Practice Midterm Exam (Statistics)

AMS572. 01 Practice Midterm Exam F completely, 2007 Instructions This is a close book exam. Any wizard who cheats in the exam shall receive a arrange of F. gratify provide complete solutions for full credit. Good luck 1 (for all students in class). In a study of hypnotic suggestion, 5 male volunteers participated in a two-phase experimental session. In the first phase, breathing was deliberate while the subject was awake and at rest.In the blurb phase, the subject was told to imagine that he was performing brawny 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 internal respiration (liters of air per minute per squargon 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 (1) drop suitable test to investigate whether there is any difference between the two experimental phases in terms of total ventilation.Please verbalise the assumption(s) of the test and report the p-value. At the substance level of 0. 05, what is your conclusion? (2) Please write up the entire SAS program unavoidable to answer questions raised in (a). Please include the data step as good as tests for testing for various assumptions. base (1) Assume that the difference picis normal. pic and pic The hypotheses are pic v. s pic. The test statistic is pic Since pic and pic, we can not reject pic at pic. pic (2) The SAS inscribe is as follows data hypnosis input subject rest work iff=work-rest datalines 1 6 6 2 7 9 3 8 9 4 7 10 5 6 7 run proc univariate data=hypnosis normal var diff run 2 (for all students in class). John Pauzke, president of Cereals Unlimited, Inc. , wants to be precise certain that the esteem weight ? of cases satisfies the tract label weight of 16 ounces. The packages are modify by a machine that is set to fill each package to a specified weight. However, the machine has random variability measured by ? 2. John would like to attain strong evidence that the regard as package weight is above 16 ounces.George Williams, quality control manager, advises him to examine a random sample of 25 packages of cereal. From his past experience, George knew that the weight of the cereal packages follows a normal distribution with standard deviation 0. 4 ounce. At the significance level ? =. 05, (1) What is the decision rule (rejection office) in terms of the sample typify pic? Please realise the general formula using the concept of eccentric I error rate. (2) What is the great power of the test when ? =16. 2 ounces? Please understand the general formula for power calculation first. 3) What is the sample size necessary to ensure a power of 80% when ? =16. 2 ounces? Please derive the general formula for sample size calculation based on the Type I and II error rates first. Solution (1) pic pi c. pic. pic. pic Hence, the rejection region is pic. (2) pic pic (3) pic picpic. pic Hence, about 25 packages of cereal should be sampled to achieve a power of 80% when (=16. 2 ounces. 3a (for all except AMS PhD students). Inference on one population mean when the population is normal, and the population magnetic variation is known.Let pic, be a random sample from the given normal population. Please assay that 1) pic. 2) pic. Solution (1) pic Thus, pic (2) pic Thus, pic 3b (for AMS PhD students ONLY). For a random sample from any population for which the mean and variance exist. Please conjure up that 1) The sample mean and sample variance are unbiased estimators of the population mean and variance respectively. 2) When the population is normal, we have learned that the sample mean and the sample variance, are indeed, independent.Please prove this for n = 2. That is, for a random sample of size 2 only. Solution (1) pic pic (2) When n=2, pic, pic If we can show that pic and pic ar e independent, then picand picare independent. This can be done easily using the mgf technique pic 4 (extra credit for all). An salutary witness in a paternity suit testifies that the length (in days) of maternalism (that is, the time from impregnation to the delivery of the child) is approximately normally distributed with parameter pic and pic.The suspect in the suit is able to prove that he was out of the field during a period that began 290 days before the birth of the child and stop 240 days before the birth. If the defendant was, in fact, the father of the child, what is the probability that the get could have had the very long or very short pregnancy indicated by the testimony? Solutionlet picpic and picpic pic(the woman had a very long or very short pregnancy) pic pic Happy Halloween