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business
statistics for business and economics

Questions and Answers of Statistics For Business And Economics

14-7. The number of accidents/month on a certain stretch of road is thought to follow a Poisson distribution. Given α = 0.05, does the following observed data set (taken over a six year period)
13-15. At a garden center two types of fertilizers (call them A and B) are being tested. Sixteen identical plots are available for the test, with four seedlings planted in each plot. Fertilizer A is
13-16. The number of sun worshipers (expressed in thousands per week) who frequented two separated but nearby beaches (call them A and B) last season, were:Beach A Beach B(June) 12.7 15 11 16 (June)
13-17. A paint manufacturer wants to determine if its new formula for exterior house paint is superior to the old formula. Sixteen large panels of wood were selected. The panels were paired into
13-18. Recently married couples were asked (independently of each other) to specify (under ideal circumstances) the number of years they would like to wait before having children:Couple Wife Husband
13-19. A production manager wishes to compare two methods for detecting defects on the surface of a finished metal product. Each method is applied to each of 25 finished items:Item Method 1 Method 2
13-20. Suppose two different inspection procedures are being compared using ten pairs of inspectors, matched by years of experience, and so on. The inspectors within each pair are randomly assigned
13-21. The marketing department of ACE Industries is investigating the sales appeal of two different display formats for a given product—one with and one without animation. The two displays are
14-1. ACE Motors surveyed 250 of its customers, asking them to indicate their preferred time for servicing their automobiles and/or trucks: morning(O1), afternoon (O2), or Saturday mornings (O3). The
14-2. John’s project for his statistics course was to toss a single six-sided die 500 times and count the number of times each face obtains in order to determine if the die is biased. (Note: A die
14-3. ACE Nice Products has been in operation for one month. It employs four salesman (denoted A, B, C, and D) who have similar territories. Sales calls made for the month are:Salesman A B C D Number
14-4. Given the following absolute frequency distribution, can we conclude forα = 0.05 that it was drawn from a normal population?X Oi 3.3–3.4 15 3.5–3.6 35 3.7–3.8 70 3.9–4.0 92 4.1–4.2
14-5. A recently printed novel was scanned to determine the number of misprints of a certain type per page. The production editor feels that this type of misprint follows a Poisson distribution with
14-6. A sawmill has its cutting equipment set to produce boards of 8 feet (or 96 inches) in length. The operations manager feels that the “true” length of a board should be normally distributed.
13-14. A statistics class is randomly divided into two groups. One uses only the textbook and the other uses the textbook and study guide. The final averages for the two groups are:Textbook Only
15-6. Suppose that in Exercise 15-5 the “Neutral” category is omitted. Is the proportion of city employees in favor of the new policy the same across the four employee types? Use α = 0.05.
16-8. Our operating assumption in this chapter is that Y is a linear function of X(plus an additive error term). What is important is not that the regression model is linear in the variables but that
16-9. Use the methodology of the preceding exercise and the accompanying data set given here to estimate the parameters β0 and β1 of the expression Y = eβ0+β1X,Y > 0, where the vertical intercept
16-10. What if instead of regressing a dependent variable Y on an explanatory variable X, we regress the Z-scores of Y on the Z-scores of X? From Yi = β0 + β1Xi + εi and Y = β0 + β1X + ¯ε,
16-11. Testing for a structural break, or testing for the equality of two regression equations, can be accomplished by performing the so-called Chow test.2 For instance, suppose we have a sample of
16-12. Binary or dummy or count variables are used to depict the situation in which some attribute is either present or absent. For instance, in taking the medical history of a patient in a hospital
16-13. It is not unusual for a regression study to be performed using a data set containing outlier observations; that is, data points that are remote or distinctly separated from the main scatter of
16-14. Prove the Gauss-Markov Theorem. Hint:(1) First express ˆ β0, ˆ β1 as linear functions of the Yi, i = 1, . . . , n; that is, writeˆ β1 =ni=1 wiYi, where wi = xi ni=1 x2 i;ˆ β0 =ni=1
16-15. Verify that, for ε normal, the least squares estimators are maximum likelihood estimators.
16-16. Verify that COV( ˆ β0, ˆ β1) = −Xσ2ε >ni=1 x2 i .
16-17. Demonstrate that S2ε= ni=1 e2 i >(n − 2) is an unbiased estimator for σ2ε .(Hint: In d2 = 1nni=1 e2 i set ei = −( ˆ β1 − β1)xi + εi − ¯ε. Then correct for the bias in E(d2)
16-18. Verify that SST = SSR + SSE. (Hint: Square yi = ˆ β1xi + ei and sum over all i values. Then determine that ni=1 xiei = 0. The desired result immediately follows.)
16-19. Demonstrate that 0 ≤ R2 ≤ 1. (Hint: Use ni=1 xiyi = ˆ β1x2 i to verify thatni=1 y2i≥ ni=1 e2 i≥ 0.)
12-1. The times (in minutes) required to complete a test administered by the human resources department of ABC Co. are approximately N(78, 12).The department would like to ascertain whether
16-7. Given the following sample data set:Y 4 2 4 6 10 6 8 12 16 14 X 2 4 6 8 10 12 14 16 18 20(a) Perform a regression and correlation study. Use α = 0.05.(b) For X0 = 23, find a 95% prediction
16-6. For the Exercise 16-1 data set:(a) Find ρˆ using (16.37).(b) For α = 0.01, test H0: ρ = 0, against H1: ρ > 0.(c) Using α = 0.05, test H0: ρ = ρ0 = 0.85, against H1: ρ < 0.85.(d) Find a
15-7. Five groups were independently and randomly polled regarding the issue of stricter handgun control. Are the population proportions of individuals favoring stricter handgun control the same
15-8. Determine an estimate of the population odds ratio for Exercise 15-1.Find a 95% confidence interval for the said odds ratio. What is the value of Cramer’s phi-squared statistic?
15-9. Find the value of Cramer’s phi-squared statistic for the contingency tables appearing in Exercises 15-2, 15-3, and 15-4.
15-10. For Exercise 15-2, find estimates of λA|B and λB|A. Also determine 95%confidence intervals for these parameters.
15-11. For Exercise 15-3, determine estimates of λA|B and λB|A. Also find 99%confidence intervals for λA|B and λB|A.
15-12. For Exercise 15-4, produce estimates of λA|B and λB|A. In addition, find 95% confidence intervals for λA|B and λB|A.
15-13. Cook-Rite Inc. has developed a spray product called No-Stick that it says prevents food from sticking to the cooking surface of frying pans.Sixty pans of the same size, material, and make are
15-14. A random sample of n = 50 newspaper business editors were asked if they agreed with the newly elected President’s current domestic economic policies. The same group of editors was polled a
16-1. Given the following sample data set (with the X values held fixed), find:(a) Least squares estimates of β0 and β1(b) An estimate of σ2ε(c) Standard error of estimate(d) Estimates of the
16-2. Using the data presented in Exercise 16-1:(a) Find 95% confidence intervals for β0 and β1.(b) For α = 0.01, test H0: β1 = 0, against H1: β1 > 0.(c) Construct a hypothesis test to determine
16-3. The point elasticity of Y with respect to X is defined as η =(dY/dX)(X/Y). For the population regression equation estimated in Exercise 16-1, find an estimate of the point elasticity
16-4. Using the data presented in Exercise 16-1:(a) Find a 95% confidence band for the population regression equation using the following X values (X = 2, 4, 6, 8, 10, 12, 14, 16).(b) For X = 9, can
16-5. Using the data presented in Exercise 16-1:(a) Construct the analysis-of-variance table for the partitioned sum of squares.(b) What is the value of the coefficient of determination?(c) Use the
15-1. Agroup of males and females is asked to state their preference for driving either a mini-van or SUV on a daily basis. The responses are indicated here. Are preference and sex independent? What
13-13. Suppose 15 additional observations on incomes in each of the communities specified in the preceding exercise have been obtained. Retest using the expanded samples.Additional Community A
12-2. A soft drink company wants to test-market a new product in n = 40 randomly selected convenience stores. The product will be introduced if in excess of 10 cases are sold per week in each store.
12-18. Fenway Park needs to purchase spotlights that exhibit long life as well as uniformity of operating life. Past experience dictates that the variance of bulb life should not exceed 230 (hours)2.
12-19. A sample of n = 25 ball bearings were tested for resistance to heat due to friction. It was found that s2 = 150 (degrees)2. Is this result consistent with the claim that the true variance will
12-20. From a sample of size n = 30 it was determined that s2 = 14.7 (inches)2.Test H0 : σ2 ≥ 18 (inches)2 against H1 : σ2 < 18 (inches)2 at the 1% level.What is your conclusion?
12-21. Farmers consider uniformity of yield to be an important attribute of an agricultural commodity. Two types of seed (brands X and Y) for growing alfalfa are to be compared. They have about the
12-22. Two brands of heavy duty fan belts (call them brand X and brand Y)have about the same average durability. However, their uniformity of wear is questionable. A sample of 10 brand X belts and 15
12-23. A study of two independent samples of sizes nX = 15 and nY = 19 yielded s2X= 10.8 and s2Y= 15.9, respectively. TestH0 : σ2X≥ σ2Y againstH1 : σ2X
12-24. Let us assume that the n = 10 rankings appearing in Table 2.13 are a random sample taken from some unspecified population distribution. From this data we found that rs = 0.93. Test H0 : ρs =
12-25. From a sample of size n = 14 rankings it was determined that the realization of Spearman’s rank correlation coefficient was rs = 0.75. Test H0 : ρs = 0, versus H1 : ρs > 0 using α = 0.01.
12-26. In a test of depth perception two judges each ranked n = 10 objects in order of their approximate distance (rounded to the nearest yard) from a fixed baseline marker. The results are presented
12-27. Given that X is N(μ, 10), the null hypothesis is H0 : μ = μ0 = 27, the alternative hypothesis is H1 : μ < 27, and n = 50, what is the distribution of X? For α = 0.05, find R and R .
12-28. Recalculate β = P(TIIE) for the preceding problem when:(a) n increases to 100(b) α decreases to 0.01(c) σ increases to 15(d) μ1 decreases to 15
12-29. Suppose that in Exercise 12-27 the alternative hypothesis is replaced by H1 : μ = 27. Find R and R . If the true population mean is 22, what is the probability of not rejectingH0? What is
12-30. Suppose X is N(μ, 20) and that, for α ≤ 0.05 and β ≤ 0.05, we want to test H0 : μ = μ0 = 76, against H1 : μ > 76. How large of a sample will be needed if μ1 = 80? How large of a
12-17. A random sample of nX = 140 households from a low-income neighborhood exhibited a head of household registered as a Democrat in x = 70 cases. A similar sample of size nY = 130 taken from a
12-16. In a sample of size nX = 100 from one binomial population it was found that x = 31, and in a sample of size nY = 150 from a second binomial population it was found that y = 50. Test the
12-3. Adistributor of a particular brand of industrial heat lamps states that their average draw of current is 0.9 amps. A sample of n = 12 lamps was tested and it was determined that ¯x = 0.13 with
12-4. Hot-Shot legal services is attempting to increase reading comprehension(RC) of its new hires. Fifteen of its first-year recruits are given a standard RC test. After the test these individuals
12-5. Last year average monthly expenditure per household on a certain product was $12.97. Has there been a statistically significant change in average household expenditure for this year? A sample
12-6. In a random sample of 500 tulip bulbs taken from a normal population 476 of them bloomed. For α=0.05, would you reject the claim that at least 90% of the bulbs will bloom? What is the p-value
12-7. From a random sample of 200 voters it was found that 110 were in favor of a particular piece of legislation. Is opinion equally divided on this legislative issue? Use α = 0.05. What is the
12-8. A manufacturer of small machine parts claims that at least 98% of all parts shipped to ACE Industries conform to specifications. In a sample of 250 parts, it was found that 22 did not conform
12-9. To compare the durabilities of two premium exterior house paints (one from House Depot and the other sold by KK-Mart), nine 4 × 6 panels of each paint were exposed to the elements for a
12-10. Given the following (random) sample results:Sample 1: nX = 16, ¯x = 25, sX = 7.14 Sample 2: nY = 10, ¯y = 29, sY = 5.19 can we conclude at the 10% level of significance that μX < μY?
12-11. The following are yields (in bushels/acre) for two different varieties of winter wheat (call them A and B):A: 62.7, 71.4, 76.7, 59.3, 59.7, 64.7, 69.1, 70.5 B: 69.8, 61.5, 49.9, 53.8, 65.1,
12-12. Ten pairs of twins of a different sex made the following scores on a specialized dexterity test:Female: 92, 83, 95, 96, 85, 61, 76, 80, 92, 87 Male: 90, 93, 80, 86, 71, 91, 80, 70, 80, 81 Is
12-13. Two different methods were used to determine the fat content (expressed as a percent) in different samples of premium vanilla ice cream. Both methods were used on scoops taken from the same
12-14. Atime-and-motion expert feels that she can shorten the workers handling time for 5lb. packages in the shipping department of a candy manufacturer.Suppose n = 10 workers are chosen at random.
12-15. A random sample of n = 1000 persons consisted of 485 females and 515 males. Of the females, 250 were college graduates and 327 of the males had graduated college. Does this sample evidence
12-31. Assume that we are sampling from a binomial population and that we are testing H0 : p = p0 = 45, against H1 : p > 0.45 using α = 0.05 given n = 200. What is the distribution of ˆP ? Find R
12-32. Recalculate β for the preceding problem when:(a) n decreases to 100(b) α increases to 0.10(c) p1 increases to 0.60
12-48. Suppose X1, . . . ,Xn is a random sample taken from a N(μ, σ) random variable with both μ, σ2 unknown. Conduct the generalized likelihood ratio test of H0 : σ2 ≥ σ2 0= 0.37, against H1
12-49. Hypothesis Tests for the Coefficient of Variation Under Random Sampling.We may want to determine if a set of sample random variables{X1, . . . ,Xn} could have been drawn from a population
13-1. A group of workers was asked if they favored a new proposed contract that emphasizes fringe benefits rather than hourly wage rate increases.“Yes” answers are coded as “a” and “no”
13-2. The ACE Dairy Co. has developed what it calls an improved and more flavorful variety of vanilla ice cream. It offers a free sample to its customers and asks them if they would switch to this
13-3. Has the following sequence of observations been generated by a random process? Use α = 0.05.147 130 75 100 92 102 120 122 176 131 89 77 141 91 73 97 138 140 80 100 99 89 107 81 102 91 98 79
13-4. Does the following sequence of daily enrollment figures at Happy Trails Day Camp exhibit negative serial correlation? Use α = 0.01.35 23 25 19 24 23 27 29 25 26 20 18 25 22 27 22 31 29
13-5. Test the following sequence of sales of ice (expressed in bags/day) at a local convenience store over a three-week period for positive serial correlation.Use α = 0.05.15 17 18 22 19 16 14 10
13-6. DEF Transportation Inc. is interested in determining whether or not onehalf of its bus commuters favor the newly instituted schedule change.A sample of 20 riders was selected at random: 14
13-7. A beginning aerobics class for working women has 27 enrollees and meets three times a week. Over a four-week period the number of absences per class was 3 2 3 4 3 5 4 3 6 5 7 6 Is there any
13-8. A wine and sprits shop is interested in knowing whether more than half of its traffic is exclusively for lottery sales. A random sample of 60 customers revealed that 43 purchased only lottery
13-9. A messenger service van completes its route every 90 minutes. For n = 15 runs of this route the number of items picked up for delivery was:23 28 15 40 27 14 16 12 23 36 26 33 24 31 44 Is there
13-10. Wheels Unlimited operates a limousine service between the local airport and one of the major hotels in a large city. The round trip typically takes one hour, but one of the new drivers seems
13-11. A human resources director feels that workers with at least one year of college will score above 75 points on a certain 100 point test. A group of n = 30 workers take the test and generate the
12-47. Suppose X1, . . . ,Xn depicts a random sample drawn from a N(μ, σ) population, where both μ and σ2 are unknown. Perform the generalized likelihood ratio test of H0 : μ = μ0 = 100 versus
12-46. For a set of sample random variables {X1, . . . ,Xn} taken from the probability density function fx; l = -2lxe−lx2 , x > 0, l > 0;0 elsewhere find a uniformly most powerful test of size α
12-33. Suppose that in Exercise 12-31 the alternative hypothesis is replaced by H1 : p = 0.45. Find R and R . If the true population proportion is 0.40, what is the probability of not rejecting H0?
12-34. Assume that we are sampling from a binomial population and that we are to test H0 : p = p0 = 76, against H1 : p < 0.76. The α- and β-risks are each not to exceed 10%. How large of a sample
12-35. Suppose that X is N(μ, 20) and n = 40. Let H0 : μ = μ0 = 100, with H1 : μ < 100. For α = 0.01, find R and R . Find β(μ) for μ = μ1 = 80, 82, 84, 86, 88, 90, 92, 94, 96, 98. Determine
12-36. For X distributed as N(μ, 10), n = 130, α = 0.05, H0 : μ = 200, and H1 : μ = 200, find R and R . Use (12.42) to determine P(μ) for a variety of μ’s satisfying H1. Graph and then
12-37. Suppose a random sample of size n = 200 is extracted from a binomial population. For α = 0.01, let us test H0 : p = p0 = 0.35, against H1 : p >0.35. Find R and R . Use (12.42) to specify the
12-38. Suppose the random variable X has a probability density function of the form f (x; l) = le−lx, x > 0, l > 0. For H0 : l = l0, and H1 : l = l1, l0 > l1, determine the best test of size α of
12-39. Let X1, . . . ,Xn denote a set of sample random variables drawn from the probability mass function p(x; θ) = e−θx X! , X = 0, 1, 2, . . . , θ ≥ 0; zero elsewhere. Find the uniformly
12-40. Determine the form of the Neyman-Pearson critical region for testing H0 : l = l0 versusH1 : l > l0 for a set of n sample random variables drawn from the probability mass function f (x; l) =

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