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Unformatted text preview: AMS 312.01 Practice Final Exam #2 Spring, 2010 Name_____________________________ID_______________Signature_____________ Instructions: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. Good luck! 1. In a study of hypnotic suggestion, 5 male volunteers participated in a two-phase experimental session. In the first phase, respiration was measured while the subject was awake and at rest. In the second phase, the subject was told to imagine that he was performing muscular work, and respiration was measured again. Hypnosis was induced between the first and second phases; thus, the suggestion to imagine muscular work was “hypnotic suggestion” for these subjects. The accompanying table shows the measurements of total ventilation (liters of air per minute per square meter of body area) for all 5 subjects. Experimental Group Subject Rest Work 1 6 6 2 7 9 3 8 9 4 7 10 5 6 7 Use suitable test to investigate whether there is any difference between the two experimental phases in terms of total ventilation. Please state the assumption(s) of the test and report the p-value. At the significance level of 0.05, what is your conclusion? Solution: Assume that the difference work rest d =- is normal. 1.4, 1.14 d d s = = and 5 n = The hypotheses are : d H μ = v.s : a d H μ ≠ . The test statistic is 1.4 2.746. / 1.14/ 5 d d t s n- = = = Since 4,0.025 2.776 t = and 4,0.025 2.776 t t < = , we can not reject H at 0.05 α = . 4,0.05 4,0.025 2.132 2.746 2.776 0.05 0.1. t t p value = < < = ⇒ <- < 2. Let X 1 , X 2 , …, X n be a random sample from a normal population N(μ, σ 2 ). Furthermore, the population variance σ 2 is unknown. For a 2-sided test of H : μ = μ 0 versus Ha: μ ≠ μ , at the significance level α, (a). Derive the one-sample t-test using the pivotal quantity method. (* Please include the derivation of the pivotal quantity, the proof of its distribution, and the derivation of the rejection region for full credit.) (b). Prove that the likelihood ratio test is equivalent to the usual one sample t-test....
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- Spring '08
- Normal Distribution, Maximum likelihood, Likelihood function, Likelihood-ratio test