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Statistical Inference 1. In the context of MAT 282, provide brief answers to each of the following questions: a. Are searcher who tried to learn statistics without taking a formal course does a hypothesis test and gets a p-value of 0.024. He says, “There is a 97.6% chance that the alternative hypothesis is false, so the null hypothesis is true.” What, if anything, is wrong with his/her statement? b. You perform a hypothesis test using a sample size of four units, and you do not reject the null hypothesis. Your research colleague says this statistical test provides conclusive evidence against the hypothesis. Do you agree or disagree with his conclusion? 1 2. Decide whether each of the following statements is either true (T) or false (F) . Clearly circle your choice. You do not need to justify your answers. a. T or F: The significance level of a hypothesis test is determined by the null distribution of the test statistic. b. TorF:Toreducethewidthofaconfidenceintervalbyafactoroftwo(i.e.,inhalf),you have to quadruple the sample size. c. T or F: A p-value of 0.08 is more evidence against the null hypothesis than a p-value of 0.04. d. TorF:Iftwoindependentstudiesaredoneonthesamepopulationwiththepurposeof testing the same hypotheses, the study with the larger sample size is more likely to have a smaller p-value than the study with the smaller sample size. e. T or F: The statement “the p-value is 0.003” is equivalent to the statement “there is a 0.3% probability that the null hypothesis is true.” f. T or F: The significance level of a hypothesis test is equal to the probability of a type I error . g. T or F: If a test is rejected at significance level α , then the probability that the null hypothesis is true equals α . h. TorF:Theprobabilitythatthenullhypothesisisfalselyrejectedisequaltothepower of the test. 2 i. T or F : A type II error occurs when the test statistic falls in the rejection region of the test. j. T or F: The power of a test is determined by the null distribution of the test statistic. k. T or F: If the p-value is0.03, then the corresponding test will reject at the significance level 0.02. l. T or F: If a test rejects at significance level 0.06, then the p-value is less than or equal to 0.06. m. T or F: The p-value of a test is the probability that the null hypothesis is correct. 3 3. Assume that the outcome of an experiment is a single random variable Y , and that Y will be used as an estimator of an unknown parameter θ . The rejection region of the significance level α = 0.10 hypothesis test is H0:θ=3 versus Ha:θ≠3 RR={Y>7 or Y<2} Based on this information, construct a 90% confidence interval for θ . (Hint: Use the confidence interval–hypothesis test duality.) 4 4. When designing a hypothesis test, one of the first decisions that needs to be made is which hypothesis to assign to H0 and which hypothesis to assign to Ha . One criterion that we discussed in class was to assign the hypothesis to H0 for which the consequences of a type I error are worse than the consequences of a type II error . Consider the following hypothetical scenario. Suppose that Michael is convicted of a crime. If the judge determines that Michael is guilty of the crime, then Michael will be sent to prison. In this scenario, there are two possible hypothesis tests that can be considered. Hypothesis Test A : H0: Michael is guilty versus Ha : Michael is not guilty Hypothesis Test B : H0: Michael is not guilty versus Ha: Michael is guilty In your opinion, for which of these two hypothesis tests (Test A or Test B) are the consequences of a type I error worse than the consequences of a type II error ? Be sure to justify your decision, and to explain the practical significance (in terms of Michael being sent to prison) of the hypothesis test you have chosen. 5 5. Let Y , Y , ..., Y denote a random sample from a population having a Poisson 12n distribution with mean λ . a. Find the form of the rejection region for a most powerful test of H0 : λ = λ0 against Ha:λ=λa,where λa >λ0. n b. Recall that ∑ Y has a Poisson distribution with mean nλ . Indicate how this information i i=1 can be used to find any constants associated with the rejection region derived in part (a). c. Is the test derived in part (a) uniformly most powerful for testing H0 : λ = λ0 against H a : λ > λa ? Why? 6 6. AbiologisthashypothesizedthathighconcentrationofactinomycinDinhibitRNA synthesis in cells and thereby inhibit the production of proteins. An experiment conducted to test this theory compared the RNA synthesis in cells treated with two concentrations of actinomycin D: 0.6 and 0.7 micrograms per liter. Cells treated with the lower concentration (0.6) of actinomycin D yielded that 55 out of 70 developed normally whereas only 23 out of 70 appeared to develop normally for the higher concentration (0.7). Do these data indicate that the rate of normal RNA synthesis is lower for cells exposed to higher concentration of actinomycin D? a. Find the p-value for the test. b. If you chose to use α = 0.05 what is your conclusion? 9 7. The output voltage for an electric circuit is specified to be 130. A sample of 40 independent readings on the voltage for this circuit gave a sample mean 128.6 and standard deviation 2.1. Test the hypothesis that the average output voltage is 130 against the alternative hypothesis that it is less than 130. Use a test with level 0.05. I- Hypotheses: II- Calculations: Test statistic: Critical value: The p-value: 12 III- Decision and Conclusion: Decision: a. P-valueapproach: b. Critical value approach: Conclusion: 13 8. Shear strength measurements derived from unconfined compression tests for two types of soils gave the results shown in the following table (measurements in tons per square foot). Do the soils appear to differ with respect to average shear strength, at the 1% significance level? Assume that the underlying populations for both Soil Type I and Type II are normally distributed with unknown but equal variances. I- Hypotheses: Soil Type I Soil Type II n =23 1 y =1.65 1 s =0.26 1 n2 = 27 y2 =1.43 s2 = 0.22 14 II- Calculations: Test statistic: Critical value: The p-value: III- Decision and Conclusion: Decision: c. P-value approach: d. Critical value approach: 15

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