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Cork Institute of Technology
Higher Certificate in Science in Computing (ACCS) - Stage 1
(NFQ – Level 6)
Summer 2007
Quantitative Analysis
(Time: 3 Hours)
Answer five questions Examiners: Ms. M. MeagherMs. A. Brown Ms. J. English
Q1. The table below gives the distribution of the overtime earnings for 200 operatives during the first quarter of 2005.
Overtime Earnings (in €’s) No. of Operatives
100 but less than 125 10 125 but less than 175 14 175 but less than 200 37 200 but less than 250 28 250 but less than 350 40 350 but less than 400 32 400 but less than 450 18 450 but less than 500 12 more than 500 9
(a) Represent the data above on a histogram. (4 marks) (b) Calculate the mean and standard deviation for this distribution. (6 marks) (c) Calculate the mode and the median for this data. (6 marks) (d) Calculate a measure of skewness for this data. Explain your result. (4 marks)
Q2. The table below shows the daily mileage travelled by a hackney driver over a fifty day period. 151 173 195 187 163 210 142 168 153 199 177 182 171 149 156 211 178 197 185 158 161 173 179 213 144 151 184 176 188 195 222 232 197 243 210 178 165 185 205 190 221 187 159 197 188 210 241 155 178 183 (a) Form a frequency distribution with 7 classes of equal width. (6 marks) (b) Represent the distribution in (a) on a Cumulative Frequency Curve. (5 marks) (c) Estimate the values of the third quartile, the median and forty fourth percentile. (6 marks) (d) On how many days did the driver drive less than 172 miles? (3 marks)
Q3. (a) Chain links produced by a certain firm have breaking strengths normally distributed with mean 1100kg and standard deviation 35kg. Estimate the percentage of links with breaking strengths: (i) less than 1000kg (2 marks) (ii) more than 1150kg (2 marks) (iii) between 1025 and 1120kg. (3 marks) (b) A large store finds that 35% of its customers pay by credit card. Find the probability that of 10 randomly chosen customers: (i) exactly seven pay by credit card (2 marks) (ii) at least three pay by credit card (4 marks) (c) The number of calls received by an ambulance service during the 10 p.m. – 6 a.m. shift is, on average, 6. Find the probability that on a given shift: (i) exactly five calls are received (2 marks) (ii) less than two calls are received (3 marks) (iii) what is the probability that eight calls are received over two 10 p.m. – 6 a.m. shifts? (2 marks)
Q5. The table below shows the quarterly sales figures for a company for the past 4 years.
Sales(€000’s) YEAR Q1 Q2 Q3 Q 2003 63 51 47 65 2004 75 57 50 71 2005 79 60 54 83 2006 90 65 62 89 (i) Plot the data above on a time series graph. (4 marks) (ii) Calculate a moving average for the data and plot the trend line on the graph of part (i). (6 marks) (iii) Using the additive model, calculate a seasonal index for each of the four periods. (7 marks) (iv) Forecast the sales for the first two quarters of 2007 for the above company. (3 marks)
Q6. (a) A manufacturing process has a total cost function C = 20 + 4x and a demand function pr = 22 − 4x , where x is the quantity in hundreds and C and p (^) r are both in €000’s. (i) Derive the revenue function R and hence obtain the total profit function P. (3 marks) (ii) Calculate the quantity x, that maximises profit and find the maximum profit. (3 marks) (iii) What is the profit situation when 320 units are produced? (2 marks) (b) A farmer can produce potatoes (X) or barley (Y) with his resources. He makes a net profit of €70 per acre of potatoes (X) and €100 per acre of barley(Y). He draws up the following summary statement of his resources and input requirements. Requirements per acre:
(i) Use inequalities to state all of the constraints facing the farmer. (3 marks) (ii) Graph these constraints. (5 marks) (iii) Find the number of acres he should devote to each product in order to maximize total net profit. What is his net profit in this situation? (4 marks)
Resources X( potatoes ) Y( barley ) Amount Available Labour 18 man days 6 man days 1068 man days Equipment 3 machine days 5 machine days 450 machine days Land 1 acre 1 acre 100 acres
Q7. (a) The final grades in statistics of 40 students at a local college are recorded in the Table below.
68 84 75 82 68 90 62 88 76 93 73 79 88 63 65 91 88 87 63 75 61 65 75 87 74 62 89 90 91 76 66 78 75 94 77 68 94 69 70 76
(i) Represent the data above on a Stem-and-Leaf diagram. Hence form a frequency distribution with seven classes. Classes do not have to be of equal interval. (3 marks) (ii) Use the Stem-and-Leaf diagram to estimate the median and the upper and lower quartiles for this data. (3 marks)
(b) During an internal audit, an accountant had to sample a very large batch of invoices for value and for errors. A random sample of 200 invoices revealed a mean value of €900 with a standard deviation of €400. (i) Determine a 99% confidence interval for the mean value of the whole batch. (3 marks) (ii) If the size of sample is increased to 800 how does this affect the confidence interval and the accuracy of the answer? If 20 invoices in the random sample of 200 invoices showed errors, calculate a 95% confidence interval for the proportion of the whole batch showing errors. (5 marks) (c) Write notes on each of the following: (i) Sampling Distribution (3 marks) (ii) Discrete and Continuous Probability Distributions (3 marks)
