Presentation of dataEHERCISE 2A

The ages of the employees of a company are shown in the following table:

Age (years) Number of employees15 and under 25 2125 and under 35 3735 and under 45 2545 and under 55 1255 and under 65 5Draw a histogram for the above data.

EXERCISE 2B

A company decided to research the price of personal computers on the market. An analysis of advertisements in the press and specialised computer magezines produced the following information:

Price (£) Number of personal computers205 and under 500 5500 and under 1,000 121,000 and under 1,500 151,500 and under 2,000 92,000 and under 3,000 63,000 and under 5,000 3Draw a histogram of the data.

EXERCISE 2C

The staffs of a company are employed in either production, sales or administration. There are 180 production staff, 75 sales staff and 45 administrative staff.

Construct 2 different, suitable, well-label led diagrams to illustrate these figures.

EXERCISE 2D

Advertising expenditure and sales revenue of XYZ company

Year Advertising expenditure(£ millions) Sales revenue (£ millions)1 4 862 5 933 5.5 954 6 1025 7 1126 8 1327 10 1468 12 168

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9 15 18210 17 202EXERCISE 2E

Wealth distribution in the United Kingdom in 1988

Most wealthy (%) Marketable wealth (%)

Marketable wealth plus occupational and State pension rights (%)

1 17 105 38 2410 53 3625 75 6050 94 83All 100 100

Note:

The data is given cumulatively, ie the most wealthy 5% includes the most wealthy 1%.

On the same graph draw Lorenz curves for:

Marketable wealth

Marketable wealth plus occupational and State pension rights

Briefly comment about the distribution of wealth in the UK

EHERCISE 2F

The following table shows the value(in £ thousand) of the sales of a company for 1985 and 1986.

1985 1986January 29.5 32.2February 30.6 36.9March 38.4 42.1April 42.8 50.5May 39.3 41.3June 47.9 45.4July 48.4 51.7August 47.6 53.9September 44.2 48.3October 39.1 45.2November 38.4 33.9December 34.3 38.6Construct a Z chart to represent the data.

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Exercise 2F

1986 1986 Movingdata Cumulative data Annul total

January 32.2 32.2 483.2February 36.9 69.1 489.5March 42.1 111.2 493.2April 50.5 161.7 500.9May 41.3 203.0 502.9June 45.4 248.4 500.4July 51.7 300.1 503.7August 53.9 354.0 510.0September 48.3 402.3 514.1October 45.2 447.5 520.2November 33.9 481.4 515.7December 38.6 520.0 520.0

Monthly sales commission (£) of 100 representatives

Range of commission (£) Number of representatives under 30 230 and under 35 435 and under 40 940 and under 45 1545 and under 50 1950 and under 55 2255 and under 60 1360 and under 65 865 and under 70 570 and over 3 Total 100

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EXERCISE 3A

The ages of the employees of a company are shown in the following table:

Age (years) Number of employees15 and under 25 2125 and under 35 3735 and under 45 2545 and under 55 1255 and under 65 5

Find the mode by both the graphical method and the calculation method. (This data was used in Exercise 2A.)

(Answer: calculation method = 30.7 years )

EXERCISE 3B

The hourly earnings (£) of a random sample of 11 employees of a company were found to be as shown below:

6,9,6,12,8,9,6,7,10,4,11

From the above date find the following measures of location:

median

mode

arithmetic mean

geometric mean.

(Answer: £8, £6, £8, £7.64)

EXERCISE 3C

Using the data of exercise 3B find the:

range

interquartile range

quartile deviation

mean deviation

standard deviation.

(Answers: £8,£4,£2,£2,£2,34)EXERCISE 3D

The 1987 profit (or loss) levels of 100 companies in the food industry are shown in the following table:

Profit (£ millions) Number of companies

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Under-4 0-4 and under 0 7 0 and under 4 21 4 and under 8 35 8 and under 12 2412 and under 16 916 and under 20 420 and over 0(a) Construct a cumulative frequency curve for these figures.

(b) Using your graph, or otherwise, find the values of the median and quartile deviation.

(c) State what the median and quartile deviation measure.

Second Level

(Answers: (b) calculation method £6,514,000;£3,286,000)

EXERCISE 3E

Profit of companies

Profit(£ million)

Number of companies

under 2 302 and under 4 20 4 and under 6 10 6 and under 10 810 and under 15 5(a) Calculate the mean.

Note:

The answer assumes that the lower boundary for the first group is zero.

(Answer: £3,650,000)

EXERCISE 3F

This uses the data of Exercise 3E.

Calculate the standard deviation.

(Answer: £3,270,000)

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Skewness and coefficient of variation EXERCISE 4A

This exercise is based on the data of Exercise 3E.

Profit of companies

Profit (£ million) Number of companiesUnder 2 30

2 and under 4 204 and under 6 10 6 and under 10 810 and under 15 5Calculate the mean and the standard deviation. You could use the results of Exercises 3E and 3F,then calculate the coefficient of variation.

(Answers: £3,650,000,£3,270,000,89.6%

EXERCISE 4B

This exercise is based on the data of Exercise 3E.

Profit of companies

Profit (£million) Number of companies under 2 30 2 and under 4 20 4 and under 6 10 6 and under 10 810 and under 15 5(a) Draw a histogram to represent this information.

(b) Calculate the mean, median and the standard deviation.

(c) Calculate a measure of skewness.

(d) Comment on your histogram and the results of parts(b) and (c).

(Answers: (b)£3,650,000,£2,650,000, £3,270,000; (c) 0.917)

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EXERCISE 4C

This uses the data of Exercise 3D.

The 1987 profit (or loss)levels of 100 companies in the food industry are shown in the following table:

Profit (£ millions) Number of companies under-4 0-4 and under 0 70 and under 4 214 and under 8 358 and under 12 2412 and under 16 916 and under 20 420 and over 0(a) Find the values of the median and the quartiles.

(b) Calculate the quartile measure of skewness.

(Answer: (a) results based on calculation method £6,514,000, £3,429,000,£10,000,000;

(b) 0.061)

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Correlation and regression EXERCISE 5A

A company has a number of branch offices which vary in size and have varying numbers of employees. The following data shows the number of employees at each branch office and the annual cost of running the branch office.

Branch office Number of employees Annual cost (£1,000A 11 64B 6 70C 16 78D 9 62E 25 128F 14 74G 8 60H 19 90(a) Plot the data on a scatter diagram and comment briefly on your diagram.

Part question Second Level

EXERCISE 5B

This exercise uses the data of Exercise 5A and is a continuation of the question.

A company has a number of branch offices which vary in size and have varying numbers of employees. The following data show the number of employees at each branch office and the annual cost of running the branch office.

Branch office Number of employees Annual cost (£1,000)A 11 64B 6 70C 16 78D 9 62E 25 128F 14 74G 8 60H 19 90(b) Calculate the least squares regression line, which would enable you to forecast annual cost for a given number of employees.

(c) Use your regression line to forecast the annual cost of a new office which is about to open and will have 30 employees. Discuss the likely accuracy of your forecast.

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EXERCISE 5C

The following data refers to the floor space and the turnover of 10 small companies:

Company Turnover($1,000) Floor space (square metres)A 85 52B 150 76C 128 62D 160 70E 70 45F 290 78G 180 60H 118 54I 175 68J 136 84(a) Draw a suitable scatter diagram of the data.

(b)Calculate the product moment correlation coefficient.

EXERCISE 5D

Calculate the coefficient of determination for the data of Exercise 5C.

EXERCISE 5E

(a) Ten applicants for a post were given an aptitude test and an interview. E was placed first on the aptitude test followed by A,G,J,C,L,B,D,F and H in that order. In the interviews, G was placed first, followed by E,J,A,B,C,I,H,D and F in that order.

Calculate the value of Spearman’s rank correlation coefficient and, without performing a significance test, interpret the value obtained.

(b) State 2 situations in which Spearman’s rank correlation coefficient should be calculated rather than the product moment correlation coefficient.

EXERCISE 5F

An investigation was conducted by a company on the value of educational and aptitude tests as assessment methods for recruiting employees. It is the present practice of the company to give recruits such tests when they apply for posts.

The following data gives the educational and aptitude test scores, together with an assessment score by the personnel department of their ability one year after joining the company. One is a low score and 20 a high score.

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Employee Educational test Aptitude test Assessment by personnel officer

A 9 17 12B 10 14 14C 15 12 16D 14 13 15E 16 10 17F 11 15 10G 12 12 11H 17 16 18(a) Rank each set of data.

(b) Calculate 2 appropriate rank correlation coefficients.

(c) Write a brief report on your findings.

EXERCISE 5G

(c) Test whether the correlation coefficient found in Exercise 5C(0.8403) differs significantly from zero, and:

(d) Comment on your results.

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Index numberEXERCISE 6A

(a) Give the relative advantages and disadvantages of Laspeyres and Paasche price index numbers.

(b) The following data shows the quantities and prices of 4 components purchased by a company in Year 1 and Year 6.

Year 1 Year 6

Component Price(£) Quantity Price(£) Quantity1 5.00 200 8.50 1802 10.00 150 13.50 1203 8.00 100 10.00 954 12.50 80 15.50 70(i) Calculate a Laspeyres price index which shows the change in prices between Years 1 and 6.

(ii) Calculate a Paasche quantity index which shows the change in quantities between years 1 and 6.

Briefly comment on your results in parts (i) and (ii).

EXERCISE 6B

(a) Give the relative advantages and disadvantages of Laspeyres and Paasche price index numbers.

(b) The following data shows the revenue and prices of 4 products (A to D) sold by a company in Year 1 and Year 4.

Year 1 Year 4

Product Price(£) Revenue(£) Price(£) Revenue(£)A 5.00 400 8.00 640B 8.00 560 14.00 700C 9.00 450 12.00 360D 12.00 288 15.00 300(i) Calculate a Laspeyres price index which shows the change in prices between Years 1 and 4.

(ii) Calculate a Paasche quantity index which shows the change in quantities between Years 1 and 4.

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(iii) Briefly comment on your results in parts (i) and (ii).

EXERCISE 6C

The following data shows the 5 main groups which make up the components of the index of retail prices, together with the weights and price relatives:

Group Weights Price relative Jul 1993 (Jan 1987 = 100)

Food and catering 189 136.8Alcohol and tobacco 113 156.0Housing and household expenditure 336 140.5Personal expenditure 97 127.0Travel and leisure 265 142.9

Source: Monthly Digest of Statistics

(a) Calculate the index of retail prices for July 1993 with January 1987=100.

(b) Outline the uses of the index of retail prices to a business person.

Part question Third Level

EXERCISE 6D

Use the index obtained in the solution to Exercise 6C (140.9).

The index of retail price for January 1987 with January 974=100 was 394.5.

Calculate the index of retail prices for July 1993 with January 1974=100.

EXERCISE 6E

Use the result of Exercise 6D (555.9).

In July 1979 a person retired with a pension of £5,000 per annum. The pension was index linked, ie increased at the same rate of increase as the index of retail prices. In July 1979 the index of retail prices was 229. 1 with January 1974 =100. Calculate the pension of this person in July 1993.

EXERCISE 6F

The average salary in a company and the index of retail prices is given below.

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Year Average salary Index of retail prices(£) (Jan 1987=100)

1988 18,000 106.91989 20,000 115.21990 23,00 126.7(a)Convert the index of retail prices to a series with 1988=100.

(b) Calculate the ‘real’ salary for each year relative to 1988 prices and comment on your result.

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Time series

EXERCISES 7A

(a) Briefly describe the components which make up a time series.

(b) The following data shows the volume of gas (100 millions of thermal units) produced in the United Kingdom from 1990 to 1992.

Gas production 1990-1992

Year Jan-Mar Apr-Jun Jul-Sep Oct-Dec1990 66 37 26 621991 74 43 24 641992 70 37 27 70 Source: Monthly Digest of Statistics

(i) Plot the data on a graph.

EXERCISE 7B

This exercise is a continuation of Exercise 7A.

(a) (ii) Using the data of Exercise 7A, find, by means of a moving average, the trend .

(iii) Plot the trend on the graph found in Exercise 7A.

EXERCISE 7C

This worked example is a continuation of Exercise 7A.

(b) (iv) Using the data of Exercise 7A and your results of Exercise 7B, obtain the seasonal factors using the additive model.

EXERCISE 7D

This exercise is a continuation of Exercise 7A.

(b) (v) Using the data of Exercise 7A and your results of Exercise 7C, obtain the seasonally adjusted figures for the 4 quarters of 1992 using the additive model.

EXERCISE 7E

This exercise is a continuation of Exercise 7A.

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(b) (vi) Using the data of Exercise 7A and your results of Exercise 7C, obtain a forecast for the 4 quartersof 1993 using the additive model. Discuss the likely accuracy of your forecasts.

EXERCISE 7F

(a) Briefly describe the components of a typical time series.

(b) A large shop opens six days a week, Monday to Saturday. The sales for three weeks, in $100,

were recorded as shown below:

Mon Tue Wed Thu Fri SatWeek 1 142 154 98 128 182 285Week 2 151 166 102 141 192 302Week 3 159 172 106 146 199 320(i) Using a moving average, find the trend and the average daily adjustments.

(ii) Forecast sales for the first two days of Week 4, and briefly discuss the likely accuracy of your forecast.

EXERCISE 7G

(a) Explain why it is necessary to adjust seasonally time series data.

(b) The following data shows the output of a factory over a 3-week period. The factory works a 5-day week.

Mon Tue Wed Thu FriWeek 1 86 92 104 98 78Week 2 90 99 110 102 80Week 3 93 101 116 110 83(i) By means of a moving average, find the trend and the average daily variations.

(ii) Seasonally adjust the data for Week 3. What does this show ?

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ProbabilityEXERCISE 8A

A company submits tenders for 4 contracts. The probabilities of acceptance are 0.4, 0.6, 0.5 and 0.2 respectively. Assuming that these are independent events, calculate the probability that the company will:

(i) obtain all 4 contracts, (ii) obtain no contracts, (iii) obtain exactly one contract.

EXERCISE 8B

This is an extra part of Exercise 8A.

(iv) Find the probability that the company will obtain at least one contract.

EXERCISE 8C

A motor insurance company classifies its customers into 4 claims categories: very good (10%), good (30%), average (20%) and poor (40%). The figures in brackets show the percentage of their customers in each classification. The claims record shows that the percentage in each classification making at least one claim over the previous year is as follows:

Claims category Percentage making at least one claim Very good 5%Good 10%Average 30%Poor 60%Assuming that there is no change in these percentages, find the probability that if a customer is selected at random:

(i) the customer will make at least one claim next year.

EXERCISE 8D

This is a continuation of Exercise 8C.

Find the probability that if a customer makes at least one claim, the person will come from the ‘very good’ classification.

EXERCISE 8E

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A company planned to recruit 30 administrative staff. The company recruits from 3 categories; graduates, professionally qualified and those with administrative experience. The company received 200 applications, some not in any of the categories listed above.

An analysis of the applicants showed that 120 were graduates, 40 had professional qualifications and 60 had administrative experience. Of the graduates 20 were professionally qualified, 22 had administrative experience and 5 had professional qualifications and administrative qualifications (these 5 were counted in the earlier categories). Of those with administrative experience 18 had professions (these included the 5 mentioned above).

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