H : ; if X z, Thus the test procedure rejects the Null i favor of the alterative, if the Sample mea follows oe s expectatio uder the Alterative. Four Steps i Traditioal Testig of Hypotheses set up: (i) Formulate H ad H 1, ad specify the value of. (ii) Defie the appropriate Test Statistic ad its samplig distributio Determie the appropriate Critical Regio of Size. (iii) Collect the sample data ad calculate the value of the test statistic (iv) Check if this value falls i the critical regio, ad accordigly, Reject the ull, do ot reject the ull (accept), or reserve judgmet. I the above two sided or oe sides tests for the Normal meas, the test statistic X follows Normal distributio with mea ad variace ( / ). Uder the Null hypothesis, its mea is. After stadardizig X to Z, the critical values (cut-off poits for the critical regio) require the values of Z /, or Z respectively, from the Normal cdf Tables.

3 I the past, the Tables for area uder various samplig distributios could be calculated umerically for just a few values. So values of, like.1(oe i 1),.5 (oe i twety), ad.1 (1 i 1) became the de-facto stadard. But these values are ot ecessarily omipotet. I real applicatios, oe may wat to choose ay value of or i practice, depedig o the risk to be covered or relative cosequeces of the two types of errors. Fortuately, usig umerical aalysis, algorithms, ad high speed computig devices, oe ca ow calculate (or simulate) the area uder ay samplig distributio i ay specified regio. The curret practice ivolves computig the area uder the curve beyod the observed values of the test statistic. I the case of test for Normal mea, we calculate the area i the ( x ) tail of the stadard ormal curve. Let z deote the observed value of the stadardized test statistic Z. The area uder the appropriate tail beyod the observed value (shaded regio) is called the P-value, prob-value, tail probability, or the observed level of sigificace. This is simply ( i) P ( X x ) for H : > ; ( ii) P ( X x ) for H : < ; ( iii) P ( X x ) for H :. Check if the P-value is less tha or equal to the stated level of sigificace ad accordigly reject the ull hypothesis, accept it or reserve the judgmet.

4 I exploratory data aalysis, the p-value is also called the stregth of evidece agaist the Null Hypothesis. Sectio 13. Tests cocerig Populatio Meas If the populatio ca be assumed to be Normal, as discussed i Sectio 13.1, the test for the mea is based o the Z-statistic. If the populatio is ot ormal, but has a fiite variace, ad the sample size is large, so that the distributio of X ca be approximated by Normal distributio (usig CLT), we ca use the test based o Z-statistics. For < 3 ad ukow, usig the result of Exercise 1., the Likelihood Ratio Test for samples from ormal populatio, the x oe-sample t-test based o t, with -1degrees of s / freedom is used. It is oe-tailed or two-tailed test depedig o whether the alterative hypothesis is oe-sided or two-sided. For, >3, the t-test uses critical values from stadard ormal distributio. Sectio 13.3 Tests cocerig differece of two meas Examples: New medicie is as good as old, Two brad of tires have same mea tread life, average life times of two brads of bulbs differ by 1 hours, etc. Idepedet radom samples of size ad from two ormal populatios with meas, ad, ad kow variaces ad,

5 ad wat to test the ull hypothesis H : agaist the alteratives H : ; or ; or. 1 Usig the Likelihood Ratio test, ad the result from Ex. 8., implies that the respective critical regios ca be described as z z ; or z z ; or z z, w here / z x x If the idepedet samples are ot from ormal populatios, but both sample sizes are large eough to use CLT, oe ca use the above test. I this case if the variaces are ot kow, oe ca substitute the sample variaces s, s for, respectively. Small sample sizes ad ukow variaces: For idepedet samples from ormal populatios, eed to assume that the two populatios have same ukow variace. The Likelihood Ratio test yields a test based the pooled estimate of variace, give by ( 1) s ( 1) s 1 p s. ( ) The two-sample t-test is based o the t-statistic t x x. 1 1 s p This expressio is the value of a radom variable havig the t- distributio with degrees of freedom. Hece the critical regios for H : ; or ; or, are give 1 by t t t t /, ; ; ; or t ; or t respectively. Whe, the pooled variace simplifies to The t-statistic simplifies to s s s p t ( x x ) / ( s s ) /.

6 If the assumptio of equal variace is ot reasoable, there are other solutios available i the literature. But we will ot cosider them here. Not-Idepedet samples. (For example correlated variables from same populatio. o Paired data (before-after, sibligs, twis, etc.) o Use the mea d of the pair-wise differece, d x x ad i 1, i, i test the ull hypothesis for the mea of these differeces: H : agaist H : ; or ; or. d 1 d d d o For small, but samples from ormal populatio, we ca use the oe sample t-test based o t d s / with v 1, with the critical regios give by t t ; or t t ; or t t respectively. /, ; ; d