Question 3 (6 marks) Marks In a recent statistics survey, individuals were asked to disclose the number of televisions in their household. The following table shows number of televisions x per household and the corresponding the probability distribution P(x). Number of televisions per household P(x) x. P(x) (x - 1)2 (x - H)2P(x) X 0 0.02 0.40 2 0.35 3 0.20 4 0.03 [ x. P (x ) = [(x - 14) 2P(x) = a) Complete the columns in the table. [3] b) What is the probability that a randomly selected household has at most two televisions? [1] c) Determine the expected number of televisions per household (mean)? [1] d) By considering the above table, determine the standard deviation of the above discrete probability distribution? [1]Question 4 (8 marks) a) There are two roofing companies that service a small community (Company X and Company Y). Company X gets 45% of the roofing jobs in the community. Of Company X's customers, 70% are satisfied. Company Y's customers, 95% are satisfied. i. Construct a tree diagram to represent the above information. [2] ii. Using Bayes' theorem (or otherwise), if a customer is chosen at random and is found to be satisfied, what is the probability that the job was done by Company Y. [2] b) Carleton University found that 25% of its students withdraw without completing the introductory sociology course. Assume that 10 students registered for the course, what is the probability that at least 2 students will withdraw? Give your answer correct to 4 decimal places. [2] c) Airline passengers departs randomly and independently at the passenger- screening facility at Kingsford Smith International Airport. The mean departure rate is 12 passengers per minute. Assume that the departure of airline passengers are Poisson distributed. Compute the probability of exactly four passengers depart in a 30-second period. Give your answer correct to 4 decimal places. [2]Question 5 (3 marks) A store manager is interested in knowing the average amount their customers spent after each visit for a randomly selected a normal day (excluding holiday or any other special day). Using the store's 854 customer records, he summarized his findings in the following table. Mean $118.66 Population standard deviation $50.30 Assume the customers expenditures are approximately normally distributed. If a customer is randomly selected from the records, what is the probability that this customer will spend between $90 and $140? [3]