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

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’s ManualStatistics for Economics,Accounting and BusinessStudiesSeventh editionMichael BarrowSOLUTION MANUAL

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4ContentsChapter 1Descriptive statistics5Chapter 2Probability18Chapter 3Probability distributions25Chapter 4Estimation and confidence intervals29Chapter 5Hypothesis testing31Chapter 6Theχ2andFdistributions35Chapter 7Correlation and regression37Chapter 8Multiple regression43Chapter 9Data collection and sampling methods47Chapter 10Index numbers48Chapter 11Seasonal adjustment of time-series data52

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5A N S W E R ST OC H A P T E R1Descriptive statistics1.1(a)Comparison is complicated because the data for women only are from a survey, andhence the absolute numbers are much smaller. Also, we compare women here withthe total for men and women in the text. We have to infer what the differencebetween men and women is. The major differences apparent are that there arerelatively more women in the ‘Other qualification’ category and relatively fewer inthe ‘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 graphalone.

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual6(c)This chart brings out a little more clearly that there is a slightly higher degree ofinactivity amongst women, particularly in the lower education categories.(d)This again brings out the higher proportion with ‘Other qualification’ and the fewerwith ‘Higher education’.

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual71.2(a) These data show theaverage(the mean) in each category, not thecount. Thedata given are monetary amounts, not frequencies.(b)The bar chart shows that mean earnings decline by education category. This is truefor both males and females, with the female values being lower than that for males.Thereappearstobeaslightlynon-linearrelationshipbetweenearningsandeducation, but this is illusory: The education categories are drawn on an arbitraryscale.(c) A stacked bar chart is not appropriate because you cannot add means together asyou can with frequencies. To get the overall mean (i.e. for men and womencombined), you would need to take a weighted average of the male and femalevalues, with the weights given by the total numbers of men and women (which arenot 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 isproportionallybigger for women.(b) The median would almost certainly show a premium for a degree as well. As themedian is less affected by outliers, the difference between the top two educationcategories might be smaller though this is not certain unless we actually look at themedian values.Hourly earnings (£s) by qualification0.002.004.006.008.0010.0012.0014.0016.0018.0020.00NVQ 4NVQ 3NVQ 2<NVQ 2NoneMaleFemale

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual81.5Bar chart:Histogram:The difference between bar chart and histogram is similar to that for the 2005distribution. The overall shape of the histogram is similar (heavily skewed to right).Comparison is difficult because of different wealth levels (because of inflation), andgrouping into classes can affect the precise shape of the graph.

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual91.6Bar chart:Histogram:Both show a highly skewed distribution once more. The histogram shows that thereis 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 greatestfrequency 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) 95,469.32/25.553=5.72>0 as expected. Ask your class whether this number isrevealing to them – is there any intuition?(d) Comparison in text.0200,000400,000600,000800,0001,000,0001,200,0001,400,0001,600,0001,800,0002,000,0001-5-10-20-50-100-250-500-1000-Number of employees

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual10(e) This would increase the mean substantially (to 31.12), but the median and modewould be unaffected.1.8(a) Mean =Σfx/Σf= 31,343,887.5/2,574,230 = 12.18. The median firm number is1,287,115 in the first-class interval, and 0 + 1,287,115/1,740,685×4 = 3.96. Themode is also somewhere in the first-class interval, which has the highest frequencydensity. The differences are obviously down to the skewness of the distribution onceagain.(b) Inter quartile range: Q1 = 2.48 (firm number 643,557.5). Q3 = 7.44 (firmnumber1,930,673.5),soIQRis4.96.Variance:Σfx2=9,263,101,469,soσ2= 9,263,101,469/2,574,230 − 12.182= 3,450.1. Hence,σ= 58.7 and coefficientof variation = 58.7/12.81 = 4.8. The extreme skewness of the data is reflected in themeasures of dispersion. The IQR gives a very different picture from the standarddeviation, which is affected by the high values.(c) Coefficient of skewness:Σf(x−μ)3= 11,983,062,77,046, and dividing byNandbyσ3(= 202,654.08) gives 22.97 > 0, and hence it is right skewed. This is a largervalue than for wealth: The distribution is more skewed.1.9(33×134 + 40×139 + 25×137)/(33 + 40 + 25) = 136.8 pence/litre.1.10Sincewi=pi/Σpi(piis the number of pupils of typei), the weighted average may beexpressed asΣpx/Σp, which is total expenditure divided by pupils. Note its similarityto the formula for mean when using frequency data. The total expenditure is actually23 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 withk= 1.5, we have that at least (1 − 1/1.52) =0.56 (56%) lies within 1.5 standard deviations of the mean, so at most 0.44 (44students) lies outside the range.(c) As Chebyshev’s theorem applies tobothtails, we cannot answer this part. Youcannot halve 0.44 as the distribution may be skewed.1.12(1 − 1/k2) = 0.8 lie withinkstandard deviations. Hence,k= 2.23. Hence, 2.23s=2,000 and sos= 896.86 (as a minimum).

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual111.13(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 upwardand downward swings.(b)

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual12This form of the graph shows that recessions can be quite severe for the vehiclemarket, while upswings show smaller per annum increases. The log graphs are verysimilar 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 Manual131.14Figure 1.20 in the text shows line charts of the series and the main features weredescribed. A chart of the log values reveals:This reveals similar features, but highlights the more rapid growth rate of intangiblefixed assets, as well as their lack of volatility (presumably because it is an estimatedrather than tangible series of values). There is a suggestion of faster rates of growthup to around 1990 followed by a slower rate of increase, but this impressionisticconclusion should not be relied upon.A graph of the change in the logs (only dwellings and transport are shown, forclarity) lends some support to the idea of a more rapid rate of growth before 1990. Itis also apparent that both series (especially transport) are quite volatile. It raises thequestion 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 Manual141.15(a) (1,994.6/2,212.6)1/23= −0.0045 or −0.45% p.a. (note that this is not particularlyrepresentative – most years had positive growth, but there were sharp falls at thebeginning 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 forinvestment), so this suggests a similar level of volatility. Note that, since the meansare very different (6.37% p.a. for investment) and for registrations it is actuallynegative, there is a big difference in volatility if one relies upon the coefficient ofvariation. The latter is misleading in this case however – it reflects differences in themeans 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 awhole.(b) 0.106.(c) The coefficient of variation is 0.106/0.0602 = 1.76, slightly greater than thefigure for investment as a whole (1.15). The latter is likely to be smoother sincethere are offsetting movements amongst the various investment categories. Thegeneral point is that the more disaggregated the data is, the more likely it is tofluctuate. When aggregating, random effects often cancel out (to some extent).1.17(a) Non-linear, upward trend. It is likely to be positively autocorrelated. Variationaround the trend is likely to grow over time (heteroscedasticity).(b) Similar to (a), except that the trend would be shallower after deflation. Probablythere 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 ofautocorrelation.1.18(a) Thepricelevelgrowsexponentiallyovertime,atleastinrecenttimes.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 ofabout 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 ofthe £), interspersing periods of stability (fixed exchange rates), sudden changes(devaluation) and random fluctuations (floating exchange rate).1.19(a) UsingSt=S0(1 +r)t, henceS0=St/(1 +r)t. SettingSt= 1,000,r= 0.07 givesS0= 712.99. Price after two years: 816.30. Ifrrose to 10%, the bond would fall to1,000/1.13 = 751.31.(b) Theincomestreamshouldbediscountedtothepresentusing()()()()2345200200200200200820.04,11111rrrrr++++=+++++so the bond should sell for£820.04. It is worth more than the previous bond because the return is obtainedearlier.

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual151.20This is the same, in principle, as a compound interest calculation. Using equation(1.26)andsettingSt=3,000,S0=30,000andt=10,weget10300010.2057,30000d=−= −i.e. 20.57% depreciation per annum. The value of themachine after one year is therefore 30,000×(1 − 0.2057) = £23,830. After twoyears, its value is £23,830×(1 − 0.2057) = £18,928, and after five years its value is30,000×(1 − 0.2057)5 = £9,485. You should check that its value is £3,000 after 10years.1.21(a) 17.9% p.a. for BMW; 14% p.a. for Mercedes.(b) Depreciated values are as shown in this table:BMW 525i22,27518,28415,00812,31910,1128,300Mercedes 200E21,90018,83316,19613,92811,97710,300These are close to actual values. Depreciation is initially slower than the average,then speeds up, for both cars.1.22The discounted income stream is()21114004008,000.11rrr×++=×=++Analternative way to see this is to note that if you had £8,000 and invested it at 5%, youwould get 8,000×5% = £400 p.a., that is C×r= Y, where C is the capital sum,risthe interest rate and Y is the annual income. Rearranging this yields C = Y/r. This isa useful concept also in economics and finance. It relates income to wealth and soallows such things as the pricing of equities (discounted future profits). The sharevalue of Tottenham Hotspurs recently (April 1995) fell by £5million when they lostan FA Cup semi-final. This reflects the loss of expected future profits (expectedbecause they might have lost the final). Electricity shares also fell when the regulatorannounced a review of prices.1.23()()().xkxnkxE xkkE xknnn+++===+=+1.24()()()()()222222.kxkkxkxV kxk Vxnnnμμμ−−−====Itisusefulifstudentscangainsomeconfidenceindoingthesekindsofmanipulation – they’re notthatdifficult.1.25The mistake is comparing non-comparable averages. A first-time buyer would havean above-average mortgage and purchase a below-average-priced house, hence theamount of buyer’s equity would be small. The original argument came from theMorning Starnewspaper many years ago. With reasoning like this, no wondercommunism failed.

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual161.26If it is harder to get the popular jobs, then it is possible for Arts students to be moresuccessful at getting both types of job than Science students, yet have a higherunemployment rate. Consider this example:Popular jobsUnpopular jobsTotalapplications successes%success applications successes%success applications successes%successScience2001748740037894.560055292Arts40035087.52001909560054090Arts 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 acareers service report at my university with precisely this flaw. Note that sciencestudentsmaybe better than Arts students; the point is that the figures don’tproveit.Answers to exercises onΣnotation1A.120, 90, 400, 5, 17, 11.1A.240, 360, 1600, 25, 37, 22.1A.388, 372, 16, 85.1A.4352, 2976, 208, 349.1A.5113, 14, 110.1A.656, 8, 48.1A.7().fxkfxkffxkfff−−==−1A.8()()2222222fxxfxfxfxffffμμμμμ−+−−+==222222.fxfxffμμμ=−+=−

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Barrow,Statistics for Economics, Accounting and Business Studies, 7e,’s Manual17Answers to exercises on logarithms1C.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.50.15, 12.58925, 125.8925, 1258.925, 1012.1C.60.8, 199.5262, 1995.2623, 1995262.3.1C.715, 40.77422, 2.71828, 22026.4658.1C.833, 1,202,604.284, 3,269,017.372, 0.36788.1C.93.16228, 1.38692, 1.41421, 0.0005787, 0.008.1C.103.1071, 1.6035, 1.6818, 1, 1, 0.6934.

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