Review Exercises 1
48. Suppose that you have an intelligent friend who has not studied probability. How would you explain Jo your friend the distinction between mutually exclusive events and independent events Illustrate your answer with suitable examples. 49. State, with reasons, whether each of the following statements is true or false a The complement of the union of two events is the intersection of their complements. b The sum of the probabilities of collectively exhaustive events must equal 1. c The number...
Numerical Summary Of Grouped Data
A histogram provides a very convenient visual summary of a large set of numerical observations. However, an investigator will frequently want, in addition to this picture, some numerical summary measures of central tendency and dispersion. When the original data are available, this can be accomplished using the procedures discussed in Sections 2.2 and 2.3. Given modern computing resources, this typically provides only a modest computational burden, even for very large data sets. However, it...
Example 12
Refer to the results in Example 7.1. We can now state that the estimates of the population mean, variance, and proportion of stocks for which the price-earnings ratio exceeded 8.5 are obtained through unbiased estimation procedures. However, the estimate of the population standard deviation, sx 3.97, is not obtained through an unbiased estimation procedure. An estimator that is not unbiased is said to be biased. The extent of the bias is the difference between the mean of the estimator and the...
Info Fln
c Find the sample variance and standard deviation. d Find the interquartile range. 1 7. Consider the following four populations All of these populations have the same mean. Without doing the calculations, arrange the populations according to the magnitudes of their variances, from smallest to largest. Then check your intuition by calculating the four population variances. 18. An auditor finds that the values of a corporation's accounts receivable have mean 295 and standard deviation 63. a Find...
Worse
About the same 106 153 75 About the same 106 153 75 a Find the probability that if the forecast is for a worse performance in earnings, this outcome will result. b If the forecast is for an improvement in earnings, find the probability that this outcome fails to result. 71. A dean has found that 62 of entering freshmen and 78 of junior college transfers eventually graduate. Of all entering students, 73 are freshmen, and the remainder are junior college transfers. a What is the probability that...
Normal Approximation To The Poisson Distribution
Let the random variable X denote the number of occurrences of an event in a particular interval of time and denote by A the expected, or mean, number of occurrences in that time interval. Then X obeys the Poisson distribution discussed in Section 4.7, with mean and variance Consider now the situation in which the mean number of occurrences, A, is large. Suppose that the time interval of interest is broken down into subintervals of equal width, as in Figure 5.19. Then the total number of...
Example 92
The decision rule is to reject H0 in favor of Ht if For a 5 -level test, a .05, and za 2 2.025 1.96. Then, since 2.50 is less than -1.96, the null hypothesis is rejected at the 5 significance level. In fact, according to the decision rule, the null hypothesis will be rejected for any significance level a for which zaj2 is bigger than 2.50. From Table 3 in the Appendix, we see that when za 2 is 2.50, a 2 is equal to .0062. Hence, a .0124. This is the p-value of the test, implying that the null...
Info Mhh
a Without assuming that the population variance is known, test the null hypothesis that the population mean weight of active ingredient per tablet is 5 grams. Use a two-sided alternative and a 5 significance level. State any assumptions that you make. b Stating any assumptions that you make, test the null hypothesis that the population standard deviation is .025 gram against the alternative hypothesis that the population standard deviation exceeds .025 gram. Use a 5 ' significance level. 63....
Example 89
For a random sample of ninety-six smokers, the mean amount of short-term absenteeism from work was 2.15 hours per month, and the sample standard deviation was 2.09 hours per month. For an independent random sample of 206 employees who had never smoked, the mean amount of absenteeism was 1.69 hours per month, and the sample standard deviation was 1.91 hours per month.21 Find a 99 confidence interval for the difference between the two population means. x 2.15 nx - 96 sx 2.09 and for those who...
Exercises Ovn
43. It is hypothesized that the more expert a group of people examining a corporation's financial report, the more variable will be their predictions about its future. Independent random samples, each of thirty individuals, from groups of different levels of expertise were chosen.39 The low-expertise group consisted of people who had just completed their first intermediate accounting course. Members of the high-expertise group had completed undergraduate studies and were employed by reputable...
Example 417
An analyst predicted that 3.5 of all small corporations would file for bankruptcy in the coming year. For a random sample of 100 small corporations, estimate the probability that at least three will file for bankruptcy in the next year, assuming that the analyst's prediction is correct. The distribution of the number X of filings for bankruptcy is binomial, with n 100 and p .035, so the mean of the distribution is We will use the Poisson distribution, with mean A 3.5, to approximate our...
Review Exercises Xfk
49. 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 random variable e The covariance between a pair of continuous random variables 50. In the real world, measurements on any quantity of interest are almost invariably made on a discrete scale. Therefore, the...
Info Vtv
a Find the sample mean and variance. b Assuming a normal population distribution, find a 95 confidence interval for the mean speed of all automobiles travelling over this stretch of highway. 12. A clinic offers a weight reduction program. A review of its records found the following weight losses, in pounds, for a random sample of ten of its patients at the conclusion of the program. Assume the population distribution is normal. a Find a 99 confidence interval for the population mean. b Without...
Info Wtc
Confidence intervals for the population proportion are centered on the sample proportion. Also, it can be seen that all other things being equal, the larger the sample size n, the narrower the confidence interval. This reflects the increasing precision of the information about the population proportion obtained as the sample size becomes larger. A random sample of 344 industrial buyers was asked What is your firm's policy for purchasing personnel to follow on accepting gifts from vendors For...
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a What is the probability that a randomly chosen adult from this population voted b What is the probability that a randomly chosen adult from this population read newspapers c What is the probability that a randomly chosen adult who did not read newspapers, from this population, did not vote 41. A campus student club distributed material about membership to new students attending an orientation meeting. Of those receiving this material, 40 were men and 60 were women. Subsequently, it was found...
Info Brw
30. In a study comparing state-chartered and federally chartered credit unions, a sample of 145 matched pairs of credit unions was formed. Each pair contained one state-chartered and one federally chartered credit union. The pairings were made in such a way that the two members were as similar as possible in regard to such factors as size and age.24 The ratio of total loans outstanding to total assets was calculated for each of the credit unions. For this ratio, the sample mean difference...
Exercises Shz
1. A professor is planning to use a new book for a financial accounting course and is considering three possibilities Financial Accounting Made Easy, Financial Accounting Without Tears, and Financial Accounting for Profit and Pleasure. He contacted a random sample of sixty students who had already taken his course and asked each to review the three books, indicating a first preference. The results obtained are shown in the table. Test the null hypothesis that for this population, first...
Exercises Wka
1. A college admissions officer for an M.B.A. program has determined that historically, candidates have undergraduate grade point averages that are normally distributed with standard deviation .45. A random sample of twenty-five applications from the current year is taken, yielding a sample mean grade point average of 2.90. a Find a 95 confidence interval for the population mean. b Based on these sample results, a statistician computes for the population mean a confidence interval running from...
Info Dxl
9. A random sample of fifty students were asked to estimate how much money they spent on textbooks in a year. The sample skewness of these amounts was found to be .83 and the sample kurtosis was 3.98. Test at the 10 -level the null hypothesis that the population distribution of amounts spent is normal. 7 Data from C. F. Lee and R. M. Leuthold, Investment horizon, risk, and return in commodity futures markets An empirical analysis with daily data, Quarterly Review of Economics and Business, 23,...
Confidence Intervals For The Mean Of A Normal Population Population Variance
We can now use the Student's t distribution to derive confidence intervals for the mean of a normal population when the variance is unknown, using an argument similar to that of Section 8.2. Assume that a random sample of n observations is available from a normal population with mean fx and unknown variance and that confidence intervals for the population mean are required. Let X and s 2 denote the sample mean and variance. Then, from Section 8.3, we know that the random variable follows a...
Info Gxn
To verify these results, note that it follows from the definition of expectation that if Z takes values a bx with probabilities Px x , its mean is Then, since the first summation on the right-hand side of this equation is 1 by property ii of Section 4.2 and the second summation is, by definition, the mean of X, we have E Z a bfxx as in Eq. 4.3.1 . Further, the variance of Z is, by definition oz2 - E Z - fjizf X a bx xz 2Px x Substituting a bfxx for iz then gives oz2 X bx bixxfPx x b2 X x...
Exercises Jxe
7. Use the Wilcoxon test to analyze the data of Exercise 1. Discuss your findings. 8. A random sample of ten students were asked to rate, in a blind taste test, the quality of two brands of beer, one domestic and one imported. Ratings were on a scale from one poor to ten excellent . The accompanying table gives the results. Use the Wilcoxon test to test the null hypothesis that the distribution of the paired differences is centered on 0 against the alternative that the imported brand is...
Info Vai
Assume that the population distribution is normal. a Find the sample mean and standard deviation. b Test at the 5 significance level the null hypothesis that the population mean is 40 hours against the alternative that it is higher. c Test at the 5 significance level against a two-sided alternative the null hypothesis that the population standard deviation is 10 hours. 60. State whether each of the following is true or false. a The significance level of a test is the probability that the null...
Info Dko
a Find the mean and standard deviation of the number of weekly breakdowns. b It is estimated that each breakdown costs the company 1,500 in lost output. Find the mean and standard deviation of the weekly cost to the company from breakdown of this machine. 18. An investor is considering three strategies for a 1,000 investment. The probable returns are estimated as follows Strategy 1 c A profit of 10,000 with probability .15 and a loss of 1,000 with probability .85 Strategy 2 A profit of 1,000...
A Manufacturer Of Detergent Claims That The Contents Of Boxes Soldweigh On
1. A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weights is known to be normal, with standard deviation .4 ounce. A random sample of sixteen boxes yielded a sample mean weight of 15.84 ounces. Test at the 10 significance level the null hypothesis that the population mean weight is at least 16 ounces. 2. A company which receives shipments of batteries tests a random sample of nine of them before agreeing to take a...
Making Sense Of Numerical Information
Any manager operating in the business environment requires as much information as possible about the characteristics of that environment. In the modern era, thanks in part to the massive information storage capacities of computer systems, much of the available information is quantitative. For example, it may be necessary to assimilate movements in interest rates, stock market prices, money supply, or unemployment. Market research surveys are carried out to determine the strength of product...
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Stating carefully any assumptions you make, find a 95 confidence interval for the difference between the two population means. 72. Samples of patients were asked to assess satisfaction with health care systems on a scale from one very satisfied to four very dissatisfied .42 For a random sample of 879 health maintenance organization H.M.O. members, the mean satisfaction level was 1.48, and the sample standard deviation was .68. For an independent random sample of 801 non-H.M.O. members, the mean...
Info Uxe
a For a randomly chosen person from this group, what is the probability function for number of purchases made in the week b For a person in this group who has three cards, what is the probability function for number of purchases made in the week c Are the number of cards owned and number of purchases made statistically independent 22. A market researcher wants to determine whether a new model of a personal computer, which had been advertised on a late-night talk show, had achieved more brand...
Exercises Soj
1 8. In October 1992, ownership of the San Francisco Giants baseball team was considering a sale of the franchise that would lead to a move to St. Petersburg, Florida. A random sample of 610 San Francisco Bay Area taxpayers, carried out by The San Francisco Examiner, contained 50.7 who would be disappointed by this move. Find a 99 confidence interval for the population proportion of Bay Area taxpayers with this feeling. 19. A random sample was taken of 189 National Basketball Association games...
Info Wgm
Two of these funds are to be chosen at random. a How many possible samples of two funds are there c Find the probability function of the sampling distribution of the sample mean. d Verify directly that the mean of the sampling distribution of the sample mean is equal to the population mean. 50. Of what relevance is the central limit theorem to the sampling distribution of the sample mean 51. Refer to Exercise 49. Find the probability function of the sampling distribution of the sample...
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30. An instructor has found that times spent by students on a particular homework assignment follow a normal distribution with mean 150 minutes and standard deviation 40 minutes. a The probability is .9 that a randomly chosen student spends more than how many minutes on this assignment b The probability is .8 that a randomly chosen student spends less than how many minutes on this assignment c Two students are chosen at random. What is the probability that at least one of them spends at least...
Info Uca
35. Following a touchdown, a college football coach has the option to elect to attempt a 2-point conversion that is, 2 additional points are scored if the attempt is successful, and none if it is unsuccessful. The coach believes that the probability is .4 that his team will be successful in any attempt, and that outcomes of different attempts are independent of each other. In a particular game, the team scored four touchdowns and 2-point conversion attempts were made each time. a What is the...
For An Audience Of 600 People Attending A Concert The Average Time On The
rrr - I Cv - ixx 2 P x c Verify for this example that 14. In taking a sample of n observations from a population of N members, the variance of the sampling distribution of the sample mean is The quantity yy is called the finite population correction factor. a To get some feeling for possible magnitudes of the finite population correction factor, calculate it for samples of n 20 observations from populations of N 20, 40, 100, 1,000, and 10,000 members. b Explain why the result for N 20, found in...
Exercises Evb
1. When a production process is operating correctly, the resistance in ohms of electrical components 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 resistance. b Find the variance of the sample mean. c Find the standard error of the sample mean. d What is the probability that the sample mean exceeds 93.0 ohms 2. The lifetimes of lightbulbs produced by a...
Info Rta
a Draw the probability function. b Calculate and draw the cumulative probability function. c What is the probability that on a randomly chosen weekday, there will be at least four riders from the subdivision on this service d Two weekdays are chosen at random. What is the probability that on both these days there will be fewer than three riders from the subdivision on this service 5. a A very large shipment of parts contains 10 defectives. Two parts are chosen at ran dom from the shipment and...
Abc Acb Bac Bca Cab Cba
This example is illustrated in the tree diagram of Figure 3.9. We begin at the intersection on the left-hand side of the figure by choosing one of the three letters to fill the first position. Following each of the emerging branches, we then have two possibilities for filling the second position. For example, if the letter A is in the first position, either B or C must be placed in the second position. Finally, once the first two positions have FIGURE 3.9 Tree diagram for Example 3.1 First...
Example 34
Consider the observation of the Dow-Jones industrial average over two consecutive days. We will designate the four basic outcomes as follows Dow-Jones average rises on both days. Dow-Jones average rises on the first day but does not rise on the second day. Dow-Jones average does not rise on the first day but rises on the second day. Dow-Jones average does not rise on either day. Clearly, one of these outcomes must occur, but not more than one can occur at the same time. We can therefore write...
Info Ftx
a Find a 90 confidence interval for the population mean weight of impurities. b Without doing the calculations, state whether a 95 confidence interval for the population mean would be wider than, narrower than, or the same as that found in a . 4. A personnel manager has found that historically, the scores on aptitude tests given to applicants for entry-level positions follow a normal distribution with standard deviation 32.4 points. A random sample of nine test scores from the current group of...
Fixed Cost 2000 Plus 2000for Each Day Taken To Complete Project Find Mean And
a What is the probability that a randomly chosen project will take less than 3 days to complete b Find the expected time to complete a project. c Find the standard deviation of time required to complete a project. d The contractor's project cost is made up of two parts a fixed cost of 20,000, plus 2,000 for each day taken to complete the project. Find the mean and standard deviation of total project cost. e If three projects are undertaken, what is the probability that at least two of them...
Exercises Uhp
45. 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 minutes with the professor. c Find the probability that a given student spends between 10 and 15 minutes with the professor. 46. Times to gather preliminary information from arrivals at an outpatients...
Bags Of Chemical Produced By A Company Have Impurity Weights That Can Be
3 5. A car rental company has determined that the probability a car will need service work in any given month is .2. The company has 900 cars. a What is the probability that more than 200 cars will require service work in a particular month b What is the probability that fewer than 175 cars will need service work in a given month Use the normal approximation to the binomial distribution, without the continuity correction. 36. It is known that 10 of all the items produced by a particular...
Exercises Hhg
3 5. 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. 36. 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...
Exercises Dmo
16. Let the random variable Z follow a standard normal distribution. e Find 1.20 lt Z lt 1.33 f Find P 1.70 lt Z lt 1.20 g Find - 1.70 lt Z lt - 1.00 17. Let the random variable Z follow a standard normal distribution. a The probability is .70 that Z is less than what number b The probability is .25 that Z is Less than what number' c The probability is .2 that Z is greater than what number d The probability is .6 that Z is greater than what number 18. It is known that amounts of money spent on...
Info Wgx
6 P. Bundy and D. Norris, What accounting students consider important in the job selection process, Journal of Applied Business Research, 8, no. 2 1992 , 1-6. 7 J. M. Carey and E. M. White, The effects of graphical versus numerical response on the accuracy of graph-based forecasts, Journal of Management, 17 1991 , 77-96. Assuming a normal distribution, test the null hypothesis that the population mean prediction would be 78.5. Use a two-sided alternative and a 10 significance level. 14. A beer...
Example 54
a Draw the probability density function. b Find and draw the cumulative distribution function. c Find the probability that the best choice of the weight X is less than .25. d Find the probability that the best choice of the weight X is more than .75. e Find the probability that the best choice of the weight X is between .2 and .8. 2. The jurisdiction of a rescue team includes emergencies occurring on a stretch of river that is four miles long. Experience has shown that the distance along this...
Find The Probability That A Randomly Chosen Student Has Not Visited A Museum
a Find the probability that a randomly chosen student has not visited a museum in the last year. b Find the means of the random variables X and Y. c Find and interpret the covariance between the random variables X and Y. 65. A basketball team's star 3-point shooter takes six 3-point shots in a game. Historically, he makes 40 of all 3-point shots attempted. Answer the following questions about the outcome of the six 3-point shots taken in this game, stating at the outset what assumptions you...
A Long-distance Taxi Service Owns 4 Vehicles.these Are Of Different Ages And
a Find the expected number of cars that will be sold in the week. b Find the standard deviation of the number of cars that will be sold in the week. c The salesman receives for the week a salary of 250, plus an additional 300 for each car sold. Find the mean and standard deviation of his total salary for the week. d What is the probability that the salesman's salary for the week will be more than 1,000 61. A multiple-choice test has nine questions. For each question, there are four possible...
Exercises Gcb
28. A production manager knows that 5 of components produced by a particular manufacturing process have some defect. Six of these components, whose characteristics can be assumed to be independent of each other, were examined. a What is the probability that none of these components has a defect b What is the probability that one of these components has a defect c What is the probability that at least two of these components have a defect 29. A politician believes that 25 of all macroeconomists...
Info Fnh
a Find the probability of event A. b Find the probability of event B. c Describe the event that is the complement of A. d Find the probability of the complement of A. e Describe the event that is the intersection of A and B. f Find the probability of the intersection of A and B. g Describe the event that is the union of A and B. h Find the probability of the union of A and B. i Are A and 8 mutually exclusive j Are A and B collectively exhaustive 3. A manager has available a pool of eight...
i EMOTIVE AND LOADED STATEMENTS
Numbers, in and of themselves, contain no value judgments. Data simply provide factual material, which could, of course, be useful on one side or another of a particular 6 The title of this section is inspired by D. Huff and I. Geis, How to Lie with Statistics New York Norton, 1954 . This delightful little book is essential reading for anyone with a serious interest in the presentation of statistical information. Also see H. Wainer, How to display data badly, American Statistician, 38 1984 ,...