QUESTION NO .5 (a) (b) (c) 15 n Let X,X2,..X be a random sample of n independent observations from a population with mean and variance . Show that E(X) = and Var(X) = 02 n [6 marks] Among the students taking a module in Engineering statistics at a certain university, a random sample of 200 students is interviewed, and it is found that 180 of them have access to a personal computer. Construct a 90% confidence interval for the proportion of all students in the module who have access to a personal computer. [4 marks] The mean lifetime of electric light bulbs produced by a company has in the past been 1100 hours. A random sample of eight electric light bulbs recently chosen from a supply of newly produced bulbs showed a mean lifetime of 1050 hours and a standard deviation of 125 hours. At the 0.01 level of significance, test whether the mean lifetime of the bulb has changed. [5 marks] QUESTION NO.6 (a) Hooke's law relates the extension of a spring under load to its extended length. The following results were obtained experimentally. 2 Load(N), x Extension(mm) 2 5 23 8 11 15 62 119 223 (i) Calculate the product moment correlation coefficient (r) and comment on your answer. (ii) Test the significance of the value r using 5% level of significance. [5 +5 marks] (b) Cellulon, a manufacturer of a new type of home insulation, wants to develop guidelines for builders and consumers regarding the effects on natural gas consumption. X is the thickness of the insulation in the attic of a home, X2 is the outdoor temperature, and Y is the monthly natural gas consumption (cubic feet). In the laboratory they varied the insulation thickness and temperature and determined the monthly natural gas consumption. The multiple-regression equation obtained is Y' 62.65 1.86X1 0.52 X2 . - Interpret the regression coefficients. - END OF PAPER - 4 [5 marks] 1