Data Analysis for Business
Module leader: C J Vallely
2005-6 Semester 2
2nd June 2006
Time: 10 – 12.10
Instructions To Candidates:
This is a CLOSED BOOK examination
Time allowed: TWO hours preceded by 10 minutes reading time.
(Please note that candidates are not permitted to write anything in
the answer booklets during the 10 minutes reading time)
Silent calculators may be used
Candidates are required to attempt ANY FOUR questions
All questions carry equal marks.
Show ALL formulas used in your calculations in your answer book.
All answers must be written in the official answer book provided.
Begin each question on a new page.
Page 1 of 8
You should attempt FOUR questions
All questions carry equal marks
1. (a) A MORI opinion poll in February 2006 showed the percentage saying they
would definitely vote Labour in an immediate general election was 38%, while
the percentage saying they would vote Conservative was 35%, both based
on the same sample survey of 1958 adults. Calculate the margin of error for
these percentages, assuming an infinite population. Is there any real
difference between the two parties when the margin of error is taken into
(b) Now suppose there is to be a by-election in a single constituency with a
population known to be 55000. How large a sample will be required to
estimate the percentage likely to vote for the Labour party in the by-election
within a margin of error of 0.5% using the result from the national opinion poll
(given in part (a) above) for p? Calculate the minimum required sample size if
you had no idea of the percentage likely to vote Labour.
(c) Use the random numbers below to select a simple random sample of size
four from the following list of thirteen people having the initials:
AA, AB, BC, BD, DE, EF, GG, GH, HJ, JJ, KL, LM, MK
Random numbers 00571 19891 42965 39013 98602 80384 74414 80420
(d) You are given the following quantity and price data for spending by the
typical family in an imaginary city. Calculate both base-weighted and current-
weighted price indices for year 2, taking year 1 as the base year. Comment
on the difference in the results for the two indices.
Year 1 Year 2
Item Quantity Price Quantity Price
Drink 50 30 40 35
Food 65 15 90 10
Fuel 90 35 50 45
TOTAL 25 MARKS
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2. The percentage grades obtained by 19 students on a module (module A)
were as follows:
45, 52, 56, 61, 51, 49, 65, 58, 53, 56, 55, 50, 46, 47, 95, 53, 60,44, 66
(a) Calculate the mode, median and arithmetic mean grade for module A.
What are the advantages and disadvantages of each of these measures in
(b) Calculate the sample standard deviation for module A using the same set
of numbers. You are given that the interquartile range for module A is 11%.
Compare the results on this module with another module (B) where mean =
53.8%, median = 57%, standard deviation = 5.7 and interquartile range = 6%.
(c) Suppose a random sample of 112 school students across a range of
modules gives a mean percentage grade of 56.2 with a standard deviation of
7.9. Construct a 95% confidence interval for the mean percentage grade of
all the students in the school. Interpret your results.
(d) A large printing company has weekly customers whose numbers follow a
normal distribution with a mean of 5500 customers and a standard deviation
of 385 customers. Use the table of the standard normal distribution given at
the end of this examination paper to find the probability that the number of
customers in any one-week period will be:
(i) More than 6000;
(ii) More than 5100;
(iii) Between 5800 and 6200.
(3 marks each = 9 Marks)
TOTAL 25 MARKS
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3. (a) If a DVD recorder costs £230 including VAT at a 17.5% rate, what is the
price excluding VAT? What would be the selling price if, other things equal,
the VAT rate increased to 21%?
(b) How much will a sum of £12800 be worth in ten years time if the interest
rate earned is 5.5% per year, paid at the end of each year, with all the
interest being re-invested? How much will the final total be if tax is payable
on all interest payments at a rate of 22%, with the tax being deducted each
(c) A company buys a van for £20000 and sells it five years later for £5000.
Calculate the annual percentage rate of depreciation using the declining
balance method. What will the book value of the van be after three years at
this rate of depreciation?
(d) The same company borrows £18000 to help buy the van. The loan is for
four years at an APR of 9.5%. Assuming payment is in equal instalments,
calculate (a) the monthly payments required to repay the loan at the end of
the four years and (b) the total interest paid on the loan. (Hint: you will first
need to calculate the monthly rate of interest equivalent to 9.5% APR).
TOTAL 25 MARKS
4. A company is trying to decide whether to adopt an investment project that
has an initial cost of £14.5 million. The project is expected to produce net
cash inflows of £3.7 million per year at the end of each year for the next five
years and to have a zero scrap value at the end of this time. Find:
(a) the payback period;
(b) the accounting rate of return, assuming straight-line depreciation;
(c) the net present value of the project, given a discount rate of 9%.
(d) You are given the additional information that the internal rate of return for
the project is 8.7%. Interpret this figure. Would you recommend this project
should be undertaken if the discount rate decreased to 8%? Explain your
TOTAL 25 MARKS
Page 4 of 8
5. You are required to investigate the relationship between a company’s annual
sales and its stockholdings of components at the end of the financial year,
both measured in millions of pounds over a 30-year period.
(a) Explain which of these you would expect to be the dependent variable, Y.
(b) Find and interpret the coefficient of determination, given the information
that (Y - Y). (X - X) 160.8, (Y - Y) 2 107.1 and (X - X) 2 253.9
(c) Find the slope and the intercept of the relationship, given the additional
information that Y 4.9 and X 8.5 .
(d) Interpret the slope and the intercept, and use your results to predict the
company’s stockholdings if annual sales are expected to be £11.2 million.
TOTAL 25 MARKS
6. (a) An independent survey of customers shows how the following seven
mobile phone companies are ranked in terms of quality of customer service
and value for money, with the most preferred being ranked 1. Find the rank
correlation between quality of service and value for money and comment on
Phone company Quality of Service Rank Value for Money
A 1 2
B 2 7
C 4 5
D 6 1
E 3 6
F 5 4
G 7 3
(b) Use the simple exponential smoothing method to forecast the next
number in the time series 201, 212, 215, 220, 226. You may take as starting
value the first number in the series. Use 0.2 as the smoothing constant .
Comment on the limitations of simple exponential smoothing in this example.
TOTAL 25 MARKS
Page 5 of 8
AC1003 FORMULA SHEET
p(1 - p)
Margin of error for an infinite population is 1.96 ˆ
where p = estimated
proportion and n = sample size
p(1 - p)(1 - n )
For a finite population of size N, ME = 1.96 N
Minimum sample size (n) to meet a margin of error E is 2
E p.(1- p)
Base-weighted price index =
q p .100% where q is quantity and p is price
q p 0 0
Current-weighted price index is
q p .100% 1 1
q p 1 0
Monthly interest rate = 1 1 or 12 (1 ) 1 where APR is annual
percentage rate of interest
P.r.(1 r) n
Payment per period for n periods to repay amount P is where r=interest
[(1 r) n 1
rate per time period.
Sample mean x
where n = sample size.
For grouped data, x
f x i i
where fi is the frequency of group i.
Sample variance s = 2 (x x) 2
Sample standard deviation s = variance
Coefficient of variation = x 100%
Interquartile range = upper quartile – lower quartile
Real interest rate i is given by 1 i where r = nominal rate and p = inflation
Compound interest, amount A = P.(1 + r)n where P = principal, r = interest rate per
time period and n = number of time periods
Present value PV = where A = amount, n = number of time periods and
(1 r ) n
r = interest rate per time period
Net present value NPV = sum of the present values of the cash flows
Internal rate of return is value of r that makes NPV = zero
Page 6 of 8
average profit average profit
Accounting rate of return ARR = .100% or .100 %
average capital initial capital
Formula for regular payments/repayments:
A = P.(1 + r)n +
x. (1 r) n 1
where either P is the amount initially invested,
followed by regular additions of amount x; or P is the amount initially borrowed
(negative), followed by regular repayments each of amount x (with A = 0).
n = number of payments and r = rate of interest per time period
Depreciation by reducing balance: A = P.(1 + r)n where P = initial cost, A = any
scrap/resale value, n = number of years and r = depreciation rate
Addition rule for probability: P(A or B) = P(A) + P(B) – P(A and B)
Multiplication rule for probability : P(A and B) = P(A) x P(B|A)
For combinations nCr = For permutations nPr =
r! (n - r)! (n - r)!
Expected value = Σxi.P(xi) where x includes all possible outcomes
For a large sample, the confidence interval for the population mean μ is given by
x Z. where x is the estimated sample mean, σ is the standard deviation, n =
sample size and Z is taken from the standard normal distribution.
For a finite population, we may have to apply the finite pop. correction
For a 95% CI, Z = 1.96. For a 90% CI, Z = 1.64
Pearson correlation r =
(X - X).(Y - Y)
(X - X) . (Y - Y)
6. d 2
Spearman correlation R = 1
Coefficient of determination = r2
ˆ (X - X).(Y - Y) and a Y - b.X
For the regression equation Y = a + b.X, b ˆ ˆ
(X - X) 2
Time-series decomposition Y = T + C + S + R or Y = T x C x S x R
Mean square error MSE =
Exponential smoothing: new forecast = last forecast + α x error in last forecast
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