SECOND UNIVERSITY EXAMINATION

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1 OLLSCOIL NA héireann, GAILLIMH NATIONAL UNIVERSITY OF IRELAND, GALWAY AUTUMN EXAMINATIONS, SECOND UNIVERSITY EXAMINATION STATISTICS [MA237] Dr. D. Harrington, Dr. J.N. Sheahan, Paul Wilson, M.A., H.D.E. Time allowed: Two Hours. Answer any five questions. All questions, but not necessarily parts therein, carry equal marks. Standard Normal Distribution Tables Are Supplied. Graph Paper is Available on Request. 1. The test results of 82 students were grouped into the frequency distribution shown below. Table 1: Score Frequency (a) For these grouped data, estimate the: i. mean, ii. standard deviation, iii. median, iv. lower quartile (Q L ). (If you wish you may use a cumulative frequency curve to estimate the median and lower quartile.) (b) Draw a histogram to illustrate the frequency distribution shown in Table 1. (It is not required that you use graph paper, but you may do so if you wish.) 1

2 2. (a) A survey of ten people was taken. In this survey the people were asked to state their Sex (Male = 1, Female = 2), their preferred type of drink (1 = minerals, 2 = beer, 3 = spirits, 4 = wine, 5 = cider), their mean weekly expenditure on alcoholic beverages (to the nearest pound), and to indicate, on a scale from 0 4, the extent to which they were concerned about the amount of alcoholic beverages that they drink per week (0 = not worried, 4 = extremely worried). The results of the survey are given in Table 2 below. i. State whether each variable (column heading) is a nominal, ordinal, discrete interval or continuous interval variable. ii. Calculate an appropriate average for each variable. Table 2: Sex Pref Expenditure Concern (b) Find the mean, median, mode and variance of the following population. (Your calculations must be shown). {1, 4, 3, 4, 8} (c) A sample of the junior cert. maths results in a certain large school was taken. It was calculated that x = 56, median = 60, Q U = 68, Q L =54. Also, the three highest results were 72, 80 and 88, whereas the three lowest results were 38, 30 and 07. Draw a boxplot to illustrate the above. 2

3 3. (a) Jack and Jill go fishing. The probability that Jack catches a fish is 0.6, the probability that Jill catches a fish is 0.5 and the probability that Jack and Jill each catch a fish is 0.3. i. Are the events Jack catches a fish and Jill catches a fish mutually exclusive? Justify your answer. ii. Are the events Jack catches a fish and Jill catches a fish independent? Justify your answer. iii. What is the probability that Jill catches a fish given that Jack catches a fish? (b) Mr. Hail, Miss Rain and Mrs. Snow all apply for a job as a park keeper. Whichever one gets the job, will have to open the gates of the park at 5.30am. If Mr. Hail gets the job, he will only open the gates on time 50% of the time. If Miss Rain gets the job, she will only open the gates on time 40% the time. If Mrs. Snow gets the job she will always open the gates on time. Assume that Mr. Hail has a 40% chance of getting the job, and that Miss Rain and Mrs. Snow each have a 30% chance of getting the job. i. What is the probability that the park gates will be opened on time? ii. If the gates are opened on time, what is the probability that Miss Snow got the job? 4. (a) The number of cars that enter a given car park is known to be Poisson distributed. If, during any ten minute period, an average of ten cars enter the car park, what is the probability that: i. Exactly eight cars enter the car park in a given ten minute period. ii. No cars enter the car park during any one minute period? iii. At least one car enters the car park during any one minute period? (b) i. Under what circumstances may we use a normal distribution to approximate a Poisson distribution? ii. Use the normal approximation to the Poisson distribution to calculate the approximate probability that in the car park mentioned in part (a) of this question: A. at most thirty cars will enter the car park in a given one hour period; B. no more than 200 cars will enter the car park in an eight hour period. iii. Why would it be unwise to use the normal approximation to the Poisson to estimate the probability that at most ten cars enter the car park in a twenty minute period? 3

4 5. (a) i. Exam results in a given subject are known to be normally distributed with a mean of 60% and a standard deviation of 16%. If a person who has sat an exam in the above subject is selected at random, what is the probability that his/her exam mark is: A. between 52% and 68%, B. less than 40%? ii. Suppose that 90% of people who sit the above exam achieve a mark of at most y%. Calculate y. (b) In how many ways can the twelve letters of the word RHODODENDRON be arranged in a row if: i. there are no restrictions, ii. the arrangements must not begin or end with a H, iii. the three D s must always be adjacent? 6. (a) A factory possesses two nut making machines, X and Y. 5% of the nuts produced by X are defective, and 2% of the nuts manufactured by Y are defective. i. If six nuts manufactured by machine Y are selected at random what is the probability that exactly two of them will be defective? ii. If seven nuts manufactured by machine X are selected at random what is the probability that at least one of them will be defective? iii. If two nuts manufactured by machine X and two nuts manufactured by machine Y are selected at random, what is the probability that none of these four nuts are defective? iv. If two nuts manufactured by machine X and two nuts manufactured by machine Y are selected at random, what is the probability that exactly two of these four nuts are defective? (b) i. Under what circumstances may we use a normal distribution to approximate a binomial distribution? ii. A fair dice is rolled 600 times. Using the normal approximation to the binomial distribution calculate the probability that: A. A 4 is thrown at least 110 times. B. An even number is thrown over 330 times? 4

5 7. (a) The time taken for a truck to drive from town A to town B (in that direction) is known to be normally distributed with a mean of 6 hours and a standard deviation of 2 hours. The time taken for a truck to make the return journey from town B to town A is known to be normally distributed with a mean of 5 hours and a standard deviation of 2 hours. The time taken to unload the lorry is normally distributed with a mean of 1 hour and a standard deviation of 1 hour. A man is to drive a truck from town A to town B, unload his lorry and then return to town A. Assuming that the lengths of time taken for the outward and return journeys, and the time taken to unload the lorry are independent, what is the probability that he will make the round trip in less than 15 hours? (b) The salaries of the people of Ballgobackwards are (approximately) normally distributed with a mean of 24, 000 and a standard deviation of 4, 000. A random sample of size n is taken from the above population. What is the probability that the mean of this sample will be more than 24, 100 if: i. n = 400, ii. n = 625? 5

(a) Find the value of x. (4) Write down the standard deviation. (2) (Total 6 marks)

(a) Find the value of x. (4) Write down the standard deviation. (2) (Total 6 marks) 1. The following frequency distribution of marks has mean 4.5. Mark 1 2 3 4 5 6 7 Frequency 2 4 6 9 x 9 4 Find the value of x. (4) Write down the standard deviation. (Total 6 marks) 2. The following table