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Statistics For Business And Economics 4th Edition Paul Newbold - Solutions
A large statistics class was taken by both business and liberal arts majors. A random sam- ple of ten business majors obtained the following scores on the final examination: 62 57 85 59 64 63 71 58 77 72 An independent random sample of eight liberal arts majors obtained the following scores: 73 79
There are some estimation problems for which no unbiased estimator can be found. For example, if a sample of one observation is drawn from a population with mean and vari ance, we have an unbiased estimator of but cannot find an unbiased estimator of o Explain why this conclusion is plausible.
Let X, X, X. be a random sample from a population with mean and variance o. Also, let X be the sample mean. (a) Show that -xx- is an unbiased estimator of o (b) Show that -xx-Xin is a biased estimator of or, and find the bias. (Use the fact that the sample variance is known to be unbiased for o.)
Let X, X.X, be a random sample from a population with mean and variance o. Also, let C. C.c. be any set of fixed numbers, and consider the estimator of p (a) Show that is an unbiased estimator of if and only if(b) Show that the variance of is Var (A) (c++...+ (c) Show that if is an unbiased
A random sample of twelve blue-collar employees in a large manufacturing plant found the following figures for number of hours overtime worked in the last month: 22 16 28 12 18 36 23 11 41 11 41 29 26 31 Use an unbiased estimation procedure to find point estimates for each of the following: (a) The
A random sample of ten economists produced the following forecasts for percentage growth in real gross domestic product in the next year: 2.2 2.8 3.0 2.5 2.4 2.6 2.5 2.4 2.7 2.6 Use unbiased estimation procedures to find point estimates for each of the following: (a) The population mean. (b) The
Let 6, be an unbiased estimator of 0, and 8, an unbiased estimator of (a) Show that (0,+6) is an unbiased estimator of (8, + 8). (b) Show that (8,-) is an unbiased estimator of (6-8).
Of what practical value is the concept of efficiency in evaluating the merits of a point esti- mator?
Let X be the number of successes in n independent trials, each with probability of success. p. Find an unbiased estimator of the variance of the sample proportion of successes.
A random sample of sixty account balances contained four with at least one serious error and ten with at least one minor error. Two of the balances contained at least one serious and one minor error. Use an unbiased estimation procedure to find a point estimate of the proportion of all balances in
A random sample of twenty bond market professionals was taken. Of these sample mem- bers, seven believed that a portfolio of low-grade corporate bonds represented a good in- vestment over the next year. In an independent random sample of twelve bond market pro- fessionals, six believed that
A random sample of 100 voters in a city found sixty-two in favor of an immediate reduc- tion in property taxes. An independent random sample of 200 voters from the same popu- lation found 102 in favor of extending the services of the public library system. Let p, de- note the population proportion
On a college campus, 480 men and 370 women live in dormitories. Of a random sample of sixty men, twenty-four were satisfied with dormitory food. Of a random sample of sixty women, thirty-two were satisfied with dormitory food. Using an unbiased estimation pro- cedure, find a point estimate of the
Batches of chemical are manufactured by a production process. Samples of twenty batches from a production run are selected for testing. If the standard deviation of the percentage. impurity contents in the sampled batches exceeds 2.5%, the production process is thor- oughly checked. Assume that the
It is known that the incomes of subscribers to a particular magazine have a normal distrib- ution with standard deviation $6,600. A random sample of twenty-five subscribers is taken. (a) What is the probability that the sample standard deviation of their incomes is bigger than $4,000? (b) What is
A candidate for office intends to campaign in a state if her initial support level exceeds 30% of the voters. A random sample of 300 voters is taken, and it is decided to campaign if the sample proportion supporting the candidate exceeds .28. (a) What is the probability of a decision to campaign if
In a particular year, 40% of home sales were partially financed by the seller. A random sample of 250 sales is examined. (a) The probability is .8 that the sample proportion is bigger than what amount? (b) The probability is .9 that the sample proportion is smaller than what amount? (c) The
A manufacturer of liquid detergent claims that the mean weight of liquid in containers. sold is at least 30 ounces. It is known that the population distribution of weights is normal with standard deviation 1.3 ounces. In order to check the manufacturer's claim, a random sample of sixteen containers
Scores on a particular test, taken by a large group of students, follow a normal distribution with standard deviation 40 points. A random sample of sixteen scores was taken to esti mate the population mean score. Let X denote the sample mean. What is the probability. that the interval (X10) to
A sample of 100 students is to be taken to determine which of two brands of beer is pre- ferred in a blind taste test. Suppose that, in the whole population of students, 50% would prefer brand A.(a) What is the probability that more than 60% of the sample members prefer brand A? (b) What is the
A quality control manager was concerned about variability in the amount of active ingre- dient in pills produced by a particular process. A random sample of 21 pills was taken. What is the probability that the sample variance of the amount of active ingredient was more than twice the population
Plastic bags used for packaging produce are manufactured so that the breaking strengths of the bags are normally distributed with a standard deviation of 1.8 pounds per square inch. A random sample of sixteen bags is selected. (a) The probability is .01 that the sample standard deviation of
It was found that 80% of seniors at a particular college had accepted a job offer before graduation. For those accepting offers, salary distribution was normal with mean $29,000 and standard deviation $4,000. (a) For a random sample of sixty seniors, what is the probability that less than 70% have
It has been found that times taken by people to complete a particular tax form follow a normal distribution with mean 100 minutes and standard deviation 30 minutes. A random sample of nine people who have completed this tax form was taken. (a) What is the probability that the sample mean time taken
Refer to Appendix A6.1 in order to derive the mean of the sampling distribution of the sample variance for a sample of n observations from a population of N members, when the population variance is oy. By appropriately modifying the argument in Appendix A6.1, show that E(S)-Nox N-1) Note the
The lifetimes of a certain electronic component are known to be normally distributed with a mean of 1,600 hours and a standard deviation of 400 hours. (a) For a random sample of sixteen components, find the probability that the sample mean is more than 1,500 hours. (b) For a random sample of
In a particular year, the percentage rates of returns of U.S. common stock mutual funds had a normal distribution with mean 14.8 and standard deviation 6.3. A random sample of nine of these mutual funds was taken. (a) What is the probability that the sample mean percentage rate of retum is more
A company services home air conditioners. It has been found that times for service calls follow a normal distribution with mean 60 minutes and standard deviation 10 minutes. A random sample of four service calls was taken. (a) What is the probability that the sample mean service time is more than
The scores of all applicants taking an aptitude test required by a law school have a normal distribution with mean 420 and standard deviation 100. A random sample of twenty-five scores is taken. (a) Find the probability that the sample mean score is bigger than 450. (b) Find the probability that
Refer to Exercise 49.Find the probability function of the sampling distribution of the sam- ple proportion of funds with average maturity more than 36 days for samples of two ob servations. Also verify directly that the mean of the sampling distribution of the sample proportion is equal to the
Of what relevance is the central limit theorem to the sampling distribution of the sample. mean?
An investor is considering six different money market funds. The average number of days to maturity for these funds is 41 39 35 35 33 38 Two of these funds are to be chosen at random. (a) How many possible samples of two funds are there? (b) List all possible samples. (c) Find the probability
A manufacturer has been purchasing raw materials from a supplier whose consignments have a variance of 15.4 (in squared pounds) in impurity levels. A rival supplier claims that he can supply consignments of this raw material with the same mean impurity level but with lower variance. For a random
A drug company produces pills containing an active ingredient. The company is con- cerned about the mean weight of this ingredient per pill, but it also requires that the vari- ance (in squared milligrams) be no more than 1.5. A random sample of twenty pills is se- lected, and the sample variance
A precision instrument is checked by making twelve readings on the same quantity. The population distribution of readings is normal. (a) The probability is .95 that the sample variance is more than what percentage of the population variance? (b) The probability is 90 that the sample variance is
Each member of a random sample of fifteen business economists was asked to predict the rate of inflation for the coming year. Assume that the predictions for the whole population of business economists follow a normal distribution with standard deviation 1.8%. (a) The probability is .01 that the
A random sample of ten stock market mutual funds was taken. Suppose that rates of re- turns on the population of all stock market mutual funds follow a normal distribution. (a) The probability is 10 that the sample variance is bigger than what percentage of the population variance? (b) Find any
A production process manufactures electronic components with timing signals whose du- ration follows a normal distribution. A random sample of six components was taken, and the durations of their timing signals were measured. (a) The probability is 05 that the sample variance is bigger than what
In Table 6.1, we considered the fifteen possible samples of four observations from a popu- lation of N 6 values of years on the job for employees. The population variance for these six values is 47 12 For each of the fifteen possible samples, calculate the sample variance. Find the average of these
Numbers of hours spent watching television by students in the week before final exams have a normal distribution with standard deviation 4.5 hours. A random sample of thirty students was taken. (a) Is the probability more than .95 that the sample standard deviation exceeds 3.5 hours? (b) Is the
In a large city it was found that summer electricity bills for single family homes followed a normal distribution with standard deviation $100. A random sample of 25 bills was taken. (a) Find the probability that the sample standard deviation is less than $75. (b) Find the probability that the
A mathematics test of 100 multiple choice questions is to be given to all freshmen entering a large university. Initially, in a pilot study, the test was given to a random sample of twenty freshmen. Suppose that, for the population of all entering freshmen, the distribu- tion of number of correct
It is believed that first-year salaries for newly qualified accountants follow a normal distrib- ution with standard deviation $2,500. A random sample of sixteen observations was taken. (a) Find the probability that the sample standard deviation is more than $3,000. (b) Find the probability that
Monthly rates of return on the shares of a particular common stock are independent of one another and normally distributed with a standard deviation of 1.7 A sample of 12 months is taken. (a) Find the probability that the sample standard deviation is less than 2.5. (b) Find the probability that the
A process produces batches of a chemical whose impurity concentrations follow a normal distribution with variance 1.75. A random sample of twenty of these batches is chosen. Find the probability that the sample variance exceeds 3.10.
The annual percentage salary increases for the chief executive officers of all midsize cor- porations are normally distributed with mean 12.2% and standard deviation 3.6%. A ran- dom sample of eighty-one of these chief executive officers was taken. What is the proba- bility that more than half the
A manufacturing plant has 438 blue-collar employees. Of this group, 239 are concerned about future health care benefits. A random sample of 80 of these employees was ques- tioned to estimate the population proportion concerned about future health care benefits. (a) What is the standard error of the
A small college has an entering freshman class of 528 students. Of these, 211 have brought their own personal computers to campus. A random sample of 120 entering fresh- men was taken. (a) What is the standard error of the sample proportion bringing their own personal com- puters to campus? (b)
A journalist wanted to learn the views of the chief executive officers of the 500 largest U.S. corporations on program trading of stocks. In the time available, it was only possible to contact a random sample of eight-one of these chief executive officers. If 55% of all the population members
A company wants to estimate the proportion of people who are likely purchasers of elec- tric shavers who watch the nationally telecast baseball playoffs. A random sample ob- tained information from 120 people who were identified as likely purchasers of electric shavers. Suppose that the proportion
In Exercise 26, suppose that it is decided that a sample of 100 voters is too small to pro- vide a sufficiently reliable estimate of the population proportion. It is required instead that the probability that the sample proportion differs from the population proportion (what- ever its value) by
A random sample of 100 voters is taken to estimate the proportion of a state's electorate in favor of an increase in the level of gasoline tax to provide additional revenue for highway repairs. What is the largest value that the standard error of the sample proportion in favor of this measure can
A store has determined that 30% of all lawn mower purchasers will also purchase a ser- vice agreement. In one month, 280 lawn mowers are sold to customers who can be re- garded as a random sample of all purchasers. (a) What is the standard error of the sample proportion of those who will purchase a
A charity has found that 42% of donors from last year will donate again this year. A ran- dom sample of 300 donors from last year was taken. (a) What is the standard error of the sample proportion who will donate again this year? (b) What is the probability that more than half these sample members
A record store owner finds that 20% of customers entering her store make a purchase. One morning 180 people, who can be regarded as a random sample of all customers, enter the store. (a) What is the mean of the sample proportion of customers making a purchase? (b) What is the variance of the sample
According to the Internal Revenue Service, 75% of all tax returns lead to a refund. A ran- dom sample of 100 tax returns is taken. (a) What is the mean of the sample proportion of returns leading to refunds? (b) What is the variance of the sample proportion? (c) What is the standard error of the
In 1992. Canadians voted in a referendum on a new constitution. In the province of Quebec, 42.4% of those who voted were in favor of the new constitution. A random sam ple of 100 voters from the province was taken. (a) What is the mean of the sample proportion in favor of a new constitution? (b)
For an audience of 600 people attending a concert, the average time on the journey to the concert was 32 minutes, and the standard deviation was 10 minutes. A random sample of 150 audience members was taken. (a) What is the probability that the sample mean journey time was more than 31 minutes? (b)
An economics course was taken by 250 students. Each member of a random sample of 50 of these students was asked to estimate the amount of time he or she spent on the previous week's assignment. Suppose that the population standard deviation is 30 minutes. (a) What is the probability that the sample
In taking a sample of a observations from a population of N members, the variance of the sampling distribution of the sample mean is The quantity (N-n) (N-1) R N- N-1 is called the "finite population correction factor." (a) To get some feeling for possible magnitudes of the finite population
In Table 6.1 and Example 6.1, we considered samples of n = 4 observations from a popa- lation of N 6 values of years on the job for employees. The population mean is -5.5 years. (a) Confirm from the six population values that the population variance is (b) Confirm, following the approach of Example
The number of hours spent studying by students on a large campus in the week before final exams follows a normal distribution with standard deviation 8.4 hours. A random sample of these students is taken to estimate the population mean number of hours studying. (a) How large a sample is needed to
The price-eamings ratios for all companies whose shares are traded on the New York Stock Exchange follow a normal distribution with standard deviation 3.8. A random sam-ple of these companies is selected in order to estimate the population mean price-camnings ratio. (a) How large a sample is
An industrial process produces batches of a chemical whose impurity levels follow a normal distribution with standard deviation 1.6 grams per hundred grams of the chemical. A random sample of 100 batches is selected in order to estimate the population mean impurity level. (a) The probability is .05
Times spent studying by students in the week before final exams follow a normal distri bution with standard deviation 8 hours. A random sample of 4 students was taken in order to estimate the mean study time for the population of all students. (a) What is the probability that the sample mean
A company produces breakfast cereal. The true mean weight of the contents of boxes of its cereal is 20 ounces, and the standard deviation is .6 ounce. The population distribution of weights is normal. Suppose that you purchase four boxes, which can be regarded as a ran- dom sample of all those
A random sample of sixteen junior managers in the offices of corporations in a large city. center was taken in order to estimate average daily commuting times for all such man- agers. Suppose that the population times have a normal distribution with mean 87 minutes and standard deviation 22
Candidates for employment at a city fire department are required to take a written aptitude test. Scores on this test are normally distributed with mean 280 and standard deviation 60.A random sample of nine test scores was taken. (a) What is the standard error of the sample mean score? (b) What is
The mean selling price of new homes in a city over a year was $115,000. The population standard deviation was $25,000. A random sample of 100 new homes sales from this city was taken. (a) What is the probability that the sample mean selling price was more than $110,000? (b) What is the probability
The fuel consumption, in miles per gallon, of all cars of a particular model has mean 25 and standard deviation 2.The population distribution can be assumed to be normal. A ran- dom sample of these cars is taken. (a) Find the probability that sample mean fuel consumption will be less than 24 miles
The lifetimes of lightbulbs produced by a particular manufacturer have mean 1,200 hours. and standard deviation 400 hours. The population distribution is normal. Suppose that you purchase nine bulbs, which can be regarded as a random sample from the manufacturer's output. (a) What is the mean of
When a production process is operating correctly, the resistance in ohms of electrical com- ponents produced has a normal distribution with mean 92.0 and standard deviation 3.6. A random sample of four components was taken.(a) Find the mean of the sampling distribution of the sample mean
A management consultant found that the amount of time per day spent by executives per- forming tasks that could equally well be done by subordinates followed a normal distribu- tion with mean 2.4 hours. It was also found that 10% of executives spent over 3.5 hours per day on tasks of this type. For
It is estimated that major league baseball game times-to-completion follow a normal dis- tribution with mean 132 minutes and standard deviation 12 minutes. (a) What proportion of all games last between 120 minutes and 150 minutes? (b) Thirty-three percent of all games last longer than how many
In a large department store, a customer complaints office handles an average of six com- plaints per hour about quality of service. The distribution is Poisson (a) What is the probability that in any hour exactly six complaints will be received? (b) What is the probability that more than 20 minutes
An emergency towing service receives an average of seventy calls per day for assistance. For any given day, what is the probability that fewer than fifty calls will be received?
A market research organization has found that 40% of all supermarket shoppers refuse to cooperate when questioned by its pollsters. If 1,000 shoppers are approached, what is the probability that less than 500 will refuse to cooperate?
It is estimated that amounts of money spent on gasoline by customers at a gas station fol- low a normal distribution with standard deviation $2.50. It was also found that 10% of all customers spend more than $15. What percentage of customers spend less than $10?
A pizza delivery service delivers to a campus dormitory. Delivery times follow a normal distribution with mean 20 minutes and standard deviation 4 minutes. (a) What is the probability that a delivery will take between 15 and 25 minutes? (b) The service does not charge for the pizza if delivery
It has been found that times taken by people to complete a particular tax form follow a normal distribution with mean 100 minutes and standard deviation 30 minutes. (a) What is the probability that a randomly chosen person takes less than 85 minutes to complete this form? (b) What is the
Grade point averages of students on a large campus follow a normal distribution with mean 2.6 and standard deviation .5. (a) One student is chosen at random from this campus. What is the probability that stu- dent has a grade point average higher than 3.0? (b) One student is chosen at random from
Let X, and X; be a pair of random variables. Show that the covariance between the random variables (X + X) and (XX) is 0 if and only if X, and X: have the same variance.
A consultant knows that it will cost her $10,000 to fulfill a particular contract. The con- tract is to be put out for bids, and she believes that the lowest bid, excluding her own, can be represented by a distribution that is uniform between $8,000 and $20,000. Therefore, if the random variable X
In practice, we find probabilities for any normal distribution based on probabilities for the standard normal distribution. Suppose that you had available tables of probabilities for the normal distribution with mean 1 and standard deviation 10, rather than tables for the stan- dard normal
"Many quantities we measure-such as height, weight, and distance-are necessarily pos- itive. It therefore follows that the mean of a sample of observations on such a quantity must also be positive. However, the normal distribution supposes a range of values from minus infinity to plus infinity.
Answer the following questions: (a) Why is it necessary to use tables to find probabilities for the normal distribution? (b) Why is it necessary to tabulate probabilities for only one of the infinite number of nor- mal distributions? (c) Why is the normal distribution important in the study of
"In the real world, measurements on any quantity of interest are almost invariably made on a discrete scale. Therefore, the study of continuous random variables is only of aca- demic interest and has no practical value." Comment on this statement
Explain verbally what can be learned from each of the following; (a) The cumulative distribution function of a continuous random variable (b) The probability density function of a continuous random variable (c) The mean of a continuous random variable (d) The standard deviation of a continuous
Suppose that the time between successive occurrences of an event follows an exponential distribution with mean 1/A minutes. Assume that an event occurs. (a) Show that the probability that more than 3 minutes elapses before the occurrence of the next event is e (b) Show that the probability that
A professor sees students during regular office hours. Times spent with students follow an exponential distribution with mean 10 minutes.(a) Find the probability that a given student spends less than 20 minutes with the professor. (b) Find the probability that a given student spends more than 5
A firm receives, on average, twenty-two replacement orders per week for a particular part. If it begins the week with a stock of twenty-five of these parts, what is the probability that this stock will be exhausted by the end of the week? (Use the normal approximation to the Poisson distribution,
A campus clinic receives walk-in patients at an average rate of thirty per hour, during the midday hours. (a) Find the probability that there will be fewer than twenty-five walk-in patients in a par- ticular midday hour. (b) Find the probability that there will be more than forty walk-in patients
In the hour before noon, a busy corporation switchboard receives an average of fifty-five calls. (a) What is the probability that more than seventy calls will be received during this hour? (b) What is the probability that fewer than fifty calls will be received during this hour? (c) What is the
Bags of a chemical produced by a company have (as in Exercise 22) impurity weights that can be represented by a normal distribution with mean 12.2 grams, and standard deviation 2.8 grams. A random sample of 400 of these bags is taken. What is the probability that at least 100 of them contain less
The tread life of a brand of tire can be represented (as in Exercise 20) by a normal distri- bution with mean 35,000 miles and standard deviation 4,000 miles. A sample of 100 of these tires is taken. What is the probability that more than 25 of them have tread lives of more than 38,000 miles? (Use
A hospital finds that 25% of its bills are at least one month in arrears. A random sample of forty-five bills was taken. (a) What is the probability that less than ten bills in the sample were at least one month in arrears? (b) What is the probability that the number of bills in the sample at least
Suppose that half of all students in campus dormitories are dissatisfied with the food ser- vice. A random sample of forty students was taken. (a) What is the probability that more than fifteen students in the sample were dissatisfied with the food service? (b) What is the probability that the
A sample of 100 blue-collar employees at a large corporation is taken to assess their atti- tudes on a proposed new work schedule. If 60% of all blue-collar employees at this corpo- ration favor the new schedule, what is the probability that less than 50 of the sample mem- bers will be in favor?
It is known that 10% of all the items produced by a particular manufacturing process are defective. From the very large output of a single day, 400 items are selected at random. (a) What is the probability that at least thirty-five of the selected items are defective? (b) What is the probability
It is estimated that the times a well-known rock band, the Living Ingrates, spends on-stage at their concerts follow a normal distribution with mean 200 minutes and standard devia tion 20 minutes. (a) What proportion of concerts played by this band last between 180 and 200 minutes? (b) An audience
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