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statistical techniques in business
Stats Means Business 1st Edition John Buglear - Solutions
5.1 An electrical goods retailer sold video recorders to 8200 customers last year and extended warranties to 3500 of these customers. When the retailer sells a video recorder, what is the probability that:(a) The customer will buy an extended warranty?(b) The customer will not buy an extended
Three friends start Higher Education courses at the same time in the same institution.Angela is studying Accounting, Bashir is studying Business and Charlie is studying Computing. Seventy per cent of students who begin the Accounting course complete it successfully, 80 per cent of students who
Twenty-five graduates join a company at the same time. During their induction and training programme friendships are established and they decide that when one of them has a birthday they will all dine at a restaurant chosen by the person whose birthday is being celebrated. One of the graduates asks
A financial services ombudsman is investigating the mis-selling of pension schemes some years previously. Some buyers of these pension schemes were sold the schemes on the basis of misleading information, and would have been better off had they made alternative arrangements.The pension schemes
What is the probability that a player who plays the Wheel of Fortune in Example 5.2 twice wins cash prizes both times?Ten of the thirty-six segments give a cash prize, so the probability of a cash prize in any one game is 10/36.
What is the probability that a customer at the service station in Example 5.3 pays by cash given that he or she has purchased magazines?We will use the abbreviations M for the purchase of magazines and Ca for cash.
A promotional stall in a shopping centre offers passers-by the opportunity to taste bison meat. A total of 200 people try the product and 122 of them stated that they liked it. Of these, 45 said they would buy the product. Overall 59 of the 200 passers-by said they would buy the product.Are liking
Use the multiplication rule to calculate the probability that a customer at the service station in Example 5.3 purchases sandwiches and pays by cash.
Deduce the probability that a prospective homebuyer in Example 5.8 expresses no preference using the simple probabilities of the other outcomes. P (Prospective homebuyer makes no choice) = 1-P(A)-P(B)-P(C) = 1-43/178-61/178-29/178 = 1 0.2416 0.3427-0.1629 = 1-0.7472 = 0.2528 or 25.28 per cent The
What is the probability that one of the prospective homebuyers in Example 5.8 chooses the ‘Ambience’ or the ‘Bermuda’ or the ‘Casa’ or expresses no preference? For convenience we will use the letter N to denote the latter.
One weekend a total of 178 prospective homebuyers visit a new housing development.They are offered a choice of three different types of house: the 2-bedroom ‘Ambience’, the 3-bedroom ‘Bermuda’, or the 4-bedroom ‘Casa’. When invited to select the type of house they would most like, 43
Use the addition rule to calculate the probability that a customer coming into the service station in Example 5.3 purchases fuel or pays by debit card.Applying the addition rule:P (F or D) = P (F) + P(D) – P (F and D)
Use the data in Example 5.3 to find the probability that a customer who has purchased magazines pays by cash.Another way of describing this is as the probability that a customer pays by cash given that he or she has purchased magazines. It is represented as:P (Ca|M)
Use the data in Example 5.3 to find the probability that a customer purchases fuel or pays by debit card.The probability that one (and by implication, both) of these outcomes occurs is based on the relative frequency of all the people who are in one or other category. This implies that we should
What is the probability that a customer in the service station in Example 5.3 purchases sandwiches and pays by cash?
A survey of the type of goods purchased and methods of payment of 500 customers at a service station produced the following results.What is the probability that a customer will pay by credit card?What is the probability that a customer will purchase fuel? Payment Sandwiches Magazines Fuel Total
A ‘Wheel of Fortune’ machine in an amusement arcade has thirty-six segments. Ten of the segments would give the player a cash prize. What is the probability that you win a cash prize if you play the game?
The horse ‘Starikon’ has entered sixteen races and won five of them. What is the probability that it will win its next race?
4.20 Select the appropriate definition for each term on the lefthand side from the list on the right hand side. (a) A regression equation (b) A correlation coefficient (c) A moving average (d) An inverse relationship (e) A predictor (f) A response (g) A trend (h) A weighted price index (i) is
4.19 A High Street chemist sells travel first aid packs. The numbers of these sold over three years were:(a) Plot a graph to portray the data.(b) Calculate centred four-point moving averages for the time series.(c) Determine the seasonal component for each quarter.(d) Compile forecasts for the four
4.18 The amounts of gas (in thousands of gigawatt hours) sold to domestic consumers by a regional energy company over three years were:(a) Produce a graph to represent the figures.(b) Determine centred four-point moving averages for the time series.(c) Find the seasonal component for each
4.17 A body-piercing and tattoo studio is open each day of the week except Sundays and Mondays. The number of customers visiting the studio per day over a period of three weeks is:(a) Plot the time series.(b) Calculate five-point moving averages for the series.(c) Estimate daily components for the
4.16 Sales of alcoholic beverages at an off-licence during the course of three days were:(a) Plot a graph to display the time series.(b) Calculate three-point moving averages for the series.(c) Determine the recurrent components for each part of the day.(d) Find estimates for the values of the
4.15 An enthusiast paid £5000 for a classic car in 1990. Since that time the car has been kept carefully and been valued every two years. The valuations were:Year 1992 1994 1996 1998 2000 Valuation (£) 5800 5200 5500 6200 6000 The values of the Retail Price Index (RPI) for these years were:Year
4.14 The turnover figures provided in the annual accounts of a large retail grocer over the six years from 1995 to 2000 were:Year 1995 1996 1997 1998 1999 2000 Turnover (£m) 7022 7101 7350 7844 8249 8598 The values of the Retail Price Index (RPI) for this period were:Year 1995 1996 1997 1998 1999
4.13 A confectioner buys cocoa, cocoa butter, sugar, and milk solids. The prices per kilogram of these ingredients in 1994, 1997, and 2000 were:The quantities that were purchased in 1994 were 7500 kg of cocoa, 4200 kg, of cocoa butter, 12 000 kg of sugar, and 5700 kg of milk solids. The purchased
4.12 A textile manufacturer makes casual jackets. The company buys lining fabric, interfacing fabric and outer fabric. These fabrics are cut and machined to make the garments. The prices per metre of these materials in 1998, 1999 and 2000 were:In 1998 the company purchased 2500 metres of lining
4.11 A cab driver pays for fuel, vehicle servicing and maintenance every 3 months, and an annual operating licence.The prices of these in 2000 and in 1996, together with the quantity of each that was purchased during 1996 are:(a) Calculate the weighted aggregate price index for the prices in 2000
4.10 A pizza manufacturer purchases cheese, pepperoni, and tomato paste. The prices of these ingredients in 1997, 1999, and 2001 were:(a) Calculate a simple aggregate price index for the prices in 1999 and 2001 using 1997 as the base period.(b) What additional information would you need in order to
4.9 The classified advertisement columns of a local paper contain details of twelve used cars of a particular make and model. The prices of these cars (in £000) and the numbers of miles they have travelled (in 000) are:Price 6.8 6.2 5.7 4.7 6.0 5.9 7.0 4.5 5.0 3.0 3.8 3.5 Mileage 9 13 11 33 14 11
4.8 In a survey of commuting patterns ten respondents say that they cycle to work. The distances they travel (in miles)and the mean journey times (in minutes) they report are:Distance 2 2 3 4 5 6 7 8 8 10 Mean journey time 10 15 15 20 25 25 30 40 45 50(a) Portray the data on a scatter diagram.(b)
4.7 Eight prominent football teams play in the same division.The total number of goals scored by each team last season and the league position each was in at the end of the season are:Goals scored 75 60 56 44 43 40 37 52 League position 2 5 9 10 14 16 17 22(a) Produce a scatter diagram to display
4.6 A consumer group has tested ten makes of personal stereo.The prices and the score (out of 100) awarded to them by a panel of experts the group commissioned to test them, are:Price (£) 95 69 18 32 27 70 49 35 50 29 Score 74 63 28 33 37 58 38 43 50 31(a) Plot a scatter diagram to portray the
4.5 The weekly turnover and the total display area, in square metres, of eight ‘late night’ grocery shops are:Turnover (£000) 23 37 33 41 47 86 72 95 Display area (m2) 15 21 30 45 61 77 79 92(a) Identify which variable is the dependent variable.(b) Plot a scatter diagram to portray these
4.4 Car parking spaces in a city centre are in short supply. The cost of parking a car for a whole day in ten public car parks within the city (in £) and the distances between the car parks and the monument that is always considered the centre of the city (in km) are:Cost 5 5 7 8 10 12 15 15 18 20
4.3 The outstanding balances on the monthly bills of 12 credit card accounts (in £) and the household income of the account holders (in £000) are:Balance: 250 1630 970 2190 410 830 0 550 0 682 0 0 Income: 15 23 26 28 31 35 37 38 42 42 45 46 Portray these figures on a scatter diagram taking the
4.2 The cost of placing a full-page colour advertisement (in£000) and the circulation figures (in £000) of nine magazines are:Cost: 9 43 16 17 19 13 20 44 35 Circulation: 135 2100 680 470 450 105 275 2250 695(a) Which of these variables should be the dependent variable, and why?(b) Plot a scatter
4.1 Consider which of the relationships below are likely to have a positive and which are likely to have a negative correlation coefficient.(a) The distance a vehicle travels and the fuel consumed(b) The demand for laptop computers and their price(c) The average temperature of countries and the
Find a trend line equation for the moving averages in Example 4.15 and use the equation to predict the trend values on day 4 and construct forecasts for day 4 by adding the recurrent components from Example 4.18.
Find the error components for the data in Example 4.13 using the table produced in Example 4.17 and the recurrent components from Example 4.18.
Find the estimates for the recurrent daily components from the figures in Example 4.17.What do they tell us about the pattern of newspaper sales?
Subtract the centred moving averages in Example 4.14 from the observations in Example 4.13.
Calculate and centre the moving averages for the data below. They are the sales of beachwear (in £000) in a department store over two years.Winter Spring Summer Autumn Year 1 14.2 31.8 33.0 6.9 Year 2 15.3 34.7 36.2 7.3 Moving averages for these figures will be four-point moving averages.First MA
Centre the moving averages in Example 4.14. Day Time Revenue Moving average 111222333 am 320 pm 92 210 eve 218 217 2 am 341 220 2 pm 101 222 2 eve 224 228 am 359 233 pm 116 249 eve 272
Calculate moving averages for the data in Example 4.13.and so on. The first MA = (320 +92 +218)/3 = 630/3 = 210 = The second MA (92 +218 + 341)/3 = 651/3 = 217 The third MA = (218341+101)/3 = 660/3: = 220
The revenue (in pounds) from newspaper sales at a new service station for the morning, afternoon and evening periods of the first three days of operation are:Morning Afternoon Evening Day 1 320 92 218 Day 2 341 101 224 Day 3 359 116 272 Construct a time series chart and examine it for evidence of a
The annual salary of the manager of a sports goods store has changed in the following way between 1997 and 2000. Use the RPI figures for those years to see whether the increases in her salary have kept up with the cost of living.1997 1998 1999 2000 Salary (£000) 24 26 27 28 RPI (1987 = 100) 157.5
In 2000 the garage purchased 50 exhaust systems, 600 batteries, and 750 tyres. Use these figures and the price figures from Example 4.9 to produce a Paasche price index to compare the prices of 2000 to those of 1997.This result suggests that the prices have increased by 25.0 per cent between 1997
The garage records show that in 1997 50 exhaust systems, 400 batteries, and 1000 tyres were purchased. Use these figures and the price figures from Example 4.9 to produce a Laspeyre price index to compare the prices of 2000 to those of 1997.q0pcq0p0× 100 =(50 × 156) + (400 × 35) + (32 ×
The garage proprietor in Example 4.8 regularly buys exhaust systems, car batteries, and tyres. The prices of these goods in 2000 and 1997 were:1997 2000 Exhaust system £125 £156 Battery £25 £35 Tyre £28 £32 Calculate a simple aggregate price index to compare the prices in 2000 to the prices
Full exhaust systems cost a garage proprietor £156 each in 2000. They cost £125 in 1997.Calculate a simple price index to represent the change in price over the period.
Use the regression equation from Example 4.6 to find how many new jackets the shop can expect to sell if they are priced at £23.50.The regression equation tells us that: Number sold = 16.6838 – 0.507353 Price If we put the figure ‘23.5’ in where the word ‘Price’ appears in the equation
Produce a fitted line plot for the data in Example 4.1.
Find the equation of the line of best fit for the data in Example 4.1.We need to find four summations; the sum of the x values, the sum of the y values, the sum of the x squared values and the sum of the products of each pair of x and y values multiplied together. We also need to know n, the number
The sales of ice-cream in an American-style ice-cream shop (in £000) and the midday temperatures (in degrees Celsius) for three days were:Sales (Y) 4 3 6 Temperature (X) 15 20 25 Which of the two lines best fits the data, the one in Figure 4.9 or the one in Figure 4.10?The deviations between the
Calculate the coefficient of determination, R2, for the data in Example 4.1.
Calculate the correlation coefficient for the data in Example 4.1.We need to calculate the sample standard deviations for X and Y. Price (x) (x - x) (x-x) Number sold (y) (y-y) (y- y) 865227 750237 4 4 4 4 4 4 421223 0 16 4 13 222222
A shop retails six brands of light shower-proof jacket. The prices in pounds and the numbers sold in a week are:Price 18 20 25 27 28 32 Number sold 8 6 5 2 2 1 Plot a scatter diagram and calculate the covariance.In Figure 4.1 Number sold has been plotted on the Y or vertical axis and Price has been
3.20 Select which of the statements on the right-hand side best define the words on the left-hand side.(a) Median (i) the square of the standard deviation(b) Range (ii) a diagram based on order statistics(c) Variance (iii) the most frequently occurring value(d) Boxplot (iv) the difference between
3.19 A film-processing shop promises to deliver photographs in half an hour. The mean and standard deviations of the processing times are 22 minutes and 3 minutes respectively.The layout of the machines in the shop has been altered. The processing times of the first ten films to be developed after
3.18 The prices of a pint of ordinary bitter in each of 30 public houses in one city in the North of the UK and another city in the South of the UK were recorded and the following boxplot were produced to portray the two distributions.Look carefully at the diagram and say whether each of the
3.17 A report on usage of glass recycling bins contains the following grouped relative frequency distribution:(a) Compile a cumulative relative frequency graph for this distribution.(b) Determine approximate values of the median, quartile and semi-interquartile range using your graph. Weight of
3.16 The credit balances in the current accounts of customers of a bank are summarized in the following grouped relative frequency distribution:Plot a cumulative relative frequency graph to portray this distribution and use it to find approximate values of the median, quartiles and
3.15 The amounts (in pounds) spent in a month by 42 women on hair-care products were 13.23 11.19 14.49 6.51 10.07 18.91 13.14 15.90 16.11 12.89 9.12 12.27 8.16 10.84 9.33 10.36 15.02 8.45 12.92 6.79 13.02 6.85 13.76 8.92 11.40 10.88 8.99 9.46 8.67 8.52 11.17 5.79 9.17 9.16 5.24 7.11 7.37 10.60
3.14 The costs of work done at a garage on 33 vehicles to enable them to pass the MOT test of roadworthiness were 482 471 277 230 357 491 213 386 357 141 282 184 324 426 408 213 155 287 415 499 470 461 233 314 240 107 113 314 242 112 289 283 3 Identify the median and quartiles of this set of data
3.13 The stem and leaf display below shows the Friday night admission prices for 31 clubs.Find the values of the median and semi-interquartile range. Stem 0 0 0112 Leaves 44 5555677789 000224444 5555588 002 Leaf unit = 1.0
3.12 The total spend of a sample of 110 customers of the Peesharsupermarket and the total spend of a sample of 128 customers of the Peevar supermarket were analysed and the following grouped frequency distribution produced:(a) Find approximate values for the mean and standard deviation of each
3.11 The time in seconds that a sample of 79 callers trying to contact an insurance company had to wait was recorded.After introducing new procedures the waiting time for a sample of 61 callers was recorded. The results are presented in the following grouped frequency distribution:Waiting
3.10 The playing times of a sample of 57 contemporary pop albums and a sample of 48 reissued classic pop albums are summarized in the following grouped frequency distributions:Playing time(minutes)Frequency(contemporary)Frequency(reissue)30 and under 35 0 4 35 and under 40 7 9 40 and under 45 13 17
3.9 The kilocalories per portion in 32 different breakfast cereals were recorded and collated into the following grouped frequency distribution:Kcal per portion Frequency 80 and under 120 3 120 and under 160 11 160 and under 200 9 200 and under 240 7 240 and under 280 2 Determine approximate values
3.8 Two friends want to take a summer holiday before going to college in the autumn. They are looking for somewhere with plenty of clubs where they can party all night. Unfortunately they have left it rather late to book and there are only two resorts, Medlena and Bistry, available within their
3.7 Three companies supply the domestic electricity supply market in a particular region; Iskra, Moogom, and Provod.Each company has produced an analysis of the distribution of the size of its customers’ bills over the last financial year.The following results have been published:Are the
3.6 Two neighbours work at the same place. One travels to work by bus, the other cycles to work. The times taken (in minutes) by each to get to work on a sample of 8 days were Bus passenger 33 28 40 32 41 32 38 42 Cyclist 26 33 27 31 31 30 28 2 Calculate the mean and standard deviation for each set
3.5 The crowd sizes for the 22 home league games played by Athletico Almaz were 1976 2162 1502 1782 1523 2033 1564 1320 1951 1714 1841 1648 1345 1837 1718 2047 1954 2000 1479 2571 1739 1781 The crowd sizes for the 22 home fixtures played by a rival club, Red Star Rubine, were 1508 2055 2085 2098
3.4 A supermarket has one checkout for customers who wish to purchase 9 items or less. The number of items presented at this checkout by a sample of 19 customers were 5 8 7 7 6 6 10 8 9 9 9 6 5 9 8 9 5 5 6(a) Find the median and quartiles for this set of data.(b) Determine the semi-interquartile
3.3 Twenty-six dental patients require the following numbers of fillings during their current course of treatment 2 3 2 2 3 1 2 2 1 3 2 2 2 2 4 3 2 2 2 2 2 1 1 0 1 1(a) Identify the mode, find the median, and calculate the mean of these figures.(b) Compile a bar chart to display the distribution.
3.2 A supermarket sells kilogram bags of apples. The number of apples in 22 bags were 7 9 8 8 10 10 8 10 10 8 8 10 7 9 9 9 7 8 7 8 9 8(a) Find the mode, median and mean for this set of data.Compare your results.(b) Plot a simple bar chart to portray the data.
3.1 The numbers of credit cards carried by 25 shoppers are 3 5 2 0 4 3 0 1 1 7 1 4 1 2 9 4 1 4 1 5 5 2 3 1 1(a) Find the mode and median of this distribution.(b) Calculate the mean of the distribution and compare it to the mode and median. What can you conclude about the shape of the
A ‘while-you-wait’ shoe repair service offers to replace certain types of heels on ladies’shoes in three minutes. Long experience has shown that the mean replacement time for these heels is 2.8 minutes and the standard deviation is 0.15 minutes. Construct a control chart using these figures
Use the cumulative relative frequency graph shown in Example 3.13, Figure 3.3, to estimate the values of the lower and upper quartiles for the distribution and produce an approximate value of the semi-interquartile range.
Find the approximate value of the standard deviation of the data represented in the grouped frequency distribution in Example 3.12. Cost of calls Midpoint (x) Frequency (f) fx x fx 5 and under 10 7.5 10 and under 15 12.5 15 and under 20 20 and under 25 17.5 22.5 25 and under 30 27.5 f = 23 150612
Find the mean and the standard deviation of the data from Example 3.14 and compare the results for the two agencies. Agency A Mean (0+4+4+5+7+ 8+ 10 + 11+ 15)/9 = 7.11 Experience (x) Mean (x) (x-x) (x-x) 11 15 044578US 7.11 -7.11 50.55 7.11 -3.11 9.67 7.11 -3.11 9.67 7.11 -2.11 4.45 7.11 -0.11 0.01
Calculate the standard deviation for the data in Example 3.21.The sum of squared deviations is 10 and the number of observations is 6, so the standard deviation of this sample is:s = 10/5 = 2 = 1.414 to three decimal places
Find the sum of the squared deviations, (x – ¯x)2, from the mean for the data in Example 3.21. 414010 Observation (x) Mean (x) Deviation (x - x) Squared deviation (x-x) -2 2 212010 444444 256434
Six newsagents sell the following number of boxes of plain potato crisps in a particular week 2 5 6 4 3 4 The mean, ¯x = 24/6 = 4
Find the SIQR for each of the two sets of data in Example 3.14.There are nine observations in each distribution, so the median position is (9 + 1)/2 = 5th in both cases. The quartile position is (5 + 1)/2 = 3rd position.Agency A 0 4 4 5 7 8 10 11 15 Agency B 0 0 4 4 7 10 10 14 15
If lowest value in the set of monthly costs of calls for male owners was wrongly recorded as £9 but was actually £4, how does the boxplot change?
Produce boxplots for the monthly costs of calls for females and males.
Find the SIQR of the data in Example 3.5 and compare this to the SIQR of the data in Example 3.15.Array 9 12 13 14 14 14 15 15 15 16 17 17 17 17 19 19 20 20 21 21 22 24 27 There are 23 observations, so the median position is the (23 + 1)/2 = 12th position.The quartile position is the (12 + 1)/2 =
Find the semi-interquartile range for the data in Example 3.15.The lower quartile monthly cost of calls is £10 and the upper quartile monthly cost of calls is £24.5.SIQR = (£24.5 – £10)/2 = £14.5/2 = £7.25
In one month the total costs (to the nearest pound) of the calls made by 23 female mobile phone owners were 14 5 15 6 17 10 22 10 12 17 13 29 7 27 33 16 30 9 15 7 33 28 21 Find the median and upper and lower quartiles for this distribution.Array 5 6 7 7 9 10 10 12 13 14 15 15 16 17 17 21 22 27 28
Two independent estate agencies each employ 9 people. The number of years’ experience in the property sector that the employees of these companies have is Agency A 0 4 4 5 7 8 10 11 15 Agency B 0 0 4 4 7 10 10 14 15 Find the range of each set of data and compare them. Plot histograms for each set
Draw a cumulative relative frequency graph to represent the grouped frequency distribution in Example 3.12 and use it to find the approximate value of the median monthly cost of calls. Cost 5 and under 10 10 and under 15 15 and under 20 20 and under 25 25 and under 30 Frequency Cumulative frequency
Use Figure 3.2 to find the modal class, median and mean of the monthly costs of calls.The grouped frequency distribution used to construct Figure 3.2 was:Cost Frequency 5 and under 10 1 10 and under 15 5 15 and under 20 10 20 and under 25 6 25 and under 30 1 The modal class is ‘15 and under 20’
Use Figure 3.1 to find the mode, median and mean of the distribution of days absence through illness.Figure 3.1 shows the frequency with which each number of days absence occurs, in the form of a bar. By checking the height of the bar against the vertical axis we can tell exactly how many times
Construct a stem and leaf display to show the data in Example 3.5. Use the display to find the mode, median and mean of the distribution.Stem-and-leaf of cost of calls n = 23 Leaf unit = 1.0 0 9 1 2 3 4 4 4 1 5 5 5 6 7 7 7 7 8 9 9 2 0 0 1 1 2 4 2 7 The modal value is 17, the leaf digit ‘7’
Produce a histogram to portray the data in Example 3.5. Find the median and compare it to the mean.See Figure 3.2.There are 23 observations so the median is the (23 + 1)/2 = 12th observation.Array 9 12 13 14 14 14 15 15 15 16 17 17 17 17 19 19 20 20 21 21 22 24 27
One of the observed values in the data in Example 3.6 has been recorded wrongly. The figure ‘6’ should have been ‘2’. How does this affect the values of the mode, median and mean?
Produce a histogram to display the data from Example 3.6 and comment on the shape of the distribution.
The numbers of days that 16 employees were absent through illness were:1 1 6 0 2 1 1 4 0 2 4 1 4 3 2 1 Find the mode, median and mean for this set of data.The modal value is 1, which occurs six times.Array 0 0 1 1 1 1 1 1 2 2 2 3 4 4 4 6 The median position is: (16 + 1)/2 = 8.5th position,
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