Question;1) Using the identity() = [ E()] + [E() ] = [ E()] + B(),show thatM SE() = E[()2 ] = V () + (B())2.comment: Use the identity. Note that B() is a constant.2) The reading on a voltage meter connected to a test circuit is uniformly distributedover the interval (, +1), where is the true but unknown voltage of the circuit. Supposethat Y1,..., Yn denotes a random sample of such readings.a) Show that Y is a biased estimator of, and compute the bias.b) Find a function of Y that is an unbiased estimator of.3) Suppose that the random variable Y is an observation from a normal distributionwith unknown mean and variance 1. Find a 95% condence interval for.comment: Use the CI for when is known and n = 1.4) Athletes at major colleges graduated, on the whole, at virtually the same rateas other students, according to the NCAA in 1992. Suppose that in a new poll of500 athletes at major colleges, the number graduating was 268. Give a 98% condenceinterval for p, the proportion of athletes at major colleges who graduate.comment: See ex. 8.2 on p. 401.5) The administrators at a hospital wished to estimate the average number of daysrequired for in-patient treatment of patients between ages 25 and 34. A random sampleof 500 hospital patients between these ages produced a mean and standard deviationequal to 5.4 and 3.1 days, respectively. Construct a 95% condence interval for the meanlength of stay for the population of patients from which the sample was drawn.comment: See ex. 8.7 on p. 412-3.6) Suppose that independent random samples of n1 = n2 = 30 adults were selectedfrom two regions in the United States, and a days intake of selenium was recorded foreach person. The mean and standard deviation of the selenium daily intakes for the 30adults in Region 1 were y 1 = 167.1 g and s1 = 24.3 g respectively. The correspondingstatistics for the 30 adults in Region 2 were y 2 = 140.9 g and s2 = 17.6 g respectively.Find a 95% condence interval for the dierence in the mean selenium intake for the tworegions.comment: Use the formula (given in class or p. 415): (Y 1 Y 2) z/2Do not forget to square the sample standard deviations.12S1S2+ 2n1n27) In a 1983 Harris poll of n = 1250 individuals, the proportion who used seat beltswas 0.19. In a 1992 poll of n = 1251 individuals, the proportion who used seat belts was0.70.i) Find a 90% condence interval for p1 p2 where p1 corresponds to the 1983 poll.ii) Do you think that the (population) proportion was higher in 1992? Why?comment: Use the formula given in class (or ex. 8.8):p1 (1 p1) p2 (1 p2)(1 p2) z/2p+.n1n2CI is about (.54,.48), but use more digits.8) Let Y be a binomial random variable with parameter p. Find the sample sizenecessary to estimate p to within 0.05 with probability 0.95 in the following situations.a) The value of p is thought to be about 0.9.b) No information about p is known.9) In a 1986 newspaper poll of 221 children, 2/3 said they would like to travel inspace. How many children should have been interviewed to estimate the proportion ofchildren who would like to travel in space correct to within 0.02 with probability 0.99?(Use p = 2/3.)10) Suppose that you wish to estimate the dierence between the mean acidity forrainfalls at two dierent locations, one in a relatively unpolluted area along the ocean andone in an area subject to heavy air pollution. If you wish your estimate to be correct tothe nearest 0.1 PH with probability 0.90, approximately how many rainfalls (PH values)must you include in each sample? (Assume that the variance of the PH measurements isapproximately 0.25 at both locations and that the two samples are to be of equal size.)11) In 1991, the mean SAT verbal score was 422 and the mean SAT math score was474. Suppose that a random sample of test scores of 20 seniors from a large high schoolproduced the results tabled below.Verbal MathSample Mean419455Sample Standard Deviation5769a) Find a 90% condence interval for the mean verbal SAT scores for the high schoolseniors.b) Does the interval that you found in a) include the value 422, the true mean SATverbal score in 1991? What can you conclude?
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