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A manufacturing process produces components which have a 1% chance of being defective. Successive components are independent. (a) Find the probability that it takes exactly 110 components to be... In the coupon collectors problem, let X be the number of draws of n coupons required to obtain a complete set. Find the variance of X. A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each... A taxi company tests a random sample of 10 steel-belted radial tires of a certain brand and records the following tread wear: 48,000, 53,000, 45,000, 61,000, 59,000, 56,000, 63,000, 49,000, 53,000,... Suppose E[X 2 ] = 1, E[Y 2 ] = 2, and E[XY] = 3. Find E[(X + Y) 2 ]. Let X Unif{2, 1, 0, 1, 2}. (a) Find E[X]. (b) Find E[e X ]. (c) Find E[1/(X + 3)]. A psychologist completes a survey in which residents in a psychiatric hospital are given diagnostic interviews in order to determine their psychiatric diagnoses. After each interview, the researcher... Suppose E[X] = a, E[Y] = b, V[X] = c, and V[Y] = d. If X and Y are independent, find V[2X 3Y + 4]. List the null and alternative hypotheses for Exercise 6.77. Data from exercise 6.77 A researcher has a first-grade readiness test that is administered to kindergarten students and scored at the... Let X Pareto(m, a). (a) For what values of a does the mean exist? For such a, find E[X]. (b) For what values of a does the variance exist? For such a, find V[X]. Let X Beta(a, b), for a > 1. Find E[1/X]. An addictions researcher measured tolerance to alcohol in first-year and fourth-year college students. She gave participants a standard dose of alcohol and then had them walk along a narrow line... Suppose X 1 , . . . , X 1 00 are independent and uniformly distributed on (0, 1). (a) Find the probability the 25th smallest variable is less than 0.20. (b) Find E[X (95) ] and V[X ( 95) ]. (c) The... A density function is proportional to f(x) = x 3 (1 x) 7 , for 0 < x < 1. (a) Find the constant of proportionality. (b) Find the mean and variance of the distribution. (c) Use R to find P(X > 0.5). A random variable X has density function proportional to Use R to find P(1/8 < X < 1/4). f(x) = for 0 < x < 1. (1 2) a. Given the information in this summary table, what is η 2 ? b. Classify it as small, medium, or large. Source of Sum of Degrees of Mean Variability Squares Freedom Square ratio... Suppose that loglinear model (Y 0 , Y 1 ,...,Y T ) holds. Is this a Markov chain? Given the ANOVA summary table below, write the results in APA format for (a) The rows effect, (b) The columns effect, and (c) The interaction effect. Use α = .05. Degrees Source of Sum... Refer to Exercise 9. A device will be manufactured in which two of the components in Exercise 9 will be connected in series. The components function independently, and the device will function only... (a) By hand, draw a scatter diagram treating x as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing... The National Novel Writing Association states that the average novel is at least 50,000 words. A particularly ambitious writing club at a college-preparatory high school had randomly selected members... A researcher wanted to compare the pulse rates of identical twins to see whether there was any difference. Eight sets of twins were randomly selected. The rates are given in the table as number of... Solve the given system using the method of Example 3.25. x 1 - x 2 = 2 x 1 + 2x 2 = 5 Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). [2 0 -1 2 3 The confidence interval for the median of a set of values less than or equal to 25 in number can be found by ordering the data from smallest to largest, finding the median, and using Table J. For... Seventy two percent of American adults have read a book within the past year (pewresearch.org, December 3, 2014). Assume this is the true population proportion and that you plan to take a sample... 1. How is the number of members of the Accounting Department who are selected to serve on the grievance committee distributed? 2. Using the probability distribution you identified in (1), what is the... 1. Tables and/or graphical displays that show the amount of expenditures by category and percentage of total expenditures by category. 2. A table that shows the amount of expenditures by department... Solve the ODE by integration or by remembering a differentiation formula. y' = -1.5y A rocket is shot straight up from the earth, with a net acceleration (= acceleration by the rocket engine minus gravitational pullback) of 7t m/sec 2 during the initial stage of flight until the... Find the steady-state motion of the massspring system modeled by the ODE. Show the details of your work. y" + 2.5y' + 10y = -13.6 sin 4t Apply the given operator to the given functions. Show all steps in detail. (D 2 + 4.00D + 3.36I)y = 0 You are given the following information for the data displayed in the scatterplot in Exercise 5. Least squares prediction equation: Y' = .34(X) .56 SS y = 826.23 SS y |x = 145.44 Using this... If P = 0.50, which is the best conclusion? i. H 0 is definitely false. ii. H 0 is definitely true. iii. There is a 50% probability that H 0 is true. iv. H 0 is plausible, and H 1 is false. v. Both H... Using Table B in Appendix C of your textbook, find the critical t values for each of the following hypothesis tests: 1. One-tailed test, lower tail critical; α = .01 , n 1 = 15; n 2 =... The ANOVA summary table for the Exercise 3 data is presented below. Refer to this information and to the Exercise 3 information to answer the questions. Evaluate statistical significance with alpha... Find the motion of the mass??spring system in Fig. 72 with mass 0.125 kg, damping 0, spring constant 1.125 kg/sec 2 , and driving force cos t - 4 sin t nt, assuming zero initial displacement and... Identify each variable as nominal, ordinal, or interval. a. UK political party preference (Labour, Conservative, Social Democrat) b. Anxiety rating (none, mild, moderate, severe, very severe) c.... Consider three nonstandard dice. Instead of the numbers 1 through 6, die A has two 3??s, two 5??s, and two 7??s; die B has two 2??s, two 4??s, and two 9??s; and die C has two 1??s, two 6??s, and two... Refer to Table 3.10. In the same survey, gender was cross-classified with party identification. Table 3.11 shows some results. Explain how to interpret all the results on this printout. Table 3.10:... Determine which sets of vectors are orthogonal. -2 -4 3 -6 -1 -1 2 4 7. Refer to Problem 4.6. The sample mean and variance are 5.0 and 4.2 for treatment A and 9.0 and 8.4 for treatment B. a. Is there evidence of overdispersion for the Poisson model having a dummy... In the 2000 U.S. presidential election, Palm Beach County in Florida was the focus of unusual voting patterns (including a large number of illegal double votes) apparently caused by a confusing... Let B = {v 1 ,......, v n } be an orthonormal basis for R n . (a) Prove that, for any x and y in R n , x y = (x v 1 ) (y v 1 ) + (x v 2 ) (y v 2 ) + ...........+ (x v n ) (y v n ) (This... Find the orthogonal projection of v onto the subspace W spanned by the vectors u i . (You may assume that the vectors u i are orthogonal.) 1 -1 2 , u1 3 A bag has r red and b blue balls. Balls are picked at random without replacement. Let X be the number of selections required for the first red ball to be picked. (a) Explain why X does not have a... A piece of plywood is composed of five layers. The two outer layers are veneers with thickness 0.50 0.02 mm, and the three inner layers each have thickness 6.25 0.05 mm. Assume the thicknesses of... Breathing rates, in breaths per minute, were measured for a group of 10 subjects at rest, and then during moderate exercise. The results were as follows: Find a 95% confidence interval for the... Radiocarbon dating: Carbon-14 is a radioactive isotope of carbon that decays by emitting a beta particle. In the earths atmosphere, approximately one carbon atom in 1012 is carbon-14. Living... The following table presents the 100 senators of the 113th U.S. Congress on January 3, 2013, classified by political party affiliation and gender. A senator is selected at random from this group.... A car dealer sold 750 automobiles last year. The following table categorizes the cars sold by size and color and presents the number of cars in each category. A car is to be chosen at random from the... Simulate the probability of obtaining at least one match in the problem of coincidences (see Exercise 4.54). Data from Exercise 4.54. In the original matching problem, Montmort asked for the... A random variable X has density function f(x) = ce x , for 2 < x < 2. (a) Find c. (b) Find P(X An ODE may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution. The ODE y' 2 - xy' + y = 0 is of this kind. Show by... The Laplace distribution, also known as the double exponential distribution, has density function proportional to e |x| for all real x. Find the mean and variance of the distribution. A machine that fills beverage cans is supposed to put 12 ounces of beverage in each can. The variance of the amount in each can is 0.01. The machine is moved to a new location. To determine whether... Show that f(x) = e x e ex , for all x, is a probability density function. If X has such a density function find the cdf of X. Suppose f(x) and g(x) are probability density functions. Under what conditions on the constants and will the function f(x) + g(x) be a probability density function? Below is a sample of the data in the file NFLAttendance which contains the 32 teams in the National Football League, their conference affiliation, their division, and their average home attendance.... Suppose U ¼ Unif(0, 1). Find the density of Y = tan ( TU 2 Let X Unif(0, 1). Find E[e X ] two ways: (a) By finding the density of e X and then computing the expectation with respect to that distribution. (b) By using the law of the unconscious statistician. Let X Unif(a, b). Suppose Y is a linear function of X. That is Y = mX + n. Where m and n are constants. Assume also that m > 0. Show that Y is uniformly distributed on the interval (ma + n, mb + n). The time until the light in Bobs office fails is exponentially distributed with mean 2 hours. The time until the computer crashes in Bobs office is exponentially distributed with mean 3 hours.... Bianca Peterson is a marketing engineer for Hexagon Composites, a company which sells carbon composite storage tanks. In an effort to gain product adoptions from customers, Bianca goes on sales trips... Suppose B and C are independent random variables each uniformly distributed on (0,1). Find the probability that the roots of the equation x 2 + Bx + C = 0 are real. Let X be a continuous random variable with cdf F. Since F(x) is a function of x, F(X) is a random variable which is a function of X. Suppose F is invertible. Find the distribution of F(X). Zeitlers Department Stores sells its products online and through traditional brick-andmortar stores. The following parallel-coordinates plot displays data from a sample of 20 customers who purchased... Suppose X and Y are i.i.d. exponential random variables with = 1. Find the density of X/Y and use it to compute P(X/Y < 1). Table 8.18 refers to automobile accident records in Florida in 1988. a. Find a loglinear model that describes the data well. Interpret associations. b. Treating whether killed as the response, fit an... Let Y be a Poisson random variable with mean . a. For a constant c > 0, show that E[log(Y + c)] = log + (c 1/2)/ + O( 2 ) (Note that log(Y + c) = log + log[1 + (Y + c )/].) b. Cell counts in a 2 ... Use the acceptreject method to simulate points uniformly distributed on the circle of radius 1 centered at the origin. Use your simulation to approximate the expected distance of a point inside the... Let X Gamma(a, ). Find V[X] using the methods of Section 7.2.1. Let X Unif(0, 2). If X = x, let Y be uniformly distributed on (0, x). (a) Find the joint density of X and Y. (b) Find the marginal densities of X and Y. (c) Find the conditional densities of X and Y. According to Nielsen Media Research, approximately 86% of all U.S. households have High-definition television (HDTV). In addition, 49% of all U.S. households own Digital Video Recorders (DVR).... Let A and B be events such that P(A) = 0.3, P(B) = 0.5, and P(AB) = 0.2. (a) Find E[I A |I B = 0] and E[I A |I B = 1]. (b) Show that E[E[I A |I B ]] = E[I A ]. Let X Unif(0, 1). If X = x, then Y Exp(x). Find P(Y > 1) by conditioning on X. find an invertible matrix P and a matrix C of the form such that A = PCP -1 . Sketch the first six points of the trajectory for the dynamical system and classify the origin as a spiral attractor,... Repeat Exercise 3, using the S chart in place of the R chart. Repeat Exercise 3. Samples of three resistors are taken periodically, and the resistances, in ohms, are measured. The following table... The article ??Estimating Resource Requirements at Conceptual Design Stage Using Neural Networks?? (A. Elazouni, I. Nosair, et al., Journal of Computing in Civil Engineering, 1997:217??223) suggests... An engineer measures a dependent variable y and independent variables x 1 , x 2 , and x 3 . MINITAB output for the model y = β 0 + β 1 x 1 + β 2 x 2 +... Let X 1 , . . . , X n be a simple random sample from a N(μ, Ï 2 ) population. For any constant k > 0, define Consider ÏÌ 2 k as an estimator of Ï 2 . a. Compute... In a study relating the degree of warping, in mm, of a copper plate (y) to temperature in ?C (x), the foll owing summary statistics were calculated: n = 40, a. Compute the correlation r between the... A new concrete mix is being designed to provide adequate compressive strength for concrete blocks. The specification for a particular application calls for the blocks to have a mean compressive... A copper smelting process is supposed to reduce the arsenic content of the copper to less than 1000 ppm. Let denote the mean arsenic content for copper treated by this process, and assume that the... This exercise requires ideas from Section 2.6. In a two-sample experiment, when each item in one sample is paired with an item in the other, the paired t test (Section 6.8) can be used to test... Measure the shelf life of bunches of bananas (the number of days until the bananas go bad) for a large sample. (a) What are the cases? (b) What is the variable and is it quantitative or categorical? Estimate the bending strength of beams by bending 10 beams until they break and recording the force at which the beams broke. (a) What are the cases? (b) What is the variable and is it quantitative... Information about the worlds countries is given in All Countries, introduced in Data 1.2 . You can find a description of the variables in Appendix B. For the full dataset: (a) Indicate which of the... A fishing boat captain examines one days catch of fish to see if the average weight of fish in that area is large enough to make fishing there profitable. Describe the sample and describe a... To estimate the proportion of Americans who support changing the drinking age from 21 to 18, a random sample of 100 college students are asked the question Would you support a measure to lower the... To investigate the growth of the canine population in New York City, 100 dogs are randomly selected from a registry of licensed pets in the city, and it is found that 78 of them have been neutered.... To investigate interest across all residents of the US in a new type of ice skate, a random sample of 1500 people in Minnesota are asked about their interest in the product. (a) The sample. (b) The... To determine the height distribution of female high school students, the rosters are collected from 20 randomly selected high school girls basketball teams. (a) The sample. (b) The population of... The population is incoming students at a particular university. The name of each incoming student is thrown into a hat, the names are mixed, and 20 names (each corresponding to a different student)... See Example 4.21 on the St. Petersburg paradox. Modify the game so that you only receive $210 = $1024 if any number greater than or equal to 10 tosses are required to obtain the first tail. Find the... Zipfs law is a discrete distribution related to the Pareto distribution. If X has a Zipfs law distribution with parameters s > 0 and n {1, 2, . . .}, then The distribution is used to model... A company that produces and markets video game systems wishes to assess its customers level of satisfaction with a relatively new model, the XYZ Box. In the six months since the introduction of the... Construct a contingency table of automobile origin versus overall mechanical quality rating. Set up rows corresponding to the United States, the Pacific Rim (Japan/Korea), and Europe... On its website, the Statesman Journal newspaper (Salem, Oregon, 2005) reports mortgage loan interest rates for 30 year and 15-year fixed-rate mortgage loans for a number of Willamette Valley lending... 1. The timeplot in Figure 9 shows the price at the end of the month of one share of stock in Pfizer. What seems to be happening? How could you verify your answer? 2. Figure 10 tracks the value of an...
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