ch17thet-coefficientandthet-test.studentview

# ch17thet-coefficientandthet-test.studentview - Chapter17...

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Chapter 17:  The  t -Coefficient and the  t -Test A. Introduction In the previous two chapters, we studied inferences on  μ when (i) the population is normal and (ii) the standard deviation σis known. However, the assumption that σis known is NOT practical as we do not even know the μ. That we assumed σis known was mainly for the preliminary discussions in the earlier chapters, which acted as a stepping stone for a more in-depth study in this and the following sections. Note : We also assume ( and ALWAYS assume) that are random and independent. n x x x , , ......... , 2 1

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Question What shall we do if  σis really UNKNOWN? Answer It turns out that two slight modifications will lead to  adequate statistical inference procedures for  μ. These will be discussed below. First modification : Replace the unknown σ by its point estimate: This estimate was discussed in Chapter 14(C). 1 . . , 1 ) ( 2 - = - - = n f d n x x s i
Second modification Replace the  z -coefficient or  z -critical value by a  so-called  t -coefficient  or  t-critical value , taken  from a  t-table . (Table7). The  t-distribution  was invented in 1908 by an English  statistician named  William Sealy Gosset  (1876-1937). He was a researcher at Guinness Brewery. He invented the student t-distribution to handle small samples  in quality control in brewing.  The company forbade the publication of research by its  employees. So, he published his famous results in the pen- name “Student”  (meaning, that he was a humble student of  Karl Pearson  and  Ronald Alymer Fisher  who published articles in  Biometrika  -a scientific journal that principally covers  theoretical   statistics) Thus, the  t- distribution is also called the  Student t- distribution.

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Gosset                                                Fisher Pearson
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