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statistical techniques in business
Stats Means Business 1st Edition John Buglear - Solutions
On average 74.9 units of a brand alco-pop drink are sold at a city-centre pub each week.Sales are Normally distributed. The manager puts on a special promotion to boost sales of the product. During the five weeks following the promotion the mean sales were 82.4 units per week with a standard
A hair products company claims that a hair colouring treatment it produces lasts for an average of 45 days before fading. The company wants to check that its claim is valid. The hair of a random sample of 40 customers who had applied the treatment was monitored and the time that elapsed until the
According to the label on packets of mixed nuts there should be 25 grams of cashew nuts in every packet. The standard deviation of the weight of cashew nuts per packet is known to be 2.2 grams and the weights are Normally distributed.The mean weight of cashew nuts in a random sample of 15 packets
Use the sample mean in Example 8.12 to test the hypothesis that the tours in Example 8.10 take at least 60 minutes, H0: 60, at the 5 per cent level of significance.H0: 60 minutes H1: < 60 minutes The level of significance, = 0.05 The sample mean is 61.87 and in Example 8.12 we found
Test the hypothesis that the population mean duration of the bus tours in Example 8.10 is 60 minutes, H0: = 60. Use the sample mean given in Example 8.12 and apply a 5 per cent level of significance.H0: = 60 minutes H1: 60 minutes The level of significance, = 0.05 From Example 8.12 we
The mean of the random sample in Example 8.11 is 61.87 minutes. What is the z-equivalent of this sample mean, assuming it belongs to a sampling distribution with a mean of 60 and a standard error of 0.85? Use the z-equivalent to find the probability that a sample mean of this magnitude comes from
The standard deviation of the bus tours in Example 8.10 is known to be 6 minutes. If the duration of a random sample of 50 tours is to be recorded in order to investigate the operation, what can we deduce about the sampling distribution the mean of the sample belongs to?
A bus company promotes a ‘one-hour’ tour of a city. Suggest suitable null and alternative hypotheses for an investigation by:(a) A passenger who wants to know how long the journey will take.(b) A journalist from a consumer magazine who wants to see whether passengers are being cheated.
How many supermarkets would have to be included in the sample in Example 8.8 if the confidence interval of the proportion of establishments with trolleys suitable for shoppers with limited mobility has to be within 5 per cent of the actual population proportion with a 95 per cent degree of
A study of a sample of 110 supermarkets reveals that 31 offer trolleys suitable for shoppers with limited mobility. Construct a 95 per cent interval estimate of the proportion of all supermarkets that have these trolleys.
A random sample of 15 employees of a call centre was taken and each employee took a competency test. The mean of the scores achieved by these employees was 56.3 per cent with a standard deviation of 7.1 per cent. Results of this test have been found to be Normally distributed in the past. Construct
The mean weight of the cabin baggage checked in by a random sample of 40 passengers at an international airport departure terminal was 3.47 kilograms. The sample standard deviation was 0.82 kilogram. Construct a 90 per cent confidence interval for the mean weight of cabin baggage checked in by
If the researchers in Example 8.1 want to construct 99 per cent confidence intervals that are £5 wide, what sample size should they use?
Use the sample result in Example 8.2, £47.13, to produce a 98 per cent confidence interval for the population mean.
Use the sample result in Example 8.2, £47.13, to produce a 99 per cent confidence interval for the population mean.
The researchers in Example 8.1 take a random sample that yields a mean of £47.13.Calculate a 95 per cent confidence interval using this sample mean.47.13 ± (1.96 × 1.275) = 47.13 ± 2.50 = £44.63 to £49.63 to two decimal places This interval does not include the population mean of £50.
The total bill sizes of shoppers at a supermarket have a mean of £50 and a standard deviation of £12.75. A group of researchers, who do not know that the population mean bill size is £50, finds the total bill sizes of a random sample of 100 shoppers.
7.20 Select the appropriate description for each term on the lefthand side from the list on the right-hand side.(a) the Normal distribution(i) the z value for which P(Z > z) = 0.05(b) the Z distribution (ii) the estimated standard error(c) z (iii) a distribution used for small samples(d) z0.05
7.19 The times taken by buses to travel between two stops on a route follow a Normal pattern of variation with a mean of 13.9 minutes. A commuter makes this journey ten times each week. One week she times the ten journeys and works out that the standard deviation of them is 1.7 minutes.Treating the
7.18 The contents of pots of a certain brand of yoghurt are Normally distributed with a mean of 101.4 grams. The yoghurts are sold in packs of four. The contents of each pot in one pack of four were measured and the standard deviation was found to be 3.1 grams.(a) What mean amount of content will
7.17 The amount of analgesic per pill in a proprietary brand of painkiller follows a Normal pattern with a mean of 7.5 mg.The pills are sold in bottles of 50. If the standard deviation of the analgesic content of the 50 pills in one bottle is 0.4 mg, what is the probability that the mean analgesic
7.16 The delays to scheduled airline departures at an international airport are known to follow a skewed distribution with a mean of 11 minutes and a standard deviation of 7 minutes. What is the probability that the mean delay of a random sample of 40 flights is:(a) More than 12 minutes?(b) More
7.15 Multi-Purpose Vehicles (MPVs) produced by a manufacturer can accommodate seven adults. The load capacity of the passenger compartment is 1240 lb. If the weights of adults are distributed Normally with a mean of 170 lb and a standard deviation of 18 lb, what is the probability that the mean
7.14 A large administrative centre is replacing its IT facilities. As part of this operation all standard client files are to be stored temporarily on floppy disks as a precaution. The disk spaces that these files require are Normally distributed with a mean of 0.20 mb and a standard deviation of
7.13 As part of a charity event a radio DJ plans to broadcast a selection of 12 tracks chosen at random from the very large stock of music CDs stored at the radio station. The lengths of the tracks on these CDs are Normally distributed with a mean of 3 minutes 9 seconds and a standard deviation of
7.12 The playing time of C180 videocassettes produced by one manufacturer varies Normally with a mean of 182.4 minutes and a standard deviation of 1.3 minutes. What proportion of packs of five of these videocassettes will have a mean playing time of:(a) Longer than 183 minutes?(b) Longer than 182
7.11 The mean legal lifetime (the number of miles travelled before the tyre is worn down to the legal limit) of car tyres of a certain brand is 23 450 miles. The standard deviation is 1260 miles. If the lifetimes of the tyres are Normally distributed, what is the probability that a random sample of
7.10 The amount of time that visitors to a web site browse the site is assumed to be Normally distributed with a mean of 8.75 minutes and a standard deviation of 3.1 minutes. What is the probability that a randomly selected visitor browses the site for:(a) Less than 5 minutes?(b) More than 10
7.9 A drinks vending machine dispenses a variety of hot and cold beverages. The volumes of the drinks dispensed follow a Normal pattern with a mean of 149.7 ml and a standard deviation of 1.2 ml.(a) What proportion of drinks will be over 150 ml?(b) What proportion of drinks will be over 147.5
7.8 The times taken by a traditional barber to cut hair are Normally distributed with a mean of 15.6 minutes and a standard deviation of 3.4 minutes. What is the probability that he can cut a customer’s hair in:(a) Less than 20 minutes?(b) Less than 10 minutes?(c) Between 8 and 14 minutes?(d)
7.7 The weight losses that members of a slimming club can expect to achieve follow a Normal distribution with a mean of 24.7 lb and a standard deviation of 5.2 lb.(a) What proportion of members can expect to lose more than 25 lb?(b) What proportion of members can expect to lose more than 20 lb?(c)
7.6 A bed linen company manufactures duvets that have a mean TOG rating (a measure of thermal effectiveness) of 11.5 with a standard deviation of 0.18. If the TOG ratings are Normally distributed, what is the probability that a duvet selected at random has a TOG rating:(a) Above 12?(b) Above
7.5 A large company insists that all job applicants who are invited for interview take a psychometric test. The results of these tests follow a Normal distribution with a mean of 61 per cent and the standard deviation of 7.2 per cent.(a) What proportion of applications would be expected to score
7.4 An automobile manufacturer produces a certain model of car. The fuel economy figures of these cars are Normally distributed with a mean mileage per gallon (mpg) of 36.8 and a standard deviation of 1.3.(a) What is the probability that one of these cars will have an mpg of more than 37.5?(b) What
7.3 The colour dye used in the manufacture of certain garments fades after a mean of 96 machine washes, with a standard deviation of 7 washes. If the number of washes before fading follows a Normal distribution, what proportion of the garments will first fade:(a) After 100 washes?(b) Before 90
7.2 A confectionery company produces 100-gram ‘Real Chocolate’bars that have a mean chocolate content of 70 grams with a standard deviation of 0.8 gram. The variation in chocolate content follows a Normal distribution. What is the probability that a chocolate bar chosen at random contains:(a)
7.1 Find the areas of the Standard Normal Distribution that represent the following probabilities:(a) P (Z > 1.44)(b) P (Z > –0.29)(c) P (Z < 2.06)(d) P (Z < –1.73)(e) P (0.52 < Z < 1.99)(f) P (–2.31 < Z < –1.08)(g) P (–0.97 < Z < 0.65)
Use Table 4 to find:(a) t with four degrees of freedom that cuts off a tail area of 0.10, t0.10,4(b) t with ten degrees of freedom that cuts off a tail area of 0.01, t0.01,10(c) t with seventeen degrees of freedom that cuts off a tail area of 0.025, t0.025,17(d) t with one hundred degrees of
A customer visiting the pub in Example 7.10 purchases nine pints of real ale. The volumes of the pints served are known to be Normally distributed with a mean of 0.564 litres and the standard deviation of the volumes of the nine pints bought by the customer is 0.048 litre. What is the probability
The mean volume of draught beer served in pint glasses in a particular pub is known to be 0.564 litres. A consumer organization takes a random sample of 64 pints of draught beer and finds that the standard deviation of this sample is 0.025 litre. What is the probability that the mean volume of the
The times that passengers at a busy railway station have to wait to buy tickets during the rush hour follow a skewed distribution with a mean of 2 minutes 46 seconds and a standard deviation of 1 minute 20 seconds. What is the probability that a random sample of 100 passengers will, on average have
A group of four friends visit the pub in Example 7.6 and each of them orders a Steak Special. What is the probability that the mean uncooked weight of the steaks they order is more than 10 ounces?
In the population of uncooked steaks in Example 7.6, what is:(a) The minimum weight of the heaviest 20 per cent of steaks?(b) The maximum weight of the lightest 10 per cent of steaks?
A pub offers a ‘10 oz Steak Special’. If the steaks they use for these meals have uncooked weights that are Normally distributed with a mean of 9.8 ounces and a standard deviation of 0.5 ounces, what is the probability that a customer will get:(a) A steak that has an uncooked weight of more
Use Table 3 on pages 334–335 to find the specific value of Z, which we will call z, so that the area to the right of z, the probability that Z is greater than z, P (Z > z), is:(a) 0.4207(b) 0.0505(c) 0.0250
Use Table 3 on pages 334–335 to find the following:(a) The probability that Z is greater than 0.7, P (Z > 0.7).(b) The probability that Z is less than 0.7, P (Z < 0.7).(c) The probability that Z is greater than 2.27, P (Z > 2.27).(d) The probability that Z is greater than –1.35, P (Z >
Use the list of probabilities in Example 7.2 to find the proportion of customers that take:(a) Between 15 and 21 seconds(b) Between 12 and 24 seconds to use the cash dispensers.
Find the probabilities that customers of the bank in Example 7.1 take:(a) Less than 12 seconds(b) Less than 15 seconds(c) Less than 18 seconds(d) Less than 21 seconds(e) Less than 24 seconds to use the cash dispensing machines.You can do this by listing these values in a column in the MINITAB
According to a bank, the time taken by its customers to use cash dispensing machines is Normally distributed with a mean of 18 seconds and a standard deviation of 3 seconds.What is the probability that a customer selected at random takes less than 13 seconds?
6.20 Select the appropriate description for each term on the lefthand side from the list on the right-hand side.(a) cumulative probability (i) consists of probabilities of incidents over time(b) expected value (ii) the expected monetary value of a decision(c) binomial distribution (iii) the
6.19 Eagraville United are the last lower-division football club in the quarter-finals of a major cup competition. The other teams are all prominent clubs with large numbers of supporters. The commercial manager at Eagraville has to place the order for programmes to sell at the game before the draw
6.18 An arable farmer is thinking of sowing scientifically modified crops next year. She believes that if she did so her profits would be £75 000, compared to £50 000 if she sowed unmodified crops. A neighbouring farmer has made it clear that if his crops are contaminated he will demand
6.17 A member of a successful girl band is considering leaving the band to pursue a solo career. If she stays with the band she estimates that there is a 60 per cent chance that the band will continue to be successful and she would earn£1.7 m over the next three years. If the band’s success does
6.16 A sportswear company markets a premium brand of trainers. At present the revenue from sales of these trainers is £17 m a year. They have been negotiating with representatives of a top US sports star in order to obtain his endorsement of the product. The cost to the company of the endorsement
6.15 A film production company is about to release a new movie. They estimate that there is 5 per cent chance that the film will make a profit of $14 m, a 30 per cent chance that it will make a $1 m profit, a 25 per cent chance that it will break even, and a 40 per cent chance that it will make a
6.14 An insurance company calculates that the probability that in a year a motor insurance policyholder makes a claim arising from a major accident is 0.03, the probability that he or she makes a claim as a result of a minor accident is 0.1, and the probability that he or she makes a claim as a
6.13 A graduate wants to pursue a career in a profession. She will need to gain membership of the professional institute by passing three stages of examinations. If she becomes a full member of the institute she anticipates that she will be able to earn £50 000 per year. If she fails the final
6.12 A pharmaceuticals company marketed a drug that proved to be an effective treatment but unfortunately resulted in side effects for some patients. On the basis of initial clinical research the probability that a patient who is treated with the drug suffers no side effect is 0.85, the probability
6.11 A company produces mouse mats that are 20 cm long and 20 cm wide. These mats are cut from sheets of material that are one metre long and one metre wide. Precise machine setting means that 25 mouse mats can be cut from each sheet of material. Problems that have arisen in the laminating process
6.10 Everyone working in a large open-plan office takes cups of tea or coffee back to their desk to drink. The mean number of spilled cups of hot drinks in an eight-hour working day is 24. What is the probability that an hour passes in which:(a) There are no spillages?(b) There are two
6.9 The mean number of violent incidents at the Koshmar pub is 14 per week. As a result of being injured at work the pub bouncer will be off for a day.(a) What is the probability that the day passes without any violent incidents?(b) What is the probability that the day passes with no more than one
6.8 An office worker receives an average of 22.5 e-mail messages per day. If his working day lasts seven and a half hours, what is the probability that:(a) He receives no e-mails in an hour?(b) He receives one e-mail in an hour?(c) He receives two or fewer e-mails in an hour?(d) He receives more
6.7 The mean number of accidents per day at a large building site is 1.0. What is the probability that a day passes when there are:(a) No accidents?(b) Two accidents?(c) Four or fewer accidents?(d) More than three accidents?
6.6 A large party of Russian managers is to visit a car plant in the UK. They will be shown round the plant in small groups of five. Twenty per cent of the managers speak English. If the groups are selected at random, what is the probability that:(a) There is no English speaker in a group?(b) There
6.5 An egg-packing company thinks that a tenth of all the eggs they buy from farms are contaminated with harmful bacteria that could result in food poisoning. What is the probability that in a pack of ten eggs:(a) None of the eggs contain harmful bacteria?(b) More than two eggs contain harmful
6.4 The producers of a TV survival programme advertise for potential participants and then select random samples of ten people to appear in the programme. If 30 per cent of the people who apply to take part in the programme are smokers, what is the probability that:(a) A team consists entirely of
6.3 A lingerie manufacturer produces ladies’ knickers in packs of five pairs. Quality control has been erratic and they estimate that one in every ten pairs of knickers leaves the factory without the trim sewn on. What is the probability that in a pack of five pairs of knickers:(a) There are no
6.2 ‘Fast’ Gary and Mickey ‘the Mule’ import alcoholic beverages from France into the UK. They are offered a consignment of packs of bottled beers. Each pack contains ten bottles. The price for the consignment is low because due to problems at the bottling plant 20 per cent of the bottles
6.1 The employees of the Brecht Bank decide to put on a fivea-side football tournament to raise money for charity. If the teams are chosen at random and four tenths of the employees are women, what is the probability that:(a) A team consists entirely of males?(b) There are three females in a
The rival company in Example 6.11 publishes financial results that show an improvement in profits. As a result the retail grocery company that wants to open the supermarket now puts the probability that a rival establishment will be built at 0.9. Should the company change its mind about opening the
A retail grocery company is interested in opening a new supermarket in a disused cinema on the outskirts of a major town. Because of local opposition to the proposal there is only a 65 per cent chance that planning permission will be granted. If planning permission is obtained there is an 80 per
A rail operating company incurs extra costs if its long-distance trains are late. Passengers are given a voucher to put towards the cost of a future journey if the delay is between thirty minutes and two hours. If the train is more than two hours late the company refunds the cost of the ticket for
In Example 6.7 we found the probability that the number of people requiring treatment at the first aid tent was more than three in any hour was 0.143. If the music festival lasts 10 hours, in how many hours will there be more than three people seeking treatment?
In Example 6.1 the probability that a team of three members of the sales force chosen at random contained one female was 0.375. If there are enough sales staff for 200 teams, how many teams will contain exactly one female?
The first aid tent at a music festival has the capacity to deal with up to three people requiring treatment in any one hour. The mean number of people requiring treatment is two per hour. What is the probability that they will not be able to deal with all the people requiring treatment in an hour?
Calculate the mean, variance and standard deviation of the binomial distribution in Example 6.5.
Malenky Aviation operates commuter flights using aircraft that can take ten passengers.During each flight passengers are given a hot drink and a ‘Snack Pack’ that contains a ham sandwich and a cake. The company is aware that some of their passengers may be vegetarians and therefore every flight
Use the binomial equation to find the first two probabilities in the probability distribution for Example 6.1.We will begin by identifying the number of trials to insert in the binomial equation.Selecting a team of three involves conducting three ‘trials’, so n = 3.The variable X is the number
Calculate a set of cumulative probabilities from the probability distribution in Example 6.1.Suppose we want the probability that X, the number of females, is two or less than two.Another way of saying this is the probability that X is less than or equal to two. We can use the symbol ‘’ to
Calculate the mean and the standard deviation for the probability distribution in Example 6.1.The mean, , is 1.5, the total of the x P(x) column X P (x) 0123 0.125 0.375 x P (x) 0 0.375 0.375 0.750 0.125 0.375 1.500 0149 x P (x) 0 0.375 1.500 1.125 3.000
The Sales Director of a large organization requires all members of sales staff to attend a team-building event at an outward-bound centre. Teams of three staff will be selected by drawing names out of a hat. If there are equal numbers of females and males employed in the sales force, what are the
5.20 Select the appropriate definition for each term on the lefthand side from the list on the right-hand side.(a) compound probability(i) basing a probability on opinion(b) multiplication rule (ii) outcomes that cannot occur together(c) collectively exhaustive(iii) P (A and B) = P (A) × P(B|A)(d)
5.19 An insurance company offers quotations for motor insurance by telephone. The company employs permanent staff to do this work but as a result of a dramatic rise in the number of telephone inquiries 35 per cent of quotations are provided by temporary staff. Unfortunately 22 per cent of the
5.18 Heidi’s High Couture shop and Maggie’s market stall both sell a particular style of sweatshirt. They are the only outlets for the garment in the area, with Heidi’s accounting for 70 per cent of the total number sold and Maggie’s accounting for the remainder. Unfortunately colour dye
5.17 You win a prize in a charity raffle. The prize, donated by a small hotel chain, is a voucher entitling you to a free double room for a weekend at each of the three hotels in the chain, the Xerxes, the York and the Zetland. The room you stay in at each hotel will be picked at random by the
5.16 Thursday, Friday and Saturday are the busiest nights at the Jopper pub. Police records show that on 12 of the last 50 Thursdays, 15 of the last 50 Fridays, and 16 of the last 50 Saturdays they were summoned to deal with a disturbance at the pub. Construct a tree diagram and use it to find the
5.15 As a result of flood damage a supermarket has a very large stock of tins without labels. Forty per cent of the tins contain soup, 30 per cent contain carrots, 25 per cent contain raspberries, and 5 per cent contain asparagus. The tins are to be offered for sale at three for 50 pence. When a
5.14 Every morning Jack makes tea for the other four people in his section. Two of them take sugar. Jack always forgets which teas he has sugared. One of the two senior members of the section, Alicia, takes sugar in her tea; the other, Ben, does not. If Jack takes tea first to Alicia and then to
5.13 Two friends have a favourite compilation CD that contains some of their all-time favourite tracks. There are 20 tracks on the album. Sam likes five tracks and Chris likes six other tracks. If they program their CD player to play three tracks selected at random from the album, and assuming that
5.12 Bella is taking her friend out for the evening but has forgotten to take some cash out of her bank account. She reaches the cash machine, but can’t remember the exact sequence of her PIN number. She knows that the digits in her PIN number are 2, 7, 8, and 9.(a) What is the probability that
5.11 Declan, Emily and Farid each intend to start their own business after leaving college. Declan wants to start a computer software company, Emily intends to open a recruitment agency, and Farid would like to launch a graphic design business. According to available evidence 60 per cent of
5.10 A commuter’s journey to work consists of driving her car to her local train station, taking a train to the city, and catching a bus to her place of work. If her car trip is delayed she misses the train and is late for work. If the train is delayed, she misses the bus and is late for work. If
5.9 A final-year undergraduate applies for a well-paid job with a large and reputable organization in a highly desirable location. Competition for the job is expected to be fierce.The probability that a candidate is selected for first interview is 0.10. The probability that a candidate is then
5.8 The following survey results show the social class and type of main holiday destination of 250 adult holidaymakers.Use these figures to estimate (a) The probability that a holidaymaker belongs to social class DE.(b) The probability that a holidaymaker takes a main holiday in the UK.(c) The
5.7 A safety agency analysed road traffic accidents involving injury to pedestrians by private cars and produced the following table:What is the probability that:(a) An injury to a pedestrian proves fatal?(b) An accident involved a sports car?(c) An accident involved a 4×4 car or resulted in a
5.6 The 120 employees at a factory were asked which would best improve their working life: better promotion prospects, higher pay or more respect from other staff. The results are tabulated below.(a) What is the probability that an employee selected more respect?(b) What is the probability that an
5.5 Last year 12 966 people opened new accounts at a building society. Of these 5314 were branch-based accounts, 4056 were postal accounts, and 3596 were e-accounts. What is the probability that when a customer opens a new account:(a) It is a postal account?(b) It is an e-account?(c) It is either
5.4 A garage offers a breakdown recovery service for motorists that is available every day of the year. According to their records the number of call-outs they received per day last year were:Number of call-outs 0 1 2 3 4 Number of days 68 103 145 37 12 What is the probability that:(a) They receive
5.3 A direct marketing company produces leaflets offering membership of a book club. These leaflets are then inserted into the magazine sections of two Sunday newspapers, the Citizen and the Despatch. 360 000 leaflets are put in copies of the Citizen and 2 130 000 are put in copies of the
5.2 Since it was set up 73 825 people have visited the website of a fashion designer and 6301 of them purchased goods online.When someone visits the site what is the probability that:(a) they do not purchase goods on-line?(b) they do purchase goods on-line?
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