Question 1 (Past Exam 2010 – 2011)
(a) Consider a simple regression model
= + ( ) + , = 1,2, … , ,
where 1, 2, … , are mutually independent mean-zero normal random variables with a common variance 2. All 1, 2, … , are fixed values.
(i)Find the maximum likelihood estimators and of and respectively.
(ii) Show that
(iii) Compute the covariance of 2 and 3.
(b) Using the Instron 4206, rectangular strips of plexi-glass were stretched to failure in a tensile test. The following data give the change in length, in , before breaking () and the crosssectional area in 2 ():
(5.28, 52.36), (5.40,52.58), (4.65,51.07), (4.76,52.28), (5.55,53.02),
(5.73, 52.10), (5.84,52.61), (4.97,52.21), (5.50,52.39), (6.24,53.77)
Find the equation of the least square regression line. Also construct a twosided 95% confidence interval for the slope of the regression line.
Question 2 (Past Exam 2010 – 2011)
(a) Let be a binomial random variable Bbinomial(100, ). To test 0: =0.08 against 1: < 0.08, we reject 0 and accept 1 if an only if ≤ 6.Determine the significant level of the test. Also find the probability of the Type II error if in fact = 0.04.
(b) In a college health fitness program, let equal the weight in kilograms of a female freshman at the beginning of the program and let equal her change in weight during the semester. Assume that and follow a bivariate normal distribution. Use the following data for = 16 observations of (, ) to test the null hypothesis 0: = 0 against a twosided alternative hypothesis:
(61.4, 3.2), (62.9,1.4), (58.7,1.3), (49.3, 0.6),
(71.3,0.2), (81.5, 2.2), (60.8, 0.9), (50.2, 0.2),
(60.3,2.0), (54.6,0.3), (51.1, 3.7), (53.3, 0.2),
(81, 0.5), (67.6, 0.8), (71.4, 0.1), (72.1, 0.1).
(c) Let equal the distance between bad records on a used computer tape.Letting = 0.05 and taking = 42.2 as an estimate of , use the following 90 observations of and 10 classes of equal probability to test the hypothesis that the distribution of is exponential with mean :
30, 79, 38, 47, 22, 52, 36, 36, 7, 57,
3, 22, 30, 14, 8, 32, 15, 21, 12, 12,
6, 67, 6, 7, 35, 78, 28, 74, 5, 9,
37, 1, 3, 3, 44, 160, 50, 27, 61, 15,
39, 44, 130, 18, 6, 1, 32, 116, 23, 12,
58, 101, 68, 53, 58, 21, 21, 7, 79, 41,
80, 33, 71, 81, 17, 10, 13, 49, 21, 56,
107, 21, 17, 64, 14, 36, 26, 1, 54, 207,
64, 238, 25, 51, 82, 8, 2, 3, 43, 87
