Questions and Answers of Model Based Testing For Embedded Systems

Generate data from a bivariate normal distribution with the R command x=rmul(200).Then enter the R command y=x[,1]+x[,2]+x[,1]*x[,2]+rnorm(200) and examine the plot returned by the R command
Generate 25 pairs of observations from a bivariate normal distribution having correlation zero and store them in x. (The R function rmul, written for this book, can be used.)Generate 25 more
Generate 25 observations from a standard normal distribution and store the results in the R variable x. Generate 25 more observations and store them in y. Use rungen to plot a smooth based on the
For the experimental group of the Pygmalion data in Section 11.2.1, create a plot of the smooth using f = 1 and the function rplot. Recreate the plot, but this time omit the scatterplot of the points
For the reading data in the upper right panel of Figure 11.5, recreate the smooth. If you wanted to find a parametric regression equation, what might be tried? Examine how well your suggestions
Using the Pygmalion data, compare the slope of the regression line of the experimental group to the control group using the biweight midregression estimator.
The data in the lower left panel of Figure 11.5 are stored in the file agegesell.dat.Remove the two pairs of points having the largest x value and create a running interval smoother using the data
For the reading data in the file read.dat, use the R function rplot to investigate the shape of the regression surface when predicting the 20% trimmed mean of WWISST2 (the data in column 8) with
For the reading data in file read.dat, let x be the data in column 2 (TAAST1), and suppose it is desired to predict y, the data in column 8 (WWISST2). Speculate on whether there are situations where
Use the function ancova and the Pygmalion data to compare the control group to the experimental group using means. What might be affecting power?
Based on the results of Exercise 6, speculate about what a nonrobust smoother might look like. Check your answer with the smoother lowess using the R lplot.
Use rplot to plot a smooth of the Pygmalion data using f = 0.75 and 20% trimmed means. Create a plot for both the control and experimental groups when the goal is to predict post IQ scores with
For the Pygmalion data in Section 11.2.1, use the function reglev to determine which points, if any, are regression outliers. (The data for the control group are stored in pygc.dat, and the data for
Use the function winreg to estimate the slope and intercept of the star data using 20%Winsorization. (The data are stored in the file star.dat. See Section 1.8 on how to obtain the data.)
Test H0 : β1 = β2 = 0 using the R function regtest and βˆm. 3. For the data in Exercise 1, test H0 : β1 = 0 with the functions regci and regtest.Comment on the results.
Section 8.6.2 reports data on the effects of consuming alcohol on three different occasions. Using the data for group 1, suppose it is desired to predict the response at time 1 using the responses at
For the data in Exercise 1 of Chapter 10, the 0.95 confidence interval for the slope, based on the least squares regression line, is (0.0022, 0.0062). Using R, the 0.95 confidence interval for the
For the star data in Figure 6.3, which are stored in the file star.dat, eliminate the four outliers in the upper left corner of the plot by restricting the range of the x values. Then using the
Describe the relative merits of the OP and MGV estimators in Section 10.10.
Graphically illustrate the difference between a regression outlier and a good leverage point. That is, plot some points for which y = β1x +β0, and then add some points that represent regression
For the data in Exercise 13, identify any leverage points using the hat matrix. Next, identify leverage points with the function reglev. How do the results compare?
For the data used in Exercise 11, RAN1T1 and RAN2T1 (stored in columns 4 and 5) are measures of digit naming speed and letter naming speed. Use M regression with Schweppe weights to estimate the
For the data used in Exercise 11, compute the hat matrix and identify any leverage points. Also check for leverage points with the R function reglev. How do the results compare?
The file read.dat contains reading data collected by L. Doi. Of interest is predicting WWISST2, a word identification score (stored in column 8), using TAAST1, a measure of phonological awareness
For the data in Exercise 6, verify that the 0.95 confidence interval for the regression parameters, using the R function regci with M regression and Schweppe weights, are(−0.2357, 0.3761) and
Referring to Exercise 6, how do the results compare to the results obtained with the R function reglev?
For the data used in the previous exercise, compute 0.95 confidence intervals for the parameters using OLS as well as M regression with Schweppe weights.
The example in Section 6.6.1 reports the results of drinking alcohol for two groups of subjects measured at three different times. Using the group 1 data, compute an OLS estimate of the regression
Compute the hat matrix for the data in Exercise 1. Which x values are identified as leverage points? Relate the result to the previous exercise.
For the data in Exercise 1, use the R function reglev to comment on the advisability of using M regression with Schweppe weights.
Let T be any regression estimator that is affine equivariant. Let A be any nonsingular square matrix. Argue that the predicted y values, yˆi, remain unchanged when xi is replaced by xiA.
Using the data in Exercise 1, show that the estimate of the slope given by βˆch is 0.0057.In contrast, the OLS estimate is 0.0045, and βˆm = 0.0042. Comment on the difference among the three
Discuss the relative merits of βˆch.
The average LSAT scores (x) for the 1973 entering classes of 15 American law schools, and the corresponding grade point averages (y), are as follows.x: 576 635 558 578 666 580 555 661 651 605 653 575
Let X be a standard normal random variable, and suppose Y is contaminated normal with probability density function given by Eq. (1.1) of Chapter 1. Let Q = ρX +p 1−ρ2Y , −1 ≤ ρ ≤ 1. Verify
If in the definition of the biweight midcovariance, the median is replaced by the biweight measure of location, the biweight midcovariance is equal to zero under independence. Describe some negative
The definition of the percentage bend correlation coefficient, ρpb, involves a measure of scale, ωx , that is estimated with ωˆ = W(m), where Wi = |Xi − Mx | and m = [(1−β)n], and 0 ≤ β
For the data in the file read.dat, test for independence using the data in columns 4 and 5 and β = 0.1.
For the data used in the last two exercises, test the hypothesis of independence using the function indt. Why might indt find an association not detected by any of the correlations covered in this
Examine the variables in the last exercise using the R functions mscor.
Using the data in the file read.dat, test for independence using the data in columns 2, 3, and 10 and the R function pball. Try β = 0.1, 0.3, and 0.5. Comment on any discrepancies.
The method of detecting outliers, described in Section 6.4.3, could be modified by replacing the MVE estimator with the Winsorized mean and covariance matrix. Discuss how this would be done and its
Repeat the previous problem using the data for Group 2.
Using the Group 1 alcohol data in Section 8.6.2, compute the MVE estimate of correlation and compare the results with the biweight midcorrelation, the percentage bend correlation using β = 0.1, 0.2,
Use the function cov.mve(m,cor=T) to compute the MVE correlation for the star data in Figure 9.2. Compare the results with the Winsorized, percentage bend, skipped, and biweight correlations, as well
Demonstrate that heteroscedasticity affects the probability of a type I error when testing the hypothesis of a zero correlation based on any type M correlation and nonbootstrap method covered in this
Repeat Exercise 1 with Spearman’s rho, the percentage bend correlation, and the Winsorized correlation.
Generate 20 observations from a standard normal distribution and store them in the R variable ep. Repeat this and store the values in x. Compute y=x+ep and compute Kendall’s tau. Generally, what
Repeat Exercise 6, only now use the rank-based method in Section 8.6.12.
Analyze the data in Table 6.1 using the methods in Sections 8.6.1 and 8.6.4.
Repeat Exercises 3 and 4 using the data for the murderers in Table 6.1.
Repeat Exercise 3 using the rank-based method in Section 8.5. How do the results compare to using a measure of location?
Analyze the data for the control group reported in Table 6.1 using the methods in Sections 8.1 and 8.2. Compare and contrast the results.
For the data used in Exercise 1, compute confidence intervals for all pairs of trimmed means using the R function pairdepb.
Section 8.6.2 reports data on hangover symptoms. For group 2, use the R function rmanova to compare the trimmed means corresponding to times 1, 2, and 3.
For the schizophrenia data in Section 7.8.4, compare the groups with t1way and pbadepth.
Generate data for a 2-by-3 design and use the function pbad2way. Note the contrast coefficients for interactions. If you again use pbad2way, but with conall=F, what will happen to these contrast
Snedecor and Cochran (1967) report weight gains for rats randomly assigned to one of four diets that varied in the amount and source of protein. The results were as follows:Verify the results based
For the data in the previous two exercises, perform all pairwise comparisons using the Harrell–Davis estimate of the median.
For the data in the previous exercise, compare the groups using both the Rust–Fligner and Brunner–Dette–Munk methods.
Suppose three different drugs are being considered for treating some disorder, and it is desired to check for side effects related to liver damage. Further suppose that the following data are
Using the data from the previous two exercises, compare the 20% trimmed means of the experimental group to the control taking into account grade. Also test for no interactions using lincon and
Using the data in the previous exercise, use the function lincon to compare the experimental group with the control group taking into account grade and the two tracking abilities. (Again, tracking
Some psychologists have suggested that teachers’ expectancies influence intellectual functioning. The file VIQ.dat contains pretest verbal IQ scores for students in grades 1 and 2 who were assigned
From well-known results on the random effects model (e.g., Graybill, 1976;Jeyaratnam and Othman, 1985), it follows that Use these result to derive an alternative estimate of ρWI. estimates BSSW=
If data are generated from exponential distributions, what problems would you expect in terms of probability coverage when computing confidence intervals? What problems with power might arise?
Describe how M-measures of location might be compared in a two-way design with a percentile bootstrap method. What practical problem might arise when using the bootstrap and sample sizes are small?
For the data in Table 6.6, compare the groups using the method in Section 6.8.
Argue that when testing Eq. (6.27), this provides a metric-free method for comparing groups based on scatter.
For the data in Table 6.1, compare the two groups with the method in Section 6.12.
For the data in Table 6.1, compare the two groups with the method in Section 6.11.
For the data in Table 6.1, compare the two groups with the method in Section 6.10.
For the data in Table 6.1, compare the two groups with the method in Section 6.8.
For the cork boring data in Table 6.5, imagine that the goal is to compare the north, east and south sides to the west side How might this be done with the software in Section 6.6.1? Perform the
The file read.dat contains data from a reading study conducted by L. Doi. Columns 4 and 5 contain measures of digit naming speed and letter naming speed. Use both the relplot and the MVE method to
The MVE method of detecting outliers, described in Section 6.4.3, could be modified by replacing the MVE estimator of location with the Winsorized mean, and replacing the covariances with the
The average LSAT scores (X) for the 1973 entering classes of 15 American law schools, and the corresponding grade point averages (Y ), are as follows.Use a boxplot to determine whether any of the X
Give a general description of a situation where for n = 20, the minimum depth among all points is 3/20.
Suppose that for each row of an n-by-p matrix, its depth is computed relative to all n points in the matrix. What are the possible values that the depths might be?
Repeat the last two exercises, but now use the data in Table 6.2.
Repeat the last exercise using the data for group 2.
For the data in Table 6.1, check for outliers among the first group using the methods in Section 6.4. Comment on why the number of outliers found differs among the methods.
Repeat the last exercise using the data for group 2.
For the EEG data in Table 6.1, compute the MVE, MCD, OP, and the Donoho–Gasko.2 trimmed mean for group 1.
Using R, generate 30 observations from a standard normal distribution and store the values in x. Generate 20 observations from a chi-squared distribution with one degree of freedom and store them in
Let D = X −Y , let θD be the population median associated with D, and let θX and θY be the population medians associated with X and Y , respectively. Verify that under general conditions, θD 6=
The file tumor.dat contains data on the number of days to occurrence of a mammary tumor in 48 rats injected with a carcinogen and subsequently randomized to receive either the treatment or the
Continuing the last exercise, examine a boxplot of the data. What would you expect to happen if the 0.95 confidence interval is computed using a bootstrap-t method? Verify your answer using the R
The file pyge.dat (see Section 1.8) contains pretest reasoning IQ scores for students in grades 1 and 2 who were assigned to one of three ability tracks. (The data are from Elashoff & Snow, 1970, and
Section 5.9.6 used some hypothetical data to illustrate the R function yuend with 20%trimming. Use the function to compare the means. Verify that the estimated standard error of the difference
The example at the end of Section 5.3.3 examined some data from an experiment on the effects of drinking alcohol. Another portion of the study consisted of measuring the effects of alcohol over 3
Compute a confidence interval for p using the data in Table 5.1.
Comment on the relative merits of testing H0: p = 1/2 with Mee’s method versus comparing two independent groups with the Kolmogorov–Smirnov test.
Describe a situation where testing H0: p = 1/2 with Mee’s method can have lower power than the Yuen–Welch procedure.
Apply the Yuen–Welch method to the data in Table 5.1 where the amount of trimming is 0, 0.05, 0.1, and 0.2. Compare the estimated standard errors of the difference between the trimmed means.
Verify that if X and Y are independent, the third moment about the mean of X −Y isµx[3] −µy[3].
Compare the deciles only, using the Harrell–Davis estimator, using the data in Table 5.1.
Consider two independent groups having identical distributions. Suppose four observations are randomly sampled from the first and three from the second. Determine P(D = 1) and P(D = 0.75), where D is
Summarize the relative merits of using the weighted versus unweighted Kolmogorov–Smirnov test. Also discuss the merits of the Kolmogorov–Smirnov test relative to comparing measures of location.
Compare the two groups of data in Table 5.3 using the weighted Kolmogorov–Smirnov test. Plot the shift function and its 0.95 confidence band. Compare the results with the unweighted test.

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