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Cork Institute of Technology
Bachelor of Science in Computing โ Stage 1
(NFQ Level 7)
Summer 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 a random sample of householders to examine the monthly cost of house insurance in a certain city. Details of the findings are set out in the table below:
Monthly House Insurance (โฌโs) No. of Households
20 but less than 24 5 24 but less than 32 21 32 but less than 36 39 36 but less than 40 41 40 but less than 44 27 44 but less than 52 14 52 but less than 60 3
(i) Calculate the mean monthly cost of house insurance ( x ) and the standard deviation ( s ) from the mean. (ii) Calculate the median and hence a measure of skewness for the distribution. (12 marks)
(b) Establish a cumulative frequency table for the data in part (a) and draw the corresponding cumulative frequency polygon (ogive). Use your graph to estimate the percentage of monthly house insurances which fall in the range ( x โ s )to ( x + s ).
Determine the cost above which the highest 10% of insurances fall. (8 marks)
- The average hours worked per person in Ireland have fallen steadily since 1990. The table below shows the hours worked (in thousands) from 1990 to 2006 at two-yearly intervals:
Year 1990 1992 1994 1996 1998 2000 2002 2004 2006
Period , x 1 2 3 4 5 6 7 8 9
Average Hours, y 1.93 1.85 1.83 1.84 1.72 1.68 1.67 1.64 1. (000โs)
(i) Plot a scatter graph of the data. (You may break the y -axis.) (4 marks)
(ii) Calculate the equation of the regression line of y on x and fit this line to your scatter graph. (10 marks)
(iii) Calculate a coefficient of correlation for the data and interpret your result. (4 marks)
(iv) If the declining trend continues how many hours on average would you expect a person to be working in 2010? (2 marks)
- There has been a substantial growth in the construction industry in Ireland over the last number of years. The table below shows the number of new houses completed during each quarter of the period 2004 to 2006 inclusive. The numbers are recorded in hundreds.
Number of House Completions: 00โs
YEAR QUARTER
I II III IV
2004 166 191 182 228 2005 176 177 191 266 2006 183 202 201 292
(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 = 166 +5.9 x [ x units: 1 quarter, x = 1 gives 1 st^ quarter 2004] Use the linear trend equation and the seasonal index to forecast figures for the four quarters of 2007. (3 marks) (v) It is felt that the trend of recent past years may not continue through 2007 and that house completions could be as much as 15% lower than forecasted figures would suggest. If this is the case, then how many houses in total should be completed in 2007? (2 marks)
6.(a) โBEST BUZZโ is a new monthly entertainment magazine that has come on the market.
A particular newsagent places an order for 30 copies to be delivered to his shop every month. He pays the publisher โฌ3.40 per copy and sells the magazine for โฌ5.05 per copy. Any copies
unsold at the end of the month can be returned by the newsagent for a refund of โฌ1.85. He will have to pay a penalty cost of โฌ1.05 for each dissatisfied customer.
The demand for the magazine has the following probabilities:
Demand per month: 26 27 28 29 30 31 3 2
Probability: 0.04 0.16 0.20 0.28 0.12 0.14 0.
Using the following set of random numbers: 11, 51, 29, 65, 91, 04, 88, 43, 47, 95
(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. (12 marks)
(b) The newsagent has records of his weekly profits from the sale of a wide range of magazines
over a 40 week period. Figures are to the nearest euro.
(i) Represent the data on a stem and leaf diagram. Hence form a frequency
distribution with seven classes at most. Classes do not have to be of equal interval. Comment on the shape of the distribution.
(ii) Find the first and third quartile values. (8 marks)
7.(a) A third level college employs several lecturing staff on a part-time basis. A random sample of 40 part-time staff showed a mean weekly earning of โฌ384 per person and a standard deviation of โฌ90.
(i) Calculate a 99% confidence interval for the mean weekly earning of all part-time staff.
(ii) If the management wish to claim that the weekly earnings is within ฮผ โฌ20 of the sample mean and make this claim with 95% confidence, what sample size should be examined? Assume the standard deviation of โฌ90.
(iii)A survey of 160 staff members showed that 56 of them were female. Use this information
to estimate, with 99 per cent confidence, the proportion of all staff that is female. (12 marks)
(b) Explain the terms (i) Mutually Exclusive Events and (ii) Independent Events and in each case state the associated rule.
A firm is working independently on two separate jobs. The probability that job A will be finished on time is 0.3 while the probability that job B will be finished on time is 0.6.
Find the probability that: (i) both
(ii) just one (iii) at least one
of the jobs is finished on time. (8 marks)
Standard Error of the proportion = n
pq
- Regression and Correlation y ' = a + bx
โ^ y^ =^ na^ +^ b^ โ x
โ^ xy^ =^ a^ โ x^ +^ b^ โ x^2
a
y b x n
b
n xy x n x x
y
โ โ โ โ 2 โ^2
a
y x x xy n x x
โ โ โ โ โ โ
2 2 2 y^ =^ a^ + bx
Coefficient of Correlation :
( ) (^) ( )
r
n xy x n x x n y y
y
โ โ โ โ 2 โ^2. โ^2 โ^2 (^1 )
2
โ
โ n n
d R (^) S
