Answer all questions. Show all working. Question 1 [11 marks] a) The discrete probability distribution is given by X 0 1 2 3 P(X = X) 2m 3m 2m i. Show that m = = [2] ii. Find the mean of the probability distribution. [2] iii. Calculate P(X > /) [2] b) A bag contains 10 cards, of which 2 are marked two, I is marked three, 4 are marked four and 3 are marked five. 12 4 5 2 4 5 UT 1. Construct a probability distribution table, showing X, the number on the card. [2] ii. Use your table in i) to find the variance of this probability distribution. [3] Question 2 [23 marks] a) Research has shown at 55% of subscribers share their password with someone outside of their household. A random sample of 12 persons were sampled. 1. Define the variable of interest for this scenario. [1] ii. What probability distribution do you think best describes the situation? Why? [5] iii. Calculate the probability that less than 3 subscribers will share their password. [2] iv. If 500 persons were sampled, determine the number of persons who are expected to share their password with someone outside of their household [2] b) The number of bacterial colonies on a petri dish can has an average of 2.5 per cm?. i. Define the variable of interest for this scenario. [1] ii. What probability distribution do you think best describes the situation? Why? [4] iii. Calculate the mean and standard deviation of the distribution [2] iv. Find the probability that a) in 1 cm there are no bacterial colonies [2] b) in 4 cm there are more than three bacterial colonies. [4]Question 3 [12 marks] a) A college principal decides to consult the students about a proposed change in the time of lectures. She finds that out of a random sample of 80 students, 57 are in favour of the change. i. Calculate a 95 % confidence interval for the proportion of students who are not in favour of the change. [4] ii. State the effect on the width of such a confidence larval when the confidence level is increased. [1] b) A random variable of 7 independent observations of a normal variable gave Ex = 35.9 Ex2 = 186.19 Calculate i. The sample mean ii. The sample variance (21 iii. A 90% confidence interval for the population mean Question 4 [14 marks] a) The final grades of forty students were randomly selected, recorded and inputted into MINITAB. The results are summarized in Exhibit 1 below. One-Sample Z: Grade Test of mu = 85 vs # 85 The assumed standard deviation = 5.25 Variable N Mean StDev SE Mean Grade 40 87 5.25 1 . Calculate the value of *? ii. State the null and alternative hypotheses for the test. iii. What is the value of test statistic for this test.? iv. Calculate the p-value for this test. V. State the conclusion of this test. Give a reason for your answer. b) A machine fills cans with soft drinks so that their contents have a volume of 330ml. Over a period of time, it has been established that the volume of liquid in the cans follow a normal distribution with mean of 335 ml and standard deviation of 3 ml. A setting on the machine is altered, following which the operator suspects that the mean volume of liquid discharged by the machine into the cans has decreased. He takes a random sample of 50 cans and finds that the mean volume of liquid in these cans is 344.6 ml. Does this confirm his suspicion? Test at the 5% level of significance. [6]