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Summary Of Test Statistis For brevity, we onsider only tests involving the mean(s) of normal random variables, assuming we are always dealing with simple random samples. 1 Testing for the mean of a population In general, the variane will be unknown, hene we use the following statisti √ x−µ n s where and the µ x is the sample mean, s is the sample standard deviation, n is the ount, E = √sn , is the true mean, as speied in the null hypothesis. Dening standard error, this an also be written as x−µ E This value (the t-sore ) is (under the Null Hypothesis) distributed aording It will be heked against the ritial to a (Student) tn−1 distribution. value(s) as determined by the signiane level tailed or two-tailed (one tail: 1.1 tn−1,α , two tails: α and whether tn−1,α/2 ). the test is one- Testing two paired samples Two paired samples are handled as before, with the adjustment that the data we will be alulating with are the 2 dierenes between the paired data. Testing Proportions The null hypothesis speies an assumed true proportion p, implying a true p(1−p) . Hene the statisti (assuming the sample is large enough to n justify a normal approximation) is variane of pb − p √ p n p(1 − p) (p b is, as usual, the sample mean, i.e., the proportion of suesses in the sample) whih is (essentially) distributed as a standard normal. Hene its value will be ompared with the ritial value(s) of he standard normal distribution as α, zα/2 ) determined by the signiane level two-tailed (one tail: zα , two tails: 1 and whether the test is one-tailed or 3 Testing two independent samples 3 2 Testing two independent samples Again, the varianes are generally unknown. 3.1 Varianes unknown but assumed to be equal The test statisti, E= (x1 and anes ) x2 s x1 − x2 − (µ1 − µ2 ) E (n1 − 1) s21 + (n2 − 1) s22 n1 + n2 − 2 are the two sample means, and s21 r and 1 1 + n1 n2 s22 are the two sample vari- is, under our usual assumptions, and under the null hypothesis that µ1 − µ2 is the dierene of the true means, distributed aording to a tn1 +n2 −2 Student distribution. From this point on we proeed as in setion 1. 3.2 Varianes unknown and assumed to be unequal This is an approximate test, as the test statisti x1 − x2 − (µ1 − µ2 ) E with E= s s2 s21 + 2 n1 n2 has a distribution unt for our usual methods, sine its spei form atually depends on the (unknown) values of the varianes. However, it has been heked that treating it as if it was distributed aording to a tmin(n1 −1,n2 −1) Student distribution produes aeptable results. A more rened hoie for the number of degrees of freedom (but the test is still only approximate, as the exat distribution is not of STudent type) is usually implemented in software, but is impratial for table-based work: 1 n1 −1 s22 s21 n1 + n2 2 2 s 1 n1 + 2 1 n2 −1 s22 n2 2 Again, with these hoies we an now proeed as in setion 1.