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...

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...

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....

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

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.

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...

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?

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...

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 ...

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....

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).

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).

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)....

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.

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,...

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.

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...

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 ].

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...

Let X Unif(0, 1). If X = x, then Y Exp(x). Find P(Y > 1) by conditioning on X.

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...

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...

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...

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...

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...

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...

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)...

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...

Carry out a posterior analysis of the control device problem. That is, decide whether the engineers should be sent, and find the expected monetary value associated with either sending or not sending...

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...

If used to model data that have limits (such as the age of a customer using a credit card or amount spent by a customer), what problem does a normal model have?

Cramers V (Chapter 5) measures dependence and varies from 0 to 1. The correlation also measures dependence but varies from - 1 to + 1. Why doesnt it make sense for V also to take on negative values?

Can experiences of parents affect future children? New studies suggest that they can: Early life experiences of parents appear to cause permanent changes in sperm and eggs. In one study, some male...

In Exercise 1.17, we ask whether experiences of parents can affect future children, and describe a study that suggests the answer is yes. A second study, described in the same reference, shows...

On January 14, 2012, Andrew Brown of Great Britain set the world record time (40 days) for rowing solo across the northern Atlantic Ocean. On March 14, 2010, Katie Spotz of the United States became...

New research supports the idea that people who get a good nights sleep look more attractive. In the study, subjects ages 18 to 31 were photographed twice, once after a good nights sleep and once...

Mosquitos carry numerous tropical diseases. A public health agency wants to identify the extent of mosquito breeding areas (e.g., standing water) within its community. Comment on the following...

Match each five number summary with one of the boxplots in Figure 2.37. (a) (2, 12, 14, 17, 25) (b) (5, 15, 18, 20, 23) (c) (10, 12, 13, 18, 25) (d) (12, 12, 15, 20, 24) Match the five number...

Which histograms are approximately symmetric? Refer to histograms A through H in Figure 2.12. 2.0 -50 0.0 1.0 3 150 5 15 25 D -1 -3 8 14 -7 -4 -1 G Figure 2.12 Eight histograms LO 09 007 01 08 01...

Students in the ACHANCHA survey were asked two questions about alcohol use, one about their own personal consumption of alcohol and one about their perception of other students consumption of...

Scientists are working to train dogs to smell cancer, including early stage cancer that might not be detected with other means. In previous studies, dogs have been able to distinguish the smell of...

Consider a dataset giving the adult weight of species of insects. Most species of insects weigh less than 5 grams, but there are a few species that weigh a great deal, including the largest insect...

Are natural languages neutrally, positively, or negatively biased? That is the question a recent study set out to answer. They found the top 5000 words used in English in each of four different...

Proportion of US adults who own a cell phone. In a survey of 2252 US adults, 82% said they had a cell phone. Give the correct notation for the quantity described and give its value.

Estimate values at the 10th percentile and the 90th percentile for the data in Figure 2.25. Estimate the summary statistics requested, using the histogram in Figure 2.25. There are n = 100 data...

Proportion of people who use an electric toothbrush, using data from a sample of 300 adults. State whether the quantity described is a parameter or a statistic and give the correct notation.

Correlation between height and weight for players on the 2010 Brazil World Cup team, using data from all 23 players on the roster. State whether the quantity described is a parameter or a statistic...

Table 2.24 shows the means and standard deviations for four of the variables in the NBAPlayers2011 dataset. FGPct is the field goal percentage, Points is total number of points scored during the...

Every Fourth of July, Nathans Famous in New York City holds a hot dog eating contest in which contestants try to eat as many hot dogs as possible in 10 minutes. In 2011, over 30,000 people watched...

Is there a home field advantage in soccer? We are specifically interested in the Football Association (FA) premier league, a football (soccer) league in Great Britain known for having especially...

Using the sampling distribution shown in Figure 3.5(b), how likely are these sample means: Figure 3.5 b. (a) xÌ = 70 (b) xÌ = 100 (c) xÌ = 140 Refer to the sampling distributions...

In the 2008 Olympic Mens Marathon, 76 athletes finished the race. Their times are stored in the file Olympic Marathon. Use the times stored in the Minutes column. (a) Use StatKey, other technology,...

A type of small snail is very widespread in Japan, and colonies of the snails that are genetically similar have been found very far apart. Scientists wondered how the snails could travel such long...

Proportion of families in the US who were homeless in 2010. The number of homeless families in 2010 was about 170,000 while the total number of families is given in the 2010 Census at 78 million....

Average household income for all houses in the US, using data from the US Census. State whether the quantity described is a parameter or a statistic and give the correct notation.

How many ants will climb on a piece of a peanut butter sandwich left on the ground near an ant hill? To study this, a student in Australia left a piece of a sandwich for several minutes, then covered...

(a) The area to the right of the endpoint is about 0.02. (b) The area to the left of the endpoint is about 0.40. Find endpoint(s) on a N(0, 1) density with the given property.

(a) The area to the right of the endpoint is about 0.90. (b) The area to the left of the endpoint is about 0.65. Find endpoint(s) on a N(0, 1) density with the given property.

Figure 2.71 shows the data and regression line for using weight to predict body fat percentage. For the case with the largest positive residual, estimate the values of both variables. In addition,...

A 95% confidence interval for a proportion p if the sample has n = 100 with p = 0.43, and the standard error is SE = 0.05 Find the indicated confidence interval. Assume the standard error comes from...

A 95% confidence interval for a difference in proportions p 1 p 2 if the samples have n 1 = 50 with p 1 = 0.68 and n 2 = 80 with p 2 = 0.61, and the standard error is SE = 0.085 Find the indicated...

The StudentSurvey dataset includes scores on the Math and Verbal portions of the SAT exam. (a) What would a positive relationship between these two variables imply about SAT scores? What would a...

Figure 2.44 displays the relationship between vitamin use and the concentration of retinol (a micronutrient) in the blood. Does there seem to be an association between these two variables? Figure...

Find the terminal point Q of the vector v with components as given and initial point P. Find |v|. 0, -3, 3; P: (0, 3, -3)

Compute the standard error for sample proportions from a population with proportion p = 0.4 for sample sizes of n = 30, n = 200, and n = 1000. What effect does increasing the sample size have on the...

In this exercise, we see that it is possible to use counts instead of proportions in testing a categorical variable. Data 4.7 describes an experiment to investigate the effectiveness of the two drugs...

Compute the standard error for sample means from a population with mean = 100 and standard deviation = 25 for sample sizes of n = 30, n = 200, and n = 1000. What effect does increasing the sample...

A margin of error within 1 with 99% confidence, assuming a sample from a similar population had s = 3.4 What sample size is needed to give the desired margin of error in estimating a population mean...

A margin of error within 12 with 95% confidence, assuming we estimate that 125 What sample size is needed to give the desired margin of error in estimating a population mean with the indicated...

Exercise 4.19 on page 232 describes a study investigating the effects of exercise on cognitive function. Separate groups of mice were exposed to running wheels for 0, 2, 4, 7, or 10 days. Cognitive...

A distribution of bootstrap means for commuting times in St. Louis is given in Figure 5.13. As in Exercise 5.36, we use a N(21.97, 0.65) distribution as a reasonable model for these bootstrap means....

Answer these problems by calculating a t value and comparing it to the critical value for a two-tailed test at the .05 level. (Note: You can save time by using the means and standard deviations you...

Hospital admissions for asthma in children younger than 15 years was studied in Scotland both before and after comprehensive smoke-free legislation was passed in March 2006. Monthly records were kept...

Figure 5.21 shows a bootstrap distribution of correlations between penalty yards and uniform malevolence using the data on 28 NFL teams in Malevolent-UniformsNFL. We see from the percentiles of the...

Determine whether it is appropriate to use the normal distribution to estimate the p-value. If it is appropriate, use the normal distribution and the given sample results to complete the test of the...

Determine the principal value of the argument and graph it as in Fig. 325. -5, -5 - i, -5 + i pl+i /4

Find (a) The Fourier cosine series (b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. 4 2.

Find (a) The Fourier cosine series (b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. kIN

Find (a) The Fourier cosine series (b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. L L

Represent f (x, y) by a series (15), where f (x, y) = 1, = b = 1