Case 3 r 284 introduction to the science of

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Introduction to the Science of Statistics Composite Hypotheses Repeat the argument in Case 2 to conclude that P { X ( n ) ˜ } = ˜ ! n and that P { X ( n ) R } = 1 - P { X ( n ) < R } = 1 - R n and therefore ( ) = ( ˜ / ) n + 1 - ( R / ) n . The size of the test is the maximum value of the power function under the null hypothesis. This is case 2. Here, the power function ( ) = ˜ ! n decreases as a function of . Thus, its maximum value takes place at L and = ( L ) = ˜ L ! n To achieve this level, we solve for ˜ , obtaining ˜ = L n p . Note that ˜ increases with . Consequently, we must expand the critical region in order to reduce the significance level. Also, ˜ increases with n and we can reduce the critical region while maintaining significance if we increase the sample size. 18.3 The p -value The report of reject the null hypothesis does not describe the strength of the evidence because it fails to give us the sense of whether or not a small change in the values in the data could have resulted in a different decision. Consequently, one common method is not to choose, in advance, a significance level of the test and then report “reject” or “fail to reject”, but rather to report the value of the test statistic and to give all the values for that would lead to the rejection of H 0 . The p -value is the probability of obtaining a result at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. In this way, we provide an assessment of the strength of evidence against H 0 . Consequently, a very low p -value indicates strong evidence against the null hypothesis. Example 18.8. For the one-sided hypothesis test to see if the mimic had invaded, H 0 : μ μ 0 versus H 1 : μ < μ 0 . with μ 0 = 10 cm, σ 0 = 3 cm and n = 16 observations. The test statistics is the sample mean ¯ x and the critical region is C = { x ; ¯ x k } Our data had sample mean ¯ x = 8 . 93125 cm. The maximum value of the power function ( μ ) for μ in the subset of the parameter space determined by the null hypothesis occurs for μ = μ 0 . Consequently, the p -value is P μ 0 { ¯ X 8 . 93125 } . With the parameter value μ 0 = 10 cm, ¯ X has mean 10 cm and standard deviation 3 / p 16 = 3 / 4 . We can compute the p -value using R . > pnorm(8.93125,10,3/4) [1] 0.0770786 285
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Introduction to the Science of Statistics Composite Hypotheses 7 8 9 10 11 12 13 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x dnorm(x, 10, 3/4) Figure 18.6: Under the null hypothesis, ¯ X has a normal distribution mean μ 0 = 10 cm, standard deviation 3 / p 16 = 3 / 4 cm. The p -value, 0.077, is the area under the density curve to the left of the observed value of 8.931 for ¯ x , The critical value, 8.767, for an = 0 . 05 level test is indicated by the red line. Because the p -vlaue is greater than the significance level, we cannot reject H 0 .
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  • Spring '17
  • KASMIS MISGANEW
  • Null hypothesis, Statistical hypothesis testing, Type I and type II errors, power function, Science of Statistics

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