# ssrs 2012 barcode font In Problem 11.65, test the null hypothesis H0:l Bayes 0.05 test. in Software Drawing QR-Code in Software In Problem 11.65, test the null hypothesis H0:l Bayes 0.05 test.

11.73. In Problem 11.65, test the null hypothesis H0:l Bayes 0.05 test.
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1 against the alternative hypothesis H1:l
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CHAPTER 11 Bayesian Methods
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The Bayes 0.05 test would reject the null hypothesis if the posterior probability of the hypothesis l 1 is less than 0.05. In our case, this probability is given by the area to the left of 1 under a gamma distribution with parameters 20.5 and 0.1 and is 0.002. Since this is less than 0.05, we reject the null hypothesis.
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11.74. In Problem 11.6, assume that n 40 and x 10 and test the null hypothesis H0:u alternative H1:u 0.2 using a Bayes 0.05 test.
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0.2 against the
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(a) The posterior probability of the null hypothesis is given by the area from 0 to 0.2 under a beta density with parameters 12 and 31, which is determined to be 0.12 using computer software. Since this is not less than 0.05, we cannot reject the null hypothesis. (b) The posterior probability is the area from 0 to 0.2 under a beta density with parameters 13 and 31, which is 0.07. Since this is not less than 0.05, we cannot reject the null hypothesis. (c) The posterior probability is the area from 0 to 0.2 under a beta density with parameters 14 and 31, which is 0.04. Since this is less than 0.05, we reject the null hypothesis.
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11.75. In Problem 11.48, test the null hypothesis H0:u
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0.7 against H1:u
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0.7 using a Bayes 0.025 test.
The posterior distribution of u is gamma with parameters 10.16 and 0.04. Therefore, the posterior probability of the null hypothesis is 0.022. Since this is less than 0.025, we reject the null hypothesis.
11.76. The life-length X of a computer component has the exponential density given by (see page 118) f (x uu) ue ux, x 0 with unknown mean 1>u. Suppose that the prior density of u is gamma with parameters a 0.2 and b 0.15. If a random sample of 10 observations on X yielded an average lifelength of 7 years, use a Bayes 0.05 test to test the null hypothesis that the expected life-length is at least 12 years against the alternative hypothesis that it is under 12 years.
The null and alternative hypothesis are respectively equivalent to H0 : u 1>12 0.083 and H1 : u 0.083. From Theorem 11-4, the posterior distribution of u is gamma with parameters 10.2 and 0.013. The posterior probability of the null hypothesis is 0.10. Since this is larger than 0.05, we cannot reject the null hypothesis.
Bayes factor 11.77. In Example 11.4, find the Bayes factor of H0 : l
BF 5P(H0 u x)>[1 P(H0 u x)]6 5P(H0)>[1
1 relative to H1 : l 2 1.
P(H0)]6 (0.49>0.51) ((1>3)>(2>3)) < 1.92
11.78. It is desired to test the null hypothesis u 0.6 against the alternative u 0.6, where u is the probability of success for a Bernoulli trial. Assume that u has a uniform prior distribution on [0, 1] and that in 40 trials there were 24 successes. What is your conclusion if you decide to reject the null hypothesis if BF 1
The posterior density of u is beta with a 25 and b 17. The posterior probability of the null hypothesis is 0.52. Posterior odds ratio is 0.52>0.48 1.0833 and prior odds ratio is 6>4 1.5. BF 0.72. We reject the null hypothesis.
11.79. Prove that the ad hoc rule (see the Remark following Theorem 11-10) to reject H0 if BF lent to the Bayes a test with a P(H0).
BF 13 P(H0 u x) P(H0) ^ P(H1) P(H1 u x) 1 3 P(H0 u x)[1 P(H0)] [1
1 is equiva-
P(H0 ux)]P(H0) 3 P(H0 u x)
P(H0)
11.80. In the preceding problem, find c such that the Bayes factor criterion to reject the null hypothesis if BF is equivalent to the Bayes 0.05 rule.
By Theorem 11-10, c a[1 P(H0)] (1 a)P(H0) (0.05)(1 0.6) < 0.035. (1 0.05)(0.6)
11.81. Work Problem 11.71 using the decision to reject the null hypothesis if the Bayes factor is less than 1. We know from Problem 11.79 that the rule to reject H0 if BF 1 is equivalent to rejecting the null hypothesis if P(H0 ux) P(H0). We know from Problem 11.71 that P(H0 ux) 0.88. From Example 11.18,