Probability And Statistics For Engineers 5th Edition Richard L Scheaffer, Madhuri Mulekar, James T McClave, Cecie Starr - Solutions

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Conduct a regression analysis of the linear and interactive effects of PREP and FAMI on RTEN using the lm() function. Interpret the results in your own words, making sure to report the outcomes of the significance tests.
Copy USJudgeRatings into a new data frame object and center the variables. Use an appropriate R function to report the means of the variables and confirm that they are all 0.Report the means and say in your own words why centering is important when testing regression interactions.
Using similar code as that which appeared in the regression interactions section of this chapter, plot regression lines showing PREP (on the Xaxis) and RTEN (on the Yaxis). Plot two regression lines, one for the half of the data set where FAMI is below its median value and one where FAMI is at or
The USJudgeRatings data sets package in R contains a set of 43 ratings of the com‑petence of state judges in the U.S. Superior Court (from the 1970s). Reveal the vari‑able names and data types in the data set using appropriate R commands. In this exercise, we are particularly interested in two
Use anovaBF() to examine the main effects and interaction as described in Exer‑cise 3.Interpret the results in your own words. If you also did Exercise 3, contrast the results from the traditional ANOVA analysis with the results of the Bayes Factor ANOVA. Important note: The anovaBF() function is
Use aov() to produce significance tests of the main effects of Type and Treatment and the interaction between Type and Treatment using uptake as the dependent vari‑able. Make sure to state each null hypothesis and then use the correct language to describe the outcomes of the significance tests.
Use the interaction.plot() command in the HSAUR package to display a means plot of the “uptake” variable, using “Type” and “Treatment” as the factors. Interpret the results: Without knowing any of the statistics, would you guess that there is or is not an interaction between Type and
The data sets package in R contains a data frame called “C02.” Reveal the variable names and data types in the data set using appropriate R commands and identify the names and levels of all the independent variables that are factors. What are the possible twoway interactions that could be
Bonus Statistical Thinking Exercise: Here is a block of code that does something similar to what we did at the beginning of the chapter with the madeup GPA data set:set.seed(1)betaVar
The car package that you loaded for Exercise 6 contains several additional data sets that are suitable for multiple regression analysis. Run data(Duncan) to load the Dun‑can data on the prestige of occupations. Analyze the hypothesis that the “income”variable and the “education” variable
Run vif() on the results of the model from Exercise 2.Interpret the results. Then run a model that predicts mpg from all five of the predictors in myCars. Run vif() on those results and interpret what you find.
Run install.packages() and library() for the “car” package. The car package is “com‑panion to applied regression” rather than more data about automobiles. Read the help file for the vif() procedure and then look up more information online about how to interpret the results. Then write
Run lmBF() with the same model as for Exercise 4, but with the options posterior=TRUE and iterations=10000. Interpret the resulting information about the coefficients.
Run a multiple regression analysis on the myCars data with lmBF(), using mpg as the dependent variable and wt (weight) and hp (horsepower) as the predictors. Interpret the resulting Bayes factor in terms of the odds in favor of the alternative hypothesis. If you did Exercise 2, do these results
Using the results of the analysis from Exercise 2, construct a prediction equation for mpg using all three of the coefficients from the analysis (the intercept along with the two Bweights). Pretend that an automobile designer has asked you to predict the mpg for a car with 110 horsepower and a
Run a multiple regression analysis on the myCars data with lm(), using mpg as the dependent variable and wt (weight) and hp (horsepower) as the predictors. Make sure to say whether or not the overall Rsquared was significant. If it was significant, report the value and say in your own words whether
Create and interpret a bivariate correlation matrix using cor(myCars) keeping in mind the idea that you will be trying to predict the mpg variable. Which other variable might be the single best predictor of mpg?
The data sets package in R contains a small data set called mtcars that contains n = 32 observations of the characteristics of different automobiles. Create a new data frame from part of this data set using this command: myCars < data.frame(mtcars[,1:6]).
Using the UCBA data, run contingencyTableBF() with posterior sampling. Use the results to calculate a 95% HDI of the difference in proportions between the columns.
Use contingencyTableBF() to conduct a Bayes factor analysis on the UCB admissions data. Report and interpret the Bayes factor.
Not unexpectedly, there is a data set in R that contains these data. The data set is called UCBAdmissions and you can access the department mentioned above like this: UCBAdmissions[ , ,1]. Make sure you put two commas before the 1: this is a three dimensional contingency table that we are
Let’s say a department within a university admits 512 male students and 89 female students. At the same time, they reject 313 male students and 19 female students.Put those numbers in a 2 × 2 contingency table. Calculate marginal totals. Use the marginal totals to construct a new 2 × 2 table of
Bonus statistical thinking question: Run set.seed(123456) to set the random number seed and then run cor.test( rnorm(100), rnorm(100) ). Describe what this cor.test()command accomplishes with these particular inputs. Describe and interpret the out‑put from this analysis. What cautionary tale is
Bonus research question: cor.test() does not work on a whole data set of values—it cannot display significance tests for a whole correlation matrix. Can you find and install a package that will calculate a whole matrix of significance tests on a numeric data set? Once you find and install the
Create a copy of the bfCorTest() custom function presented in this chapter. Don’t for‑get to “source” it (meaning that you have to run the code that defines the function one time to make R aware of it). Conduct a Bayesian analysis of the correlation between“area” and “perm” in the
Run cor.test() on the correlation between “area” and “perm” in the rock data set and interpret the results. Note that you will have to use the “$” accessor to get at each of the two variables (like this: rock$area). Make sure that you interpret both the confi‑dence interval and the
Run cor(rock) and make sense of the results. Why are there two copies of the correlation between “area” and “perm”? What is meant by the numbers on the diagonal?
The R data sets() package actually contains a data set called “rock” that has data something like that described in exercise
A geologist took 48 samples of rock and measured the area of pores in the rock and the permeability of the rock (the extent to which fluid can pass through). The correla‑tion came out at about r = –.40. Say briefly in your own words what that correlation value means. A statistician conducted a
Repeat Exercise 8, with the builtin attenu earthquake data set. The research question is to find out whether the events differed from one another in their mean acceleration.Hint: You will need to coerce the numeric “event” variable into a factor.
Repeat Exercises 5, 6, 7, but this time using the builtin PlantGrowth data set. Create a written report of the results (preferably with graphics) that would be of high enough quality to be used in a professional presentation.
In situations where the alternative hypothesis for an ANOVA is supported and there are more than two groups, it is possible to do posthoc testing to uncover which pairs of groups are substantially different from one another. Using the InsectSprays data, conduct a ttest to compare groups C and F
Load the BayesFactor package and run the anovaBF() command on the InsectSprays data set. You will have to specify the model correctly using the “~” character to sepa‑rate the dependent variable from the independent variable. Produce posterior distribu‑tions with the posterior() command and
Use R or RStudio to run the aov() command on the InsectSprays data set. You will have to specify the model correctly using the “~” character to separate the dependent variable from the independent variable. Place the results of the aov() command into a new object called insectResults. Run the
Continuing with the InsectSprays example, there are six groups where each one has n= 12 observations. Calculate the degrees of freedom between groups and the degrees of freedom within groups. Explain why the sum of these two values adds up to one less than the total number of observations in the
Based on the information in question 2 and your response to that question, calculate an Fratio by hand or using a calculator. Given everything you have earned about Fratios, what do you think of this one? Hint: If you had all the information you needed for a Null Hypothesis Significance Test, would
After running the aov() procedure on the InsectSprays data set, the “Mean Sq” for spray is 533.8 and the “Mean Sq” for Residuals is 15.4. Which one of these is the between groups variance and which one is the withingroups variance? Explain your answers briefly in your own words.
The data sets package (installed in R by default) contains a data set called InsectSprays that shows the results of an experiment with six different kinds of insecticide. For each kind of insecticide, n = 12 observations were conducted. Each observation repre‑sented the count of insects killed by
Consider this ttest, which compares two groups of n = 100,000 observations each:t.test(rnorm(100000,mean=17.1,sd=3.8),rnorm(100000,mean=17.2,sd=3.8))For each of the groups, the rnorm() command was used to generate a random normal distribution of observations similar to those for the automatic
Using the same PlantGrowth data set, compare the “ctrl” group to the “trt2” group.Use all of the methods described earlier (ttest, confidence interval, and Bayesian method) and explain all of the results.
Compare and contrast the results of Exercise 6 and Exercise 7.You have three types of evidence: the results of the null hypothesis test, the confidence interval, and the HDI from the BESTmcmc() procedure. Each one adds something, in turn, to the understanding of the difference between groups.
Install and library() the BEST package. Note that you may need to install a program called JAGS onto your computer before you try to install the BEST package inside of R. Use BESTmcmc() to compare the PlantGrowth control group (“ctrl”) to treatment group 1 (“trt1”). Plot the result and
The PlantGrowth data set contains three different groups, with each representing various plant food diets (you may need to type data(PlantGrowth) to activate it). The group labeled “ctrl” is the control group, while “trt1” and “trt2” are different types of experimental treatment. As a
Using the same premise as Exercise 3, the scientist conducts a Bayesian comparison between the two groups. The procedure produces a highest density interval (HDI)plot. The mean difference at the center of the plot is 1.23 meters. The 95% HDI has a lower boundary of –0.1 meters and an upper
Using the same premise as Exercise 3, the scientist also obtains a 95% confidence interval, where the lower bound is 0.2 meters of difference between the two groups and the upper bound is 2.2 meters of difference. Comment on this result. What can you say about the implications of the study?
A scientist compares two groups of trees that have grown under different conditions.She measures and averages the heights of the trees within each group and conducts a ttest to compare them. There are n = 10 trees in both groups, so there are 18 degrees of freedom. The scientist calculates an
Mark the point on the t distribution corresponding to the null hypothesis and label it. Draw vertical lines to mark the start of the tails at the appropri‑ate spots to leave 2.5% in the lower tail and 2.5% in the upper tail.
Draw or plot a t distribution (hint: it looks like a normal distribution with thick tails).Don’t worry about the degrees of freedom, but if you have to choose a value of df to create a plot, use df =
In your own words, describe the process of Bayesian thinking. Make up an example that uses a twobytwo contingency table to document your ideas. Put real numbers in the contingency table and convert them to probabilities so that you can explain what happens when you focus attention on just one row
Run a ttest to compare the means of ctrl and trt2 in the PlantGrowth data. Report and interpret the confidence interval.
Run a ttest to compare the means of ctrl and trt1 in the PlantGrowth data. Report and interpret the confidence interval. Make sure to include a carefully worded statement about what the confidence interval implies with respect to the population mean differ‑ence between the ctrl and trt1 groups.
Create a boxplot of the plant growth data, using the model “weight ~ group.” What can you say about the differences in the groups by looking at the boxplots for the dif‑ferent groups?
The builtin PlantGrowth data set contains three different groups, each representing a different plant food diet (you may need to type data(PlantGrowth) to activate it). The group labeled “ctrl” is the control group, while the other two groups are each a dif‑ferent type of experimental
Use the set.seed() command with the value of 5 to control randomization, and then calculate a confidence interval using the rnorm() command to generate two samples, like this:set.seed(5)t.test(rnorm(20,mean=100,sd=10),rnorm(20,mean=100,sd=10))The set.seed() function controls the sequencing of
Assume the same conditions as for the previous question, but consider two new drugs, X and Y. When comparing Drug X to placebo, the confidence interval was–15 to +5. When comparing Drug Y to placebo, the confidence interval was –7 to –3.Which drug is better at providing pain relief and why?
Some doctors conducted clinical trials on each of two new pain relievers. In the first trial, Drug A was compared to a placebo. In the second trial, Drug B was also com‑pared to a placebo. In both trials, patients rated their pain relief such that a more negative number, such as –10, signified
Run the code shown in this chapter that created the animation of confidence intervals:install.packages(“animation”)library(animation)conf.int(level=0.95)Once the animation has finished running, comment on your results. Pay particular attention to the number of times that the confidence interval
Answer the following true/false questions about confidence intervals: (a) The center of the confidence interval is the population value; (b) The confidence interval always contains the population value; (c) A wider confidence interval is better, because it signals more certainty; (d) When we say
In your own words, write a definition of a 95% confidence interval.
Redo Exercises 4 and 5, but this time use the precip data set that is built into R. The precip data only includes one variable, so you will not need to use the $ subsetting operator. Use a sample size of n = 1,000. Hint: When you use the sample() command, make sure to include “replace=TRUE” as
Redo Exercise 5, but this time use a sample size of n = 100 (instead of the original sample size of n = 11 used in Exercise 5). Explain why the 2.5% and 97.5% quantiles are different from the results you got for Exercise 5.As a hint, think about what makes a sample “better.”
If you did Exercise 4, you calculated some quantiles for a distribution of raw data. If you did Exercise 5, you calculated some quantiles for a sampling distribution of means.Briefly describe, from a conceptual perspective and in your own words, what the differ‑ence is between a distribution of
Write R code that will construct a sampling distribution of means from the weight data (as noted above, if you did exercise 3 you can use myChkWts instead of ChickWeight$weight to save yourself some typing). Make sure that the sampling distribution contains at least 1,000 means. Store the sampling
In the second to last command of the previous exercise, you created a copy of the weight data from the ChickWeight data set and put it in a new vector called myChkWts.You can continue to use this myChkWts variable for the rest of the exercises below. Cre‑ate a histogram for that variable. Then
When a data set contains more than one variable, R offers another subsetting operator,$, to access each variable individually. For the exercises below, we are interested only in the contents of one of the variables in the data set, called weight. We can access the weight variable by itself, using
For the remaining exercises in this set, we will use one of R’s builtin data sets, called the “ChickWeight” data set. According to the documentation for R, the ChickWeight data set contains information on the weight of chicks in grams up to 21 days after hatch‑ing. Use the
Marlboro College in Vermont is one of the smallest liberal arts colleges in the United States, with an enrollment of about 240 students. Consider this whole group to be the population of interest for a study of the heights of students. In a few sentences, describe how you would draw a random sample
The incidence of HIV in the U.S. population is 0.00036 (0.036%). There is a blood test for HIV, but it is not perfect in its ability to detect the disease. If a person has HIV, the probability is 89% that the blood test will be “positive” (meaning that it has detected the disease). If a person
Imagine that Barclays deploys the screening test from Exercise 6 on a new customer and the new customer fails the test. What is the probability that this customer will actu‑ally default on his or her mortgage? Show your work and especially show the tables that you set up to help with your
In a typical year, 71 out of 100,000 homes in the United Kingdom is repossessed by the bank because of mortgage default (the owners did not pay their mortgage for many months). Barclays Bank has developed a screening test that they want to use to predict whether a mortgagee will default. The bank
One hundred students took a statistics test. Fifty of them are high school students and 50 are college students. Eighty students passed and 20 students failed. You now have enough information to create a twobytwo contingency table with all of the marginal totals specified (although the four main
Convert all of the cells in the contingency table that you developed for Exercise 4(including the marginal totals and the grand total) into probabilities. Explain in your own words which cell has the highest probability and why.
Run 50 trials in which you flip a fair coin four times. As a hint, you should use rbinom()and table() together. Save the results or write them down. Then, using the “prob=0.8”argument with the rbinom() command, run another set of 50 trials (with four flips) with an “unfair” coin (i.e., a
Using the data and results from Exercises 1 and 2, create a bar plot of cumulative prob‑abilities. Write an interpretation of that bar plot as well, making sure to explain how cumulative probabilities differ from the results you described in Exercise 2.
Using the output from Exercise 1, summarize the results of your 100,000 trials of nine flips each in a bar plot using the appropriate commands in R. Convert the results to probabilities and represent that in a bar plot as well. Write a brief interpretive analysis that describes what each of these
Flip a fair coin nine times and write down the number of heads obtained. Now repeat this process 100,000 times. Obviously you don’t want to have to do that by hand, so cre‑ate the necessary lines of R code to do it for you. Hint: You will need both the rbinom()function and the table() function.
As in the previous exercise, use the data() function to get a list of the data sets that are included with the basic installation of R. Choose a data set that includes just one variable, for example, the LakeHuron data set (levels of Lake Huron in the years 1875 through 1972). Use the hist()
Use the data() function to get a list of the data sets that are included with the basic installation of R: just type “data()” at the command line and press enter. Choose a data set from the list that contains at least one numeric variable—for example, the Bio‑chemical Oxygen Demand (BOD)
Write the equations, using the appropriate Greek letters, for the population mean and population standard deviation. Explain briefly what each Greek letter means. The R environment offers about 20 different kinds of statistical distributions. Choose any one of these distributions other than the
Using the material from this chapter and possibly other information that you look up, write a brief definition of these terms in your own words: mean, median, mode, variance, standard deviation, histogram, normal distribution, and Poisson distribution.
9. Do the following until i = n:(a) If i = n, stop.(b) Simulate a trial value of x from the importance function’s associated two-dimensional (bivariate) distribution. For Morgenstern’s distribution this is accomplished through the following steps:i. Generate two independent Uniform(0,1)
8. Set i = 0.
7. If n = 0, stop.
6. Generate an n deviate from the Poisson (μ) distribution.
5. Computeα = max x∈Rλ(x)μg(x). (6.17)
4. Set β ∈ (−1, 1).
3. Select an importance function, g(x). This function needs to be the PDF of a two-dimensional distribution function. Here, Morgenstern’s two-dimensional distribution is employed (Johnson, 1987, p. 185) and hence g(x) = 1 +β(2x1/L − 1)(2x2/W − 1), x ∈ R.
2. Compute μ = Volu˜me(λ(x),R) using Simpson’s rule in two dimensions.
1. Compute the area, A, of a rectangular region, R, that has length L and width W.
5. If R2 is close to 1, is it reasonable to think that there is very little noise?
4. What are the smallest and largest values that R2 can have?
3. For the example, at what α-level would you reject the hypothesis that β1= 0 based on the F-test?
2. For the example, at α = 0.05, would you reject or not reject H0 : β1= 0?
1. Why are βˆ0 and βˆ1 unbiased estimators of β0 and β1?
4. Find the probability of the complement of the event in step 3.
3. Assuming no hunting effects, find the probability that none of the three main regions are hunting-allowed regions. This can be done as follows.Randomly draw three regions from the pool of seven regions. Record if none of the drawn regions allows hunting. Repeat these two steps 1000
2. Define a trial and all elementary events.
1. List the sample space.
9.51 The yield stresses were studied for two sizes of steel rods, with results as follows:Estimate the average increase in yield stress of the 14-mm rod over the 10-mm rod, with a confidence coefficient of 0.90. 10-mm-diameter Rods 14-mm-diameter Rods n = 51 12=12 X = 485 N/mm S = 17.2 2=499 N/mm
9.47 The total amount of hydrogen chloride (HCl) in columns above an altitude of 12 kilometers was measured at selected sites over the El Chichon volcano both before and after an eruption. (See Mankin and Coffey, Science, October 12 ,1984, p. 170.) Before-and-after data for one location were as

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Probability And Statistics For Engineers 5th Edition Richard L Scheaffer, Madhuri Mulekar, James T McClave, Cecie Starr - Solutions

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