Cumulative frequency Diploma in Business Administration Part Quantitative Methods Examiner s Suggested Answers Question 1 Cumulative Frequency Curve 1 9 8 7 6 5 4 3 1 5 1 15 5 3 35 4 45 Weeks 1
(b) x f fx fx 5 5 15 315 75 1 9 675 15 5 315 175 6 455 7965 5 1 5 565 75 1 75 7565 35 7 75 739375 375 7 61 984375 45 3 175 5418 1 195 4775 195 x = = 1 19 5 4775 sd = 19 5 = 1 1 1 (c) 95% confidence interval is: 195 ± (196-1 1 ) = 195 ± 196 1 (d) H : µ = 15 H 1 : µ 15 At the 5% level, the critical value of z = 196 z = x µ 19 5 15 = = 45 σ 1 1 n 1 Reject H Question (i) A random sample is an unbiased subset of a population in which every member of the population has an equal chance of being selected (ii) To collect a systematic sample of size n from a sampling frame of size N, choose one item at random from the sampling frame and then choose every (n/n)th item from that point (b) According to sampling theory (specifically, the central limit theorem), the sampling distribution of the mean (and other parameters) is normal for sufficiently large samples This means that the normal probability distribution can be used to construct confidence limits in estimation and to test hypotheses at given levels of significance
(c) Px ( < 1 ) = Pz 1 3 < = 1 33 15 = 918 from the normal table (d) H : µ 1 = µ H 1 : µ 1 µ At the 5% level, the critical value of z = 196 z = x σ1 n x 1 1 σ + n = 11 1 1 5 + 4 6 = 1 = 586 17 Reject H Question 3 Let y be CIS and x be IR Then, x = 13, y = 3, xy = 579, x = 935, y = 94, n = 1 R = ( 1 579 ) ( 3 13 ) { }{ } = 18 = 18 ( 1 935) 13 ( 1 94) 3 611 9116 36 6 = -94 This represents relatively strong, negative linear correlation (b) b = 18 = 363 611 a = y bx = 6 833 (( 3 63) 8 583) = 57 99 So, y = 57 99 3 63x 3
(c) Scatter diagram Investment 45 4 35 3 5 15 1 Investment/Interest Rate y = 5799-363x 5 5 1 15 Interest Rate (d) When IR = 11, CIS = 186 When IR = 1, CIS = 1443 So the prediction is that the 1% interest rate rise will cause capital investment to fall by 363b The prediction is not likely to be very accurate as the sample size is small, other influences on investment are ignored and the prediction is outside the range of the sample data Question 4 A time series consists of four main components: trend, seasonal variation, cyclical variation and residual variation In the additive model, it is assumed that the time-series variable is the sum of these components (y = T + S + C + R) In the multiplicative model, it is assumed that the time-series variable is the product of the components (y = T _ S _ C _ R) (b) (i) Let y be the number of new mortgage loans issued: y Trend (centred 4-pt MA) 3 46 5 39 6 3975 36 4 48 45 5 415 3 415 38 4175 48 45 5 34 4
(ii) Time-series and trend No of loans 6 5 4 3 1 4 8 1 Quarters Number of loans Trend (iii) Qtrs 1 3 4 Years 1 11-1375 -4 75 875-115 3-375 55 Average: -3875 65 9875-165 Adjusted: -3844 6531 996-1594 (iv) Average increase in trend = 4 5 39 7 = 5 Question 5 Forecasts for Year 4: Q1 : 45 + (3 _ 5) - 3844 = 4156 Q : 45 + (4 _ 5) + 6531 = 5131 Q3 : 45 + (5 _ 5) + 996 = 5496 Q4 : 45 + (6 _ 5) - 1594 = 396 The ways of estimating the demand for a new product include: Market research by means of interviewing or a questionnaire survey Pilot studies to test demand in selected regions Investigations of the markets for similar products (b) (i) At the breakeven level, R = C Therefore: 5Q = + 4Q 1Q = Q = 5
So the breakeven level of output is units (ii) Profit = R - C = 1Q - So when Q = 1, profit is: 1 - = -1 11 So a loss of 1, would be made (iii) When 1Q - =, Q = 4 So the firm should produce 4 units to make a profit of, (iv) When 1Q - = -6, Q = 14 Since the breakeven level is, an additional 6 units are required to break even Question 6 A one-tailed (or one-sided) test investigates whether a sample value is significantly greater than (or less than) an assumed population value at a given level of significance The test uses an area in one tail of the normal curve A two-tailed (or two-sided) test investigates whether a sample value is significantly different from an assumed population value and so uses areas in both tails of the normal curve (b) H : π = 9 H 1 : π 9 p = 41/5 = 84 At the 5% level, the critical value of z = 196 in a two-tailed test z = p π ( ) = 84 9 π π 1 9 1 n 5 = 76 = 566 134 Reject H The local authority s claim is not supported by the sample data (c) H : π > 5 H 1 : π 5 p = 495/1 = 495 At the 5% level, the critical value of z = 1645 in a one-tailed test 6
z = p π ( ) = 495 5 π π 1 5 5 n 1 = 5 = 316 158 Do not reject H The survey results do not allow us to reject the chocolate company s claim Question 7 Aggregate price index = p1 5 1 1 113 6 p = = (b) Price relatives (p 1 /p ) 9/5 = 18 5/6 = 833 5/6 = 833 6/5 = 1 4666 So the arithmetic mean of price relatives is 4 666 4 1 = 116 7 (c) p q p 1 q p q 1 p 1 q 1 45 81 4 7 6 5 1 1 6 5 1 1 15 15 1 1 641 115 63 95 Laspeyres Index = pq pq 1 115 1 = 1 = 158 3 641 (d) Paasche Index = pq 1 1 1 pq = 95 1 15 3 1 63 = (e) The Laspeyres price index uses base-year quantities as weights, while the Paasche price index uses current-period quantities as weights By using base-year quantities as weights, the Laspeyres price index tends to give relatively more weight to goods whose prices have risen and less weight to goods whose prices have fallen The Paasche index does the opposite So the Laspeyres index tends to overestimate price rises, while the Paasche index tends to underestimate them 7
Question 8 (i) Discrete data can only take integer values (such as the number of children in families), while continuous data can take any real number values (such as the weight of biscuits in packets) (ii) (iii) The mean deviation and standard deviation are both measures of dispersion The mean deviation is the average absolute deviation of each value from the mean The standard deviation is the square root of the average squared deviation of each value from the mean The coefficient of variation is a measure of relative dispersion, calculated as a percentage by taking the standard deviation divided by the mean and multiplied by 1 The coefficient of skewness measures whether a distribution is symmetrical or skewed to the left or right (b) The central limit theorem states that for sufficiently large samples (n > 3), the sampling distribution of the mean will be a normal distribution regardless of the shape of the population distribution It is important because it means that, given a sufficiently large random sample, the normal distribution can be used to construct confidence intervals and to test hypotheses Similar results apply to other population parameters, such as proportions and estimated regression coefficients 8