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business statistics using excel
Business Statistics Using Excel 2nd Edition Glyn Davis, Branko Pecar - Solutions
The following coins are placed in a bag: 1p, 2p, 5p, and 10p. A coin is taken at random and then replaced. A second coin is taken at random and then replaced. Calculate the following probabilities: (a) P(1p chosen first and 2p chosen second); (b) P(sum is 3p);(c) P(at least one 10p).
Ten discs with a different number (0, 1, 2 . . .. 9) printed on them are placed in a bag.Two discs are taken out of the bag one at a time at random to form a two digit number (where 08 is counted as the number 8). Assuming the first disc is replaced before the second is chosen, fi nd the following
A sample of 50 married women was asked how many children they had in their family. The results are presented in Table 3.6.Estimate the probability that if any married woman is asked the same question, she will answer: (a) None; (b) Between 1 and 3 inclusive; (c) More than 3; (d) Neither 3 nor
Five cards are labelled A1, B2, C3, D3, and E3 respectively. A card is selected at random and then a second is selected again before the first is replaced. (a) Show by listing the sample space that there are just 20 possible outcomes. (b) Find the following probabilities: (i) The first card
For each question indicate whether the events are mutually exclusive: (a) Thermometers are inspected and rejected if any of the following are found: poor calibration; inability to withstand extreme temperatures without breaking; and not within specified size tolerances; (b) A manager will reject
Consider two events, A and B, of an experiment which is not empty. Display this information in a Venn diagram and shade the area representing the event {A or B’}.
A survey shows that 80% of all households have a colour television and 30% have a microwave oven. If 20% have both a colour television and a microwave, what percentage has neither?
In a group of 50 students, 30 study French or German. If 20 study French and 15 study German fi nd the probability that a student studies French and German.
A bowl contains three red chips and five blue chips. Two chips are drawn successively, at random and without replacement. Calculate the probability that the first chip drawn is red and the second blue.
Two events, D and E, are found to have the following probability relationships: P(D) = 1/3, P(E) = 1/4, and P(D or E) = 1/2. Calculate the following probabilities: (a) P(D and E), (b) P(D/E), (c) P(E/D).
Two events A and B are found to have the following probability relationships: P(A) = 1/3, P(B) = 1/2, and P(A or B) = 3/4. Calculate the following probabilities: (a) P(A/B), (b) P(B/A), (c) P(B’/A’), (d) P(A’/B’).
A bag contains four red counters and six black counters. A counter is picked at random from the bag and not replaced. A second counter is then picked. Calculate the following probabilities: (a) The second counter is red, given that the first is red; (b) Both the counters are red;(c) The counters
The Gompertz Oil Company drills for oil in old oil fields that large companies have stated are uneconomic. The decision to drill will depend upon a number of factors, including the geology of the proposed sites. Drilling experience shows that there is a 0.40 probability of a type A structure
A dart is thrown at a board and is equally likely to land in any one of eight squares numbered 1–8 inclusive. Let A = Event dart lands in square 5 or 8; B = Event dart lands in square 2, 3, or 4; and C = Event dart lands in square 1, 2, 5, or 6. From this information calculate the following
Each month DINGO Ltd receives a shipment of 100 parts from its supplier, which will be checked on delivery for defective parts. Historically, the average number of defective parts was 5. The new quality assurance procedure involves randomly selecting a sample of three items (without replacement)
Susan takes examinations in Mathematics, French, and History. The probability that she passes Mathematics is 0.7; the corresponding probabilities for French and History are 0.8 and 0.6. Given that her performances in each subject are independent, draw a tree diagram to show the possible outcomes.
A manufacturing firm quality assures components manufactured and historically the length of a tube is found to be normally distributed with a population mean of 100 cm and a population standard deviation of 5 cm.Calculate the probability that a random sample of one tube will have a length of at
A bag contains six white and four red counters, three of which are drawn at random and without replacement. If X can take on the values of 0, 1, 2, 3 red counters, construct the probability distribution of X. If the experiment was repeated 60 times, how many times would we expect to draw more than
In a game you are offered the chance to toss a fair coin until a ‘tail’ appears. If a tail appears on the first toss you win £2. If the first tail appears on the second toss you win £4. If the first tail appears on the third toss you win £8. How much should you be willing to pay to
Use the NORM.DIST function to calculate the following probabilities, X ~ N(100, 25):(a) P(X ≤ 95); (b) P(95 ≤ X ≤ 105); (c) P(105 ≤ X ≤ 115); (d) P(93 ≤ X ≤ 99). For each probability identify the region to be found by shading the area on the normal probability distribution graph.
Skodel Ltd is developing a low calorie lager for the European market with a mean designed calorie count of 43 calories per 100 ml. The new product development team are having problems with the production process and have collected an independent random sample to assess whether the target calorie
Bakers Ltd is currently in the process of reviewing the credit line available to supermarkets who they have defined as a ‘good’ or ‘bad’ risk. Based upon a £100,000 credit line the profi t is estimated to be £25,000 with a standard deviation of £5000. Calculate the probability that the
The manager at BIG JIMS restaurant is concerned about the time it takes to process credit card payments at the counter by counter staff .The manager has collected the following processing time data (time in minutes/seconds) and requested that the data be checked to see if it is normally distributed
Given that a normal variable has a mean of 10 and a variance of 25, calculate the probability that a member chosen at random is: (a) ≥ 11, (b) ≤ 11, (c) ≤ 5, (d) ≥ 5, (e) between 5 and 11.
The lifetimes of certain types of car battery are normally distributed with a mean of 1248 days and standard deviation of 185 days. If the supplier guarantees them for 1080 days, what proportion of batteries will be replaced under guarantee?
A local authority surveyed the travel preferences of people who travelled to work by train or bus. The initial analysis suggested that one in five people travelled by train to work. If five people are interviewed, what is the probability that: (a) Exactly three prefer travelling by train, P(X =
Electrical resistors have a design resistance of 500 ohms. The resistors are produced by a machine with an output that is normally distributed N(501,9) ohms. Resistances below 498 ohms and above 508 ohms are rejected. Find: (a) The proportion that will be rejected; (b) The proportion which would
A manufacturing company regularly conducts quality control checks at specifi ed periods on all products manufactured. A new order for 2000 light bulbs is due to be delivered to a national do-it-yourself store. Historically, the manufacturing process has a failure rate of 15% and the sample to be
A binomial model has n = 4 and p = 0.6.(a) Find the probabilities of each of the five possible outcomes (i.e. P(0), P(1) . . . P(4)).(b) Construct a histogram of this data.
A company is reviewing the number of telephone lines available for customer support. The average number of calls received per day is three calls during a five-minute period. Estimate the proportion of phone calls that cannot be answered during a five-minute period: (a) If the company installs four
Attendance at a cinema has been analysed and shows that audiences consist of 60% men and 40% women for a particular film. If a random sample of six people was selected from the audience during a performance, fi nd the following probabilities:(a) All women are selected (b) Three men are
In a large consignment of apples 3% are rotten. What is the probability that a carton of 60 apples will contain fewer than 2 rotten apples? We have here a binomial experiment and therefore could easily apply the binomial distribution with p = 0.03, q = 0.97 and n = 60.
Assume you have a fair coin and wish to know the probability that you would get eight heads out of ten flips. The binomial distribution has a mean of μ = np = 10 * 0.5 = 5 and a variance of σ2 = npq = 10 * 0.5 * 0.5 = 2.5. The standard deviation is therefore 1.5811. A total of 8 heads is 1.8973
A quality control system selects a sample of three items from a production line. If one or more is defective, a second sample is taken (also of size three), and if one or more of these are defective then the whole production line is stopped. Given that the probability of a defective item is 0.05,
Enquiries at a travel agent lead to a holiday booking being made only sometimes. The agent needs to make 35 bookings per week to break even. If during a week there are 100 enquiries and the probability of a booking in each case is 0.4, find the probability that the agent will at least break even in
Five people in seven voted in an election. If four of those on the roll are interviewed what is the probability that at least three voted?
A small tourist resort has a weekend traffi c problem and is considering whether or not to provide emergency services to help mitigate the congestion that results from an accident or breakdown. Past records show that the probability of a breakdown or an accident on any given day of a four-day
The average number of broken eggs per lorry is known to be 50. What is the probability that there will be more than 70 broken eggs on a particular lorry load?We may use the normal approximation to the Poisson distribution, where the mean and variance are calculated as follows: mean (μNormal ≈
Calculate P(0), P(1), P(2), P(3), P(4), P(5), P(6), and P(>6) for a Poisson variable with a mean of 1.2. Using this probability distribution determine the mean and variance.
In a machine shop the average number of machines out of operation is two. Assuming a Poisson distribution for machines out of operation, calculate the probability that at any one time there will be:(a) Exactly one machine out of operation (b) More than one machine out of operation.
A factory estimates that 0.25% of its production of small components is defective. These are sold in packets of 200. Calculate the percentage of the packets containing one or more defectives.
The average number of faults in a metre of cloth produced by a particular machine is 0.1. (a) What is the probability that a length of four metres is free from faults? (b) How long would a piece have to be before the probability that it contains no flaws is less than 0.95?
A garage has three cars available for daily hire. Calculate the following probabilities if the variable is a Poisson variable with a mean of 2: (a) Find the probability that on a given day that exactly none, one, two, and three cars will be hired, and determine the mean number of cars hired per
Accidents occur in a factory randomly and, on average, at the rate of 2.6 per month. What is the probability that in a given month: (a) No accidents will occur (b) More than one accident will occur?
A new telephone directory is to be published. Before publication entries are proofread for errors and any corrections made. Experience suggests that, on average, 0.1% of the entries require correction and that entries requiring correction are randomly distributed. The directory contains 800 pages
Given X is a discrete binomial random variable with p = 0.3 and n = 20: (a) Can we use the normal approximation to estimate the binomial probability? (b) What if n is changed to 15? (c) if n = 40 and p = 0.1 is the normal approximation appropriate?
Concerned at the time to react to customer complaints CoCo S.A. has implemented a new set of procedures for its support centre staff. The customer service director has directed that a suitable test is applied to a new sample to assess whether the new target mean time for responding to customer
A local maternity hospital has an average of 36 births per week. Use this information to calculate the following probabilities: (a) The probability that there are fewer than 30 births in a given week; (b) The probability that there will be more than 40 births in a given week;(c) The probability
Five people have all made claims for the amounts shown in Table 5.2A sample of two people is to be taken at random, with replacement, from the five. Derive the sampling distribution of the mean and prove(a)(b) Person Insurance claim, € Table 5.2 1 500 2 400 3 900 4 1000 5 1200
To illustrate this property consider the problem of tossing a fair die. The die has 6 numbers (1, 2, 3, 4, 5, and 6), with each number likely to have the same frequency of occurrence. If we then take all possible samples of size 2 from this population then we will be able to illustrate two
Bakers Ltd is currently undertaking a review of the delivery vans used to deliver products to customers. The company runs two types of delivery van (type A, recently purchased, and type B, at least 3 years old) which are supposed to be capable of achieving 20 km per litre of petrol. A new sample
Diet X runs a number of weight reduction centres within a large town in the north east of England. From the historical data it was found that the weight of participants is normally distributed with a mean of 150 lb and a standard deviation of 25 lb. This can be written in mathematical notation as X
Skodel Ltd is developing a low calorie lager for the European market with a mean designed calorie count of 43 calories per 100 ml. The new product development team are having problems with the production process and have collected two independent random samples to assess whether the target calorie
If X is a normal random variable with mean 10 and standard deviation 2, i.e. X ~ N (10, 4). Define and compare the sampling distribution for samples of size: (a) 2, (b) 4,(c) 16.
Calculate the probability that the sample mean lies between 146 and 158 pounds.
If X is any random variable with mean = 63 and standard deviation = 10. Define and compare the sampling distribution for samples of size:(a) 40, (b) 60, (c) 100.
A random sample of 30 part-time employees is chosen without replacement from a firm employing 200 part-time workers. If the mean hours worked per month is 60 hours with a standard deviation of 5 hours determine the probability that the sample mean: (a) will lie between 60 and 62 hours,(b) Be over
Consider the sampling of 50 electrical components from a production run where, historically, the component’s average lifetime was found to be 950 hours with a standard deviation of 25 hours. The population data is right-skewed and therefore cannot be considered to be normally distributed.
Assuming that the weights of 10,000 items are normally distributed and that the distribution has a mean of 115 kg and a standard deviation of 3 kg: (a) Estimate how many items have weights between 115 and 118 kg; (b) If you have to pick one item at random from the whole 10,000 items, how
It is known that 25% of workers in a factory own a personal computer. Find the probability that at least 26% of a random sample of 80 workers will own a personal computer. In this example, we have the population proportion π = 0.25 and sample size n = 80. The problem requires the calculation of
By treating the following as finite and infinite samples comment on the standard errors: (a) Find the sample mean and standard error for random samples of 1000 accounts if bank A has 5024 saving accounts with an average in each account of £512 and a standard deviation of £150; (b) Find the
Consider the problem of sampling from a population which consists of the salaries for public sector employees employed by a national government. The historical data suggests that the population data is normally distributed with mean of €45,000 and standard deviation of €10,000. We can use Excel
A sample of 100 was taken from a population with π = 0.5. Find the probability that the sample proportion will lie between: (a) 0.4 and 0.6, (b) 0.35 and 0.65, (c) 0.5 and 0.65.
An experiment on the measurement of the length of rods was performed five times, with the following results: 1.010, 1.012, 1.008, 1.013, and 1.011. Calculate the unbiased estimates of the mean and variance of possible measurements, and give an estimate for the standard error of your estimate of the
From a parliamentary constituency a sample of 100 people were asked whether they would vote Labour or Conservative. It is thought that 40% of the constituency will favour Labour. Find the approximate probability that in an election Labour will win (assume only a two-party vote).
In a sample of 400 textile workers, 184 expressed dissatisfaction regarding a prospective plan to modify working conditions. Provide a point estimate of the population proportion of total workers who would be dissatisfied and give an estimate for the standard error of your estimate.
The annual income of doctors constitutes a highly positive-skewed distribution. Suppose the population has an unknown mean and a standard deviation of £10,000. An estimate of the population mean is to be made using the sample mean. This estimate must be within £1000 either side of the true
Eight samples measuring the length of cloth are sampled from a population where the length is normally distributed with population standard deviation 0.2. Calculate a 95% confidence interval for the population mean based on a sample of 8 observations: 4.9, 4.7, 5.1, 5.4, 4.7, 5.2, 4.8, and 5.1.
The average number of Xerox copies made in a working day in a certain office is 356 with a standard deviation of 55. It costs the firm three pence per copy. During a working period of 121 days what is the probability that the average cost per day is more than £11.10?
A researcher determines that a margin of error (or sampling error, e) of no more than ± 0.5 units is desired, along with a 98% confidence interval. If we assume a normal population standard deviation of 0.2, calculate the sample size, n.
A random sample of 5 values was taken from a population: 8.1, 6.5, 4.9, 7.3, and 5.9. Estimate the population mean and standard deviation, and the standard error of the estimate for the population mean.
The mean of 10 readings of a variable was 8.7 with standard deviation 0.3. A further 5 readings were taken: 8.6, 8.5, 8.8, 8.7, and 8.9. Estimate the mean and standard deviation of the set of possible readings using all the data available.
Two samples are drawn from the same population as follows: sample 1 (0.4, 0.2, 0.2, 0.4, 0.3, and 0.3) and sample 2 (0.2, 0.2, 0.1, 0.4, 0.2, 0.3, and 0.1). Determine the best unbiased estimates of the population mean and variance.
A random sample of 100 rods from a population line were measured and found to have a mean length of 12.132 with standard deviation 0.11. A further sample of 50 is taken. Find the probability that the mean of this sample will be between 12.12 and 12.14.
A random sample of 20 children in a large school were asked a question and 12 answered correctly. Estimate the proportion of children in the school who answered correctly and the standard error of this estimate.
A random sample of 500 fish is taken from a lake and marked. After a suitable interval a second sample of 500 is taken and 25 of these are found to be marked. By considering the second sample estimate the number of fish in the lake.
The standard deviation for a method of measuring the concentration of nitrate ions in water is known to be 0.05 ppm. If 100 measurements give a mean of 1.13 ppm, calculate the 90% confidence limits for the true mean.
In trying to determine the sphere of influence of a sports centre a random sample of 100 visitors was taken. This indicated a mean travel distance (d) of 10 miles with a standard deviation of 3 miles: (a) What are the 90% confidence limits for the population mean travel distance (D), (b) What
The masses, in grams, of 13 ball bearings taken at random from a batch are: 21.4, 23.1, 25.9, 24.7, 23.4, 24.5, 25.0, 22.5, 26.9, 26.4, 25.8, 23.2, and 21.9. Calculate a 95% confidence interval for the mean mass of the population, supposed normal, from which these masses were drawn.
Concerned at the time taken to react to customer complaints, CoCo S.A. has implemented a new set of procedures for its support centre staff . The customer service director has directed that a suitable test is applied to a new sample to assess whether the new target mean time for responding to
A business analyst has been requested by the managing director of a national supermarket chain to undertake a business review of the company. One of the key objectives is to assess the level of spending of shoppers who, historically, have weekly mean levels of spending of €168.00 with a standard
The historical output by employees is a mean rate of 100 units per hour with a standard deviation of 20 units per hour. A new employee is tested on 36 separate random occasions and is found to have an output of 90 units per hour. Does this indicate that the new employee’s output is signifi cantly
Bakers Ltd are currently undertaking a review of the delivery vans used to deliver products to customers. The company runs two types of delivery van (type A, recently purchased, and type B, at least three years old) which are supposed to be capable of achieving 20 km per litre of petrol. A new
A supermarket is supplied by a consortium of milk producers. Recently, a quality assurance check suggests that the amount of milk supplied is significantly different from the quantity stated within the contract: (i) Define what we mean by significantly different; (ii) State the null and
Employees of a firm produce units at a rate of 100 per hour with a standard deviation of 20 units per hour. A new employee is tested on 36 separate random occasions and is found to have an output of 90 units per hour. Does this indicate that the new employee’s output is signifi cantly different
Skodel Ltd is developing a low calorie lager for the European market with a mean designed calorie count of 43 calories per 100 ml. The new product development team are having problems with the production process and have collected two independent random samples to assess whether the target calorie
A business analyst is attempting to understand visually the meaning of the critical test statistic and the p-value. For a z value of 2.5 and significance level of 5% provide a sketch of the normal probabilty distribution and use the sketch to illustrate the location of the following statistics:
A local car dealer wants to know if the purchasing habits of a buyer buying extras have changed. He is particularly interested in male buyers. Based upon collected data he has estimated that the distribution of extras purchased is approximately normally distributed with an average of £2000 per
What are the critical z values for a significance level of 2%: (i) Two tail, (ii) Lower one tail,(iii) Upper one tail?
A large organization produces electric light bulbs in each of its two factories (A and B). It is suspected that the quality of production from factory A is better than from factory B. To test this assertion the organization collects samples from factory A and B, and measures how long each light
A marketing manager has undertaken a hypothesis test to test for the difference between accessories purchased for two different products. The initial analysis has been performed and an upper one tail z-test has been chosen. Given that the z value was calculated to be 3.45 find the corresponding
A mobile phone company is concerned at the lifetime of phone batteries supplied by a new supplier. Based upon historical data this type of battery should last for 900 days with a standard deviation of 150 days. A recent, randomly selected sample of 40 batteries was selected and the sample battery
Concerned by the number of passengers not wearing rear seat belts in cars, a local police authority decided to undertake a series of surveys based upon two large cities. The survey consisted of two independent random samples collected from city A and B. The police authority would like to know if
A local Indian restaurant advertises home delivery times of 30 minutes. To monitor the eff ectiveness of this promise the restaurant manager monitors the time that the order was received and the time of delivery. Based upon historical data the average time for delivery is 30 minutes with a standard
A certain product of organic beans are packed in tins and sold by two local shops. The local authority have received complaints from customers that the amount of beans within the tins sold by the shop are different. To test this statistically two small random samples were collected from both shops.
Calculate the critical t values for a significance level of 1% and 12 degrees of freedom:(i) Two tail, (ii) Lower one tail, (iii) Upper one tail.
After further data collection the marketing manager (Exercise X6.4) decides to revisit the data analysis and changes the type of test to a t-test. (i) Explain under what conditions a t-test could be used rather then the z-test,(ii) Calculate the corresponding p-value if the sample size was 13 and
A certain product of organic beans is packed in tins and sold by two local shops. The local authority have received complaints from customers that the amount of beans within the tins sold by the shop is different. To test this statistically two small, random samples were collected from both shops.
A tyre manufacturer conducts quality assurance checks on the tyres that it manufactures. One of the tests consists of undertaking a test on their medium-quality tyres with an independent random sample of 12 tyres providing a sample mean and standard deviation of 14,500 km and 800 km respectively.
Suppose that Super Slim is advertising a weight reduction programme that they say will result in more than 10 lb weight loss in the first 30 days. Twenty-six subjects were independently and randomly selected for a study, and their weights before and after the weight-loss programme were recorded.
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