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Solution Manual for Statistics for Economics, Accounting and Business Studies , 7th Edition

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    ’s Manual Statistics for Economics, Accounting and Business Studies Seventh edition Michael Barrow SOLUTION MANUAL

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    4 Contents Chapter 1 Descriptive statistics 5 Chapter 2 Probability 18 Chapter 3 Probability distributions 25 Chapter 4 Estimation and confidence intervals 29 Chapter 5 Hypothesis testing 31 Chapter 6 The χ 2 and F distributions 35 Chapter 7 Correlation and regression 37 Chapter 8 Multiple regression 43 Chapter 9 Data collection and sampling methods 47 Chapter 10 Index numbers 48 Chapter 11 Seasonal adjustment of time-series data 52

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    5 A N S W E R S T O C H A P T E R 1 Descriptive statistics 1.1 (a) Comparison is complicated because the data for women only are from a survey, and hence the absolute numbers are much smaller. Also, we compare women here with the total for men and women in the text. We have to infer what the difference between men and women is. The major differences apparent are that there are relatively more women in the ‘Other qualification’ category and relatively fewer in the ‘Higher education’ category. (b) There is not a great difference apparent between this graph and the one in the text. Closer inspection of the figures suggests a higher proportion of women are ‘inactive’ and a lower proportion ‘In work’, but this detail is difficult to discern from the graph alone.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 6 (c) This chart brings out a little more clearly that there is a slightly higher degree of inactivity amongst women, particularly in the lower education categories. (d) This again brings out the higher proportion with ‘Other qualification’ and the fewer with ‘Higher education’.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 7 1.2 (a) These data show the average (the mean) in each category, not the count . The data given are monetary amounts, not frequencies. (b) The bar chart shows that mean earnings decline by education category. This is true for both males and females, with the female values being lower than that for males. There appears to be a slightly non-linear relationship between earnings and education, but this is illusory: The education categories are drawn on an arbitrary scale. (c) A stacked bar chart is not appropriate because you cannot add means together as you can with frequencies. To get the overall mean (i.e. for men and women combined), you would need to take a weighted average of the male and female values, with the weights given by the total numbers of men and women (which are not given here). 1.3 (a) Higher education, 88%. (b) In work, 20%. 1.4 (a) The premiums are £5.45 for men (44% of the earnings of someone with NVQ 3) and £5.26 for women (55%). The premium is proportionally bigger for women. (b) The median would almost certainly show a premium for a degree as well. As the median is less affected by outliers, the difference between the top two education categories might be smaller though this is not certain unless we actually look at the median values. Hourly earnings (£s) by qualification 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 NVQ 4 NVQ 3 NVQ 2 <NVQ 2 None Male Female

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 8 1.5 Bar chart: Histogram: The difference between bar chart and histogram is similar to that for the 2005 distribution. The overall shape of the histogram is similar (heavily skewed to right). Comparison is difficult because of different wealth levels (because of inflation), and grouping into classes can affect the precise shape of the graph.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 9 1.6 Bar chart: Histogram: Both show a highly skewed distribution once more. The histogram shows that there is a smooth but rapid decline in firm numbers as the number of employees rises. 1.7 (a) Mean 16.399 (£000); median 8.92; mode 0–1 (£000) group has the greatest frequency density. They differ because of skewness in the distribution. (b) Q1 = 3.295, Q3 = 18.339, IQR = 15.044; variance = 652.88; standard deviation = 25.552; coefficient of variation = 1.56. (c) 95,469.32/25.55 3 = 5.72 > 0 as expected. Ask your class whether this number is revealing to them – is there any intuition? (d) Comparison in text. 0 200,000 400,000 600,000 800,000 1,000,000 1,200,000 1,400,000 1,600,000 1,800,000 2,000,000 1- 5- 10- 20- 50- 100- 250- 500- 1000- Number of employees

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 10 (e) This would increase the mean substantially (to 31.12), but the median and mode would be unaffected. 1.8 (a) Mean = Σ fx / Σ f = 31,343,887.5/2,574,230 = 12.18. The median firm number is 1,287,115 in the first-class interval, and 0 + 1,287,115/1,740,685 × 4 = 3.96. The mode is also somewhere in the first-class interval, which has the highest frequency density. The differences are obviously down to the skewness of the distribution once again. (b) Inter quartile range: Q1 = 2.48 (firm number 643,557.5). Q3 = 7.44 (firm number 1,930,673.5), so IQR is 4.96. Variance: Σ fx 2 = 9,263,101,469, so σ 2 = 9,263,101,469/2,574,230 − 12.18 2 = 3,450.1. Hence, σ = 58.7 and coefficient of variation = 58.7/12.81 = 4.8. The extreme skewness of the data is reflected in the measures of dispersion. The IQR gives a very different picture from the standard deviation, which is affected by the high values. (c) Coefficient of skewness: Σ f ( x − μ ) 3 = 11,983,062,77,046, and dividing by N and by σ 3 (= 202,654.08) gives 22.97 > 0, and hence it is right skewed. This is a larger value than for wealth: The distribution is more skewed. 1.9 (33 × 134 + 40 × 139 + 25 × 137)/(33 + 40 + 25) = 136.8 pence/litre. 1.10 Since w i = p i / Σ p i ( p i is the number of pupils of type i ), the weighted average may be expressed as Σ px / Σ p , which is total expenditure divided by pupils. Note its similarity to the formula for mean when using frequency data. The total expenditure is actually 23 million, and dividing by 18,000 gives 1,277.8. 1.11 (a) z = 1.5 and −1.5 respectively. (b) Using Chebyshev’s theorem with k = 1.5, we have that at least (1 − 1/1.5 2 ) = 0.56 (56%) lies within 1.5 standard deviations of the mean, so at most 0.44 (44 students) lies outside the range. (c) As Chebyshev’s theorem applies to both tails, we cannot answer this part. You cannot halve 0.44 as the distribution may be skewed. 1.12 (1 − 1/ k 2 ) = 0.8 lie within k standard deviations. Hence, k = 2.23. Hence, 2.23 s = 2,000 and so s = 896.86 (as a minimum).

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 11 1.13 (a) The series shows an initial peak in 1989, then a slow recovery from 1991 to 2003. After this the market turns down again. The series is quite volatile with long upward and downward swings. (b)

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 12 This form of the graph shows that recessions can be quite severe for the vehicle market, while upswings show smaller per annum increases. The log graphs are very similar to the levels graphs. There is not always an advantage in drawing these.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 13 1.14 Figure 1.20 in the text shows line charts of the series and the main features were described. A chart of the log values reveals: This reveals similar features, but highlights the more rapid growth rate of intangible fixed assets, as well as their lack of volatility (presumably because it is an estimated rather than tangible series of values). There is a suggestion of faster rates of growth up to around 1990 followed by a slower rate of increase, but this impressionistic conclusion should not be relied upon. A graph of the change in the logs (only dwellings and transport are shown, for clarity) lends some support to the idea of a more rapid rate of growth before 1990. It is also apparent that both series (especially transport) are quite volatile. It raises the question of whether this is the most efficient form of investment for the long term.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 14 1.15 (a) (1,994.6/2,212.6) 1/23 = −0.0045 or −0.45% p.a. (note that this is not particularly representative – most years had positive growth, but there were sharp falls at the beginning and end of the period). (b) 0.084 (around the geometric mean). (c) The standard deviations of the two growth rates are similar (it was 0.0766 for investment), so this suggests a similar level of volatility. Note that, since the means are very different (6.37% p.a. for investment) and for registrations it is actually negative, there is a big difference in volatility if one relies upon the coefficient of variation. The latter is misleading in this case however – it reflects differences in the means rather than in volatility. 1.16 (a) 6.02% p.a. (use (37,044/5,699)1/32 − 1), similar to the growth of investment as a whole. (b) 0.106. (c) The coefficient of variation is 0.106/0.0602 = 1.76, slightly greater than the figure for investment as a whole (1.15). The latter is likely to be smoother since there are offsetting movements amongst the various investment categories. The general point is that the more disaggregated the data is, the more likely it is to fluctuate. When aggregating, random effects often cancel out (to some extent). 1.17 (a) Non-linear, upward trend. It is likely to be positively autocorrelated. Variation around the trend is likely to grow over time (heteroscedasticity). (b) Similar to (a), except that the trend would be shallower after deflation. Probably there will be less heteroscedasticity because price variability has been removed, which may also increase the autocorrelation of the series. (c) Unlikely to show a trend in the very long run, but there might be one over, say, five years, if inflation is increasing. Likely to be homoscedastic, with some degree of autocorrelation. 1.18 (a) The price level grows exponentially over time, at least in recent times. Historically, there have also been long periods of stable and even falling prices. (b) The inflation rate should not have a trend, but should fluctuate around a figure of about 3% p.a. (for the UK, again, for recent history). (c) Exchange rates are difficult to predict, both in the short and long term. The £/$ exchange rate has shown a rising trend over the past 50 years or so (depreciation of the £), interspersing periods of stability (fixed exchange rates), sudden changes (devaluation) and random fluctuations (floating exchange rate). 1.19 (a) Using S t = S 0 (1 + r ) t , hence S 0 = S t /(1 + r ) t . Setting S t = 1,000, r = 0.07 gives S 0 = 712.99. Price after two years: 816.30. If r rose to 10%, the bond would fall to 1,000/1.13 = 751.31. (b) The income stream should be discounted to the present using ( ) ( ) ( ) ( ) 2 3 4 5 200 200 200 200 200 820.04, 1 1 1 1 1 r r r r r + + + + = + + + + + so the bond should sell for £820.04. It is worth more than the previous bond because the return is obtained earlier.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 15 1.20 This is the same, in principle, as a compound interest calculation. Using equation (1.26) and setting S t = 3,000, S 0 = 30,000 and t = 10, we get 10 3000 1 0.2057, 30000 d = − = − i.e. 20.57% depreciation per annum. The value of the machine after one year is therefore 30,000 × (1 − 0.2057) = £23,830. After two years, its value is £23,830 × (1 − 0.2057) = £18,928, and after five years its value is 30,000 × (1 − 0.2057)5 = £9,485. You should check that its value is £3,000 after 10 years. 1.21 (a) 17.9% p.a. for BMW; 14% p.a. for Mercedes. (b) Depreciated values are as shown in this table: BMW 525i 22,275 18,284 15,008 12,319 10,112 8,300 Mercedes 200E 21,900 18,833 16,196 13,928 11,977 10,300 These are close to actual values. Depreciation is initially slower than the average, then speeds up, for both cars. 1.22 The discounted income stream is ( ) 2 1 1 1 400 400 8,000. 1 1 r r r     × + + = × =   + +      An alternative way to see this is to note that if you had £8,000 and invested it at 5%, you would get 8,000 × 5% = £400 p.a., that is C × r = Y, where C is the capital sum, r is the interest rate and Y is the annual income. Rearranging this yields C = Y/ r . This is a useful concept also in economics and finance. It relates income to wealth and so allows such things as the pricing of equities (discounted future profits). The share value of Tottenham Hotspurs recently (April 1995) fell by £5million when they lost an FA Cup semi-final. This reflects the loss of expected future profits (expected because they might have lost the final). Electricity shares also fell when the regulator announced a review of prices. 1.23 ( ) ( ) ( ) . x k x nk x E x k k E x k n n n + + + = = = + = +    1.24 ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 . kx k k x k x V kx k V x n n n μ μ μ − − − = = = =    It is useful if students can gain some confidence in doing these kinds of manipulation – they’re not that difficult. 1.25 The mistake is comparing non-comparable averages. A first-time buyer would have an above-average mortgage and purchase a below-average-priced house, hence the amount of buyer’s equity would be small. The original argument came from the Morning Star newspaper many years ago. With reasoning like this, no wonder communism failed.

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 16 1.26 If it is harder to get the popular jobs, then it is possible for Arts students to be more successful at getting both types of job than Science students, yet have a higher unemployment rate. Consider this example: Popular jobs Unpopular jobs Total applications successes % success applications successes % success applications successes % success Science 200 174 87 400 378 94.5 600 552 92 Arts 400 350 87.5 200 190 95 600 540 90 Arts students are more successful at both types of job but are less successful overall – a nice counter-intuitive conclusion. The inspiration for this question came from a careers service report at my university with precisely this flaw. Note that science students may be better than Arts students; the point is that the figures don’t prove it. Answers to exercises on Σ notation 1A.1 20, 90, 400, 5, 17, 11. 1A.2 40, 360, 1600, 25, 37, 22. 1A.3 88, 372, 16, 85. 1A.4 352, 2976, 208, 349. 1A.5 113, 14, 110. 1A.6 56, 8, 48. 1A.7 ( ) . f x k fx k f fx k f f f − − = = −        1A.8 ( ) ( ) 2 2 2 2 2 2 2 f x x f x fx fx f f f f μ μ μ μ μ − + − − + = =         2 2 2 2 2 2 . fx fx f f μ μ μ = − + = −    

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    Barrow, Statistics for Economics, Accounting and Business Studies , 7e, ’s Manual 17 Answers to exercises on logarithms 1C.1 −0.8239, 0.17609, 1.17609, 2.17609, 3.17609, 1.92284, 0.96142, impossible. 1C.2 −0.09691, 0.90309, 1.90309, 0.60206, 1.20412, impossible. 1C.3 −1.89712, 0.40547, 2.70705, 5.41610, impossible. 1C.4 −0.20397, 1, 1.09861, 3.49651, impossible. 1C.5 0.15, 12.58925, 125.8925, 1258.925, 10 12 . 1C.6 0.8, 199.5262, 1995.2623, 1995262.3. 1C.7 15, 40.77422, 2.71828, 22026.4658. 1C.8 33, 1,202,604.284, 3,269,017.372, 0.36788. 1C.9 3.16228, 1.38692, 1.41421, 0.0005787, 0.008. 1C.10 3.1071, 1.6035, 1.6818, 1, 1, 0.6934.

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