All the questions in the document and please write the steps to obtain the answers. Thanks

COMP 233, Winter 2016 CONCORDIA UNIVERSITY PROBABILITY AND STATISTICS FOR COMPUTER SCIENCE. Assignment 4. Due: April 13, 2016. No late submissions allowed. 1. (12 points) Consider the special exponential density function f (x ; ) = cx2 ex , x 0 0, x 747.5 at the signicance level of 1 = 0.05. (b) Test the null hypothesis H0 : 747.5 versus alternate hypothesis H1 : 747.5 at the signicance level of 1 = 0.01. (c) Find the p-value of the test, and use the p-value to verify your answers. 4. (12 points) A car is advertised as having a gas mileage rating of at least 30 miles/gallon in highway driving. If the miles per gallon obtained in 10 independent experiments are 26, 24, 20, 25, 27, 25, 28, 30, 26, 33, should you believe the advertisement? Hint: Note that the sample size is small. Consequently use t-distribution. 5. (12 points) All pacemakers generate a pulse of current to depolarize a bit of the heart, the wave then spontaneously spreads to the rest of that part of the heart. Pulse duration is programmable in all currently manufactured permanent pacemakers. The shorter the pulse, the greater the necessary current. The length of the pulse for a particular setting will vary due to random uctuations such that the actual pulse length is normally distributed with a mean equal to the setting and with an unknown variance 2 . A type of pacemaker is claimed to be reliable since its standard deviation is less than 0.10. If a random sample of 20 pacemakers of that type resulted in a sample standard deviation of 0.08, should this claim be accepted? Suppose the level of signicance is = 0.05. (a) State the appropriate choice of the null hypothesis, and the alternate hypothesis. COMP 233, Winter 2016 Assignment 4, Page 2 (b) Compute the value of the test statistic, 2 . 19 (c) Determine its p-value. (d) Based on this evidence, should the claim be accepted? Also, comment on the appropriate choice of a signicance level for this problem. 6. (12 points) The chlorine residual in a swimming pool at various times after being cleaned is as given: Time (hr) 2 4 6 8 10 12 Chlorine Residual (pt/million) 1.60 1.50 1.45 1.42 1.38 1.36 Fit a curve of the form Y aebx . What would you predict for the chlorine residual 15 hours after a cleaning? 7. (12 points) Give a method for generating random numbers with a distribution function F (x) = 1 (x + x2 ), 2 0