MATH177 – 2020T2 Mock test – Long question (a) A survey from a certain university showed that 50% of students have a Visa credit card (V) and 40% have a MasterCard (M), while 25% have both. Consider randomly selecting a student from that university. (i) Compute the probability that the selected student has at least one of the two types of cards. (ii) What is the probability that the selected student has neither type of card (iii) Describe, in terms of V and M , the event that the selected student has a Visa card but not a MasterCard, and then calculate the probability of this event. (iv) Given that the selected student has a MasterCard, what is the probability that they also have a Visa card (b) Machines A and B are both used for making copies of a particular textbook. Machine A produces 40% of such copies. A quality check showed that 15% of copies made by machine A fail the binding strength test while that failing percentage from machine B is 10%. (i) Display the given information on a tree diagram. (ii) What percentage of copies fail the binding strength test (iii) If a randomly selected copy fails the binding strength test, what is the probability that it is made by machine A (iv) What are the odds that a copy made by machine A fails the test (v) Are the events {fail the test} and {made by machine A} independent Explain your answer. (c) Let X denote the number of mythical creatures of a particular type captured in a force field during a given time period. Suppose that creatures are independently captured, only one at a time, and on average the force field will contain 4.5 creatures during the given time period. (i) Write down the specific distribution of X. (ii) Calculate the probability that the force field contains exactly 5 creatures. (iii) What is the expectation and standard deviation of X (iv) Calculate the probability that the number of captured creatures X is within 1 standard deviation of its mean value. (d) A survey of 200 shoppers was conducted to investigate how often people buy wine at a certain su- permarket. Each shopper was asked whether they had bought any wine in each of the last 4 weeks. If they had not bought any wine the number of weeks was set to zero. So, in the following table of results, 34 shoppers did not buy any wine, 40 bought wine only in one of those 4 weeks and so on. Number of weeks when 0 1 2 3 4 Total wine was bought (x) Number of shoppers who bought 34 40 64 56 6 200 wine in that many weeks (fi) pi 0.0915 0.2995 0.3675 0.2005 0.0410 1 ei It has been suggested that these data are binomially distributed with n = 4 and some (unknown) parameter p, which is the probability that a randomly selected shopper will buy wine in any given week. 1 (i) Calculate the sample mean and hence estimate the parameter p for the suggested binomial dis- tribution. (ii) Write down the formula to calculate the binomial probability P (X = 3) = 0.2005 (4dp), as given in the table. Show clearly how to use the answer to (a) to calculate the probability given here. (iii) Use the pi-values given in the table, which were obtained as in part (b), to calculate the cor- responding expected values (ei): write them in the spaces provided in the table above. Give your expected values to 1 decimal place. Then do a goodness-of-fit test on these data, using a 5% level of significance. Comment briefly on the result of your test. 2
