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Cork Institute of Technology
Bachelor Science in Computing – Stage 1
(NFQ Level 7)
Autumn 2007
Quantitative Analysis
(Time: 3 Hours)
Answer five questions Examiners: Ms. M. Meaghar All questions carry equal marks Ms. A. Brown Ms. F. Wood
1.(a) A survey was carried out on 150 households having broadband connection. As part of the survey each household was asked to estimate on average the amount of time per day spent using the Internet. The findings are tabulated as follows:
Time (minutes) No. of households
0 but less than 30 6 30 but less than 60 29 60 but less than 90 43 90 but less than 120 35 120 but less than 180 19 180 but less than 240 13 240 but less than 360 5
(i) Calculate the mean daily usage ( x ) and the standard deviation ( s ) from the mean. (ii) Represent the data on a histogram. (iii) Determine the median of the distribution (either from your graph or by calculation) and hence find a coefficient of skewness. Comment on your findings. (17 marks)
(b) If two households are chosen at random from the distribution in part (a), determine the probability that both households use the internet, on average, at least two hours per day? (3 marks)
- A finance company carried out an investigation into annual credit card expenditure by a number of its customers. Details of the customers’ annual incomes were also examined. The findings are listed in the table below:
Annual Income (€000’s), x 94 73 52 65 132 84 158 104
Annual Credit Card Expenditure (€000’s) , y 13.7 8.5 8.3 6.8 15.7 13.1 19.3 12.
(i) Plot the data on a scatter diagram. (4 marks)
(ii) Calculate the equation of the regression line of y on x and fit this line to your graph in (i) (10 marks)
(iii) Calculate the coefficient of correlation and interpret your result. (4 marks)
(iv) Calculate the estimated expenditure for an annual income of €200,000. Would you place much reliance on your findings? Justify your answer. (2 marks)
- A company manufacturing components for personal computers is planning its production and thus uses quarterly sales figures over the past three years to forecast future demand. The sales figures are as follows:
Quarterly sales figures (000’s)
YEAR QUARTER
I II III IV
2004 550 528 415 615 2005 648 618 524 656 2006 658 601 547 672
(i) Plot the data on a time series graph. (4 marks) (ii) Calculate a four quarter centred moving average for the data and plot this on your graph in (i) (7 marks) (iii) Using a multiplicative model, determine seasonal index values for the data. (4 marks) (iv) The trend in the data can be depicted by the linear equation y = 519 + 10. 4 x [ x units : 1 quarter , x = 1 gives 1st^ quarter 2004] Use the linear trend equation and the seasonal index to forecast figures for each quarter of 2007. (3 marks) (v) Comment on the trend for 2007. How does it compare with the trend in previous years? (2 marks)
- (a) A new monthly magazine called “Property Views” has come on the market. A particular newsagent has places an order for 40 copies to be delivered to his shop monthly. He pays the publisher €3.10 per copy and sells the magazine for €5.40 per copy. Any copies unsold at the end of the month can be returned by the newsagent for a refund of €0.80. He will have to pay a penalty cost of €1.20 for each dissatisfied customer. The demand for the magazine has the following probabilities:
Demand per month: 30 33 36 39 42 45 48 Probability: 0.06 0.21 0.28 0.24 0.12 0.07 0.
Using the following set of random numbers: 47, 63, 23, 81, 19, 78, 06, 52, 41, 96
(i) Simulate 10 months demand for the magazine. (ii) Calculate the average monthly profit or loss associated with the sale of the magazine. (iii) Calculate the expected level of demand for the magazine. (14 marks)
(b) Write notes on each of the following: (i) Sampling Distribution (ii) Discrete and Continuous Probability Distributions (6 marks)
7.(a) The number of flights booked on line using a particular web-site was recorded for a period of time during 40 consecutive days. The findings are tabulated below:
135 92 132 51 120 114 116 71 132 135 148 101 78 53 105 54 124 140 113 89 117 112 122 141 93 61 97 104 137 119 105 121 72 138 136 124 84 90 131 149
(i) Represent the data on a stem and leaf diagram. Hence form a frequency distribution with seven classes at most. Comment on the shape of the distribution.
(ii) Determine the median and the lower and upper quartile values of the data. (10 marks)
(b) A survey of the amount spent on a package holiday by a random sample of 65 customers of a travel agency shows that the average expenditure is €925 with a standard deviation of € 135. (i) Calculate a 99% confidence interval for the amount spent by all customers. (ii) If you wanted to obtain an estimate with a margin of error of no more than €30, what size of sample should be chosen?
Further surveys show that 15 out of the 65 people have an annual travel insurance policy. Use this information to estimate, with 95 per cent confidence, the proportion of all travellers who have such policies. (10 marks)
Statistical Formulae
- Descriptive Statistics
Mean :
x
fx f
∑
∑ f
c f d x a c
.
Standard Deviation :
= ∑^ −
f
f ( x x )^2
2 2
∑
∑ − ∑
∑ f
c fd f
c fd
σ c
or
2 2
f
fx f
fx
Median = L C
N F
M M f
M M
(^12) 1
Mode =
L C
f f M M f f f
M M M M M
− − +
1 (^21 )
Binomial Distribution P( r , n ) = nC p qr r^ n^ − r
Poisson Distribution P( r ) = !
r
λ r e −^ λ
: λ = np
- The Normal Distribution Standard Units Z
x x
σ
- Standard Errors
Standard Error of the mean = n
s
Standard Error of the proportion = n
pq
