Statistical Data Analysis Using SAS Intermediate Statistical Methods 2nd Edition Mervyn G. Marasinghe, Kenneth J. Koehler - Solutions

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3. Invert the formula df = 2(1 + 1∕f ) to get a formula for f = g(df ).
2. A failed attempt at a consistent estimator: Box et al. (2008) suggest only n/4 correlations should be estimated. Modify the periodogram code to use only floor(n/4) instead of n–1 terms, thus using only “reliable” correlations: ̂p( f ) =Ĉ 0 + 2Σfloor(n∕4)k=1 Ĉ kcos(2????kf ). Use
1. Use the formula for Ĉ k to show that for k close to n, only a few lags are used to estimate the value.
10. (i) Simulate 400 AR(2) with a1 = 1.559 and a2 = −0.81 and plot the periodogram(loge scale) and the true model both versus frequency.(ii) Using Exercise 6, compute where the maximum of p(f ) should occur [and that it is indeed a maximum] and that the graph matches this computation.
9. Compute the acf() plot and the periodogram for the mean maximum temperature in Melbourne [Exercise 1 (a), Chapter 3]. Identify (and/or reason to) the period for this data and explain what characteristic of each plot verifies this period.
8. Produce a periodogram for the lynx data [From Exercise 1 (d), Chapter 3] and estimate the primary period (eyeball it).Use both the na¨ıve code and R to produce the periodogram and verify that both methods are getting the same results.
7. Plots for MA(1) and MA(2) similar to Figures 8.2, 8.3, 8.4, 8.5, 8.6, and 8.7.(i) Find p(f ) for the MA(1) and MA(2) models.(ii) Simulate 300 MA(1) observations and plot the periodogram (loge scale) and the true model (derived above) both versus frequency. You can choose your own parameter, but
6. The periodogram of an AR(2) model…(i) Find the closed form formula for the maximum or minimum of h(f ) = 1 − 2a1 cos(2????f ) − 2a2 cos(4????f ) + 2a1a2 cos(2????f ) + a21+ a22 in 0 < f < 0.5.(ii) Compute the second derivative of h(f ).(iii) Use this information to discuss the graphs of
5. Show|1 − ae−2????fi|2 = 1 − 2a ⋅ cos(2????f ) + a2 and|1 − a1e−2????fi − a2e−4????fi|2 =1 − 2a1 cos(2????f ) − 2a2 cos(4????f ) + 2a1a2 cos(2????f ) + a21+ a22[this is just the real expansion of the denominator of the AR(2) power spectrum].
4. Justify the claim that the following series related to the derivation of the AR(1)spectrum are convergent:∞Σk=1(ae−2????fi)k = ae−2????fi 1 − ae−2????fi and∞Σj=1(ae2????fi)j = ae2????fi 1 − ae2????fi.
3. Show 2 ⋅ cos(2????jf ) cos(2????kf ) = cos(2????[j + k]f ) + cos(2????[j − k]f ).
2. Show ∫1∕2−1∕2 p(f ) sin(2????kf )df = 0.(i) Show sin(???? + ????) + sin(???? − ????) = 2 sin(????) cos(????).(ii) ∫1∕2−1∕2 p(f ) sin(2????kf )df = ∫1∕2−1∕2{Co + 2Σ∞j=1 Cj cos(2????jf )} sin(2????kf )df , why?(iii) Multiply out the terms of the initial integral and
1. Show Σ∞k=−∞ i ⋅ Ck sin(2????kf ) = 0. Hints: (i) recall Ck = C−k and (ii) what is the value of sin(b) + sin(−b)?
9. Use the series representations to show:(a) d dx exp (x) = exp(x).(b) d dx sin (x) = cos(x).(c) d dx cos (x) = −sin(x).
8. Show sin(2x) = 2⋅sin(x)⋅cos(x) and cos (2x) = cos2 (x) − sin2(x).
7. (i) Show Σn−1 s=0 cos (2????sf ) = cos(????f [n − 1])sin(????fn)sin(????f )(hint: use induction).(ii) Evaluate the sum at f = 0, 0.5, 2/3, 1, when n = 24.
6. (i) Show Σn−1 s=0 exp(2????sfi) =exp (2????nfi) − 1 exp (2????fi) − 1.(ii) Evaluate the sum at f = 0, 1/3, 0.5, 1, when n = 12.
5. Show cos (x) = (exp [ix] + exp[−ix])∕2 and sin (x) = (exp [ix] − exp[−ix])∕2.
4. Evaluate exp (iu) + exp(−iu) = ??
3. Compute the sum: S = 2 + 1 + 0.5 + 0.25+…
5. Simulate each series in the previous table with the specified white noise variance:(i) Verify b1 and b2 that meet the conditions for invertibility.(ii) Compute all values on the table.(iii) Simulate and plot the time series.(iv) Produce the acf() graph.(v) Verify R1, R2, R3 and the variance for
4. Solve for the values in the following table (with ????2w= 4.0).b1 b2 R1 R2 R3 ????2 MA−0.6 0.3 ?? ?? ?? ??0.1 −0.8−0.7 −0.2 0.2 0.5 ?? ??
3. Give a rationale for MA(1) = AR(∞).(i) Show ????n = −b1????n−1 − b21 wn−2 + wn.(ii) Using the idea above, develop an MA(1) formula analogous to the AR(1)formula: ????n = ak 1????n−k +Σk j=1 aj−1 1 wn−j+1.(iii) Use this formula to suggest an MA(1) process is equivalent to an
2. Derive all autocovariances and autocorrelations for the MA(2) model.
1. Derive all autocovariances and autocorrelations for the MA(1) model.
9. Write an R program plot several (perhaps 50 or 100) simulated AR(1) series with a = ±(1 + ????), where ???? is some small number. Describe the general pattern suggested.
8. Simulate each series in the previous table (Exercise 7) with the specified white noise variance:(i) Find a1 and a2 and verify they meet the conditions for stability.(ii) Compute all values on the table.(iii) Simulate and plot the time series.(iv) Produce the acf() graph.(v) Verify R1, R2, R3 and
7. Solve for the values in the following table (with ????2w= 4)a1 a2 R1 R2 R3 ????2 AR?? ?? 0.8 0.6 ?? ??0.4 −0.1−0.3 0.7?? ?? −0.5 −0.4 ?? ??
6. Show that, for the AR(2) model, ????2 AR = ????2w∕(1 − a1R1 − a2R2). hint: expand E(????j????j) = E([a1????j−1 + a2????j−2 + wj]2) and exploit the similarities between these formulas.
5. For an AR(1) process, ????n = a21????n−2 + a1wn−1 + wn (the formula from Exercise 4 with k = 2). Use this fact to re-derive the formula ????2 AR = ????2w∕(1 − a21).
4. Show that for the AR(1) process: ????n = ak 1????n−k +Σk j=1 aj−1 1 wn−j+1.
3. Show that when a2 = 0, the stability conditions for an AR(2) model are equivalent to the stability conditions for an AR(1) model.
2. For the lynx data, “LYNX.txt”, from the Exercises folder (DataMarket–Ecology–Annual number of lynx trapped, MacKenzie River, 1821–1934):(i) Plot the data versus time as a time series.(ii) Does the data appear periodic? If so, what it the period?(iii) Use R to compute an acf() plot for
1. Write the code to compute the acf() plot yourself for the file “Furnas 31 78.txt”from the Exercises folder (DataMarket–Hydrology–Monthly riverflow in cms, Furnas–vazoes medias mensais, 1931–1978). Also use the R function acf() for comparison. Since the data is monthly you should
2. Crazy Cauchy: The Cauchy distribution (aka a t distribution with one degree of freedom) is outlier prone (extremely heavy tailed). In R, Cauchy random variables can be simulated using rcauchy(). For example, the R code z
1. Compare the point estimates and interval estimates for the mean (original scale)and median (use the log scale and back transform the resulting interval) using the cloud seeding data and the seeded days.
3. Beginning to think about specialized plots.For the same data sets from Exercise 1, run the R procedures pacf(), acf(), and spec.pgram() and describe how the graphs differ from the similar white noise graphs in Exercise 2 (do not expect to be good at this yet!).
2. Simulate and store 500 observations of white noise.Using whatever name you gave these, suppose the name is “z,” run the functions acf(z), pacf(z), spec.pgram(z).This will produce three plots associated with this simulated data (do not worry too much that you do not know what you are doing).
1. Consider the following data sets that could be found in the Exercises folder of the diskette or DataMarket—Time Series Data Library.For each data set: (a)–(e)(i) Scan the data into R.(ii) Plot the data versus time.(iii) Describe what signal might fit the data. If the data appears periodic,
7. Use R to invert the matrix[7 4 1 2]
6. Use R to find the transpose of the matrix[7 4 1 2].
5. Which assumption of the simple regression model does serial correlated data violate?
4. Create a 3 by 3 identity matrix in R.
3. Use the help() function to describe the arguments that are included in the summary(), matrix(), cbind(), and solve().
2. Simulate a number of regressions (2000 or more iterations) and get the sampling distribution for ????̂. Compare the empirical sampling distribution (mean, variance, shape) to the theory.
1. An exercise for keeping in mind the true model versus the estimated model and the true error versus the estimated error (i.e., the residuals):Simulate linear regression data in R with white noise.Simulate the data for y = 5.0 – 4.0x + error, experiment with the variance until the plot clearly
4. Add log(15) + tan(????/2) in R.
3. Write R code, using a “for” loop, to find the value 20! (20 factorial) and verify that it is correct.
2. Recall Σm j=1 j = m(m + 1)∕2. Write R code to find the sum Σ20 j=1 j and verify that the sum is correct.
1. Guess the output of the following bits of R code and check your answers (by typing the code in R).(a) TRUE = FALSE; (b) TRUE == FALSE; (c) NA-NA; (d) 1/Inf; (e) Inf -Inf; (f) 1ˆInf; (g) y
The data for this exercise are from a study of the frequencies of urinary tract infections in n = 98 men infected with the HIV virus. CD4+ cell counts were also measured. They are used as an indication of how well the immune system is working in people infected with HIV. CD4+ counts are reported as
Myers (1990) presents data from a study of the growth behavior for protozoa colonies in a particular lake. Fifteen sponges were placed in the lake, and three of the sponges were removed from the lake at each of five time points, 1, 3, 6, 15, and 21 days after the sponges were put into the lake. The
Collett (2003) discusses data from a study of the specific gravity of two species of rotifers. Rotifers are microscopic organisms that make up a substantial proportion of freshwater zooplankton. When rotifers are centrifuged in a solution that has lower specific gravity than their own, they will
Lloyd (1999, Chapter 6) examines the following data on the relationship of age and marital status from a survey of 185 people in Denmark over the age of 16. A respondent is classified as divorced if the person had been divorced at any time prior to the survey, regardless of whether or not that
This problem explores an alternative model for the beetle survival data from problem (7.1) in which the log-odds for survival are related to the natural logarithm of the carbon disulfide concentration to which the beetles are exposed. The log concentration appears in the second column of the data
Chester Bliss (1935) examined data from a study of the toxic effects of exposing a certain species of beetle to various levels of concentrations of gaseous carbon disulfide for a period of 5 hours. The first column in the following table shows the eight concentrations of carbon disulfide
An experiment is designed to study pigment dispersion in paint. Four different mixes of a particular pigment are studied. The procedure consists of preparing a particular mix and then applying that mix to a panel by three application methods (brushing, spraying, and rolling). The response measured
Soy protein isolates (SPI), widely used in the food industry, are usually stored in dry powder form produced via spray drying or freeze drying, to enhance shelf life and make them easier to distribute. A study was conducted at Iowa State University (Deak and Johnson 2007) to determine how various
An experiment, conducted in a split-plot design to determine the effect of three bacterial inoculation treatments applied to two cultivars of grasses on dry weight yields, is discussed in Littell et al. (1991). The cultivar is the whole-plot factor and inoculi, the subplot factor. Each of the two
In a lab experiment carried out in a completely randomized design, a soil scientist studied the growth of barley plants under three different levels of salinity (control, 6 bars, 12 bars) in a controlled growth medium(Kuehl 2000). Two replications of each treatment were obtained, and three plants
An engineer is designing a battery for use in a device that will be subjected to extreme variations in temperature (Montgomery 1991). He considers a two-way factorial with plate material and temperature as the two factors but has a large number of feasible choices for plate material and
Consider an experiment to examine the variation in the effects of different analysts on chemical analyses for the DNA content of plaque (Montgomery 1991). Three female subjects (ages 18–20 years) were chosen for the study. Each subject was allowed to maintain her usual diet, supplemented with 30
In a health awareness study (Kutner et al. 2005), each of three states independently devised a health awareness program. Three cities within each state of similar demographics were selected at random, and five households within each city were randomly selected to evaluate the effectiveness of the
In a study reported in Dunn and Clark (1987), each of three different sprays were applied to four trees selected at random. After 1 week, the concentration of nitrogen was measured on each of six leaves obtained in a random way from each tree. The data are given in the following table.a. Write the
A manufacturing company wishes to study the variation in tensile strength of yarns produced on four different looms (Sahai and Ageel 2000). In an experiment designed for this purpose, 12 machinists were selected, and each loom was assigned to three different machinists at random. Samples from two
The objective of a case study discussed in Ott and Longnecker (2001)was to determine whether the pressure drop across the expansion joint in electric turbines was related to gas temperature. Also, the researchers wanted to assess the variation in readings from various types of pressure gauge and
A compound is sent to five randomly selected laboratories in the United States for a routine analysis. At each laboratory, four chemists are chosen at random, and each chemist makes three chemical determinations on the compound using the same method of chemical analysis. The object is to study the
In an experiment described in Dunn and Clark (1987), four brands of airplane tires are compared to assess the differences in the rate of tread wear. The data were collected on eight planes, with two tires used under each wing. The researcher uses each airplane as a block, mounting four test tires,
In an experiment to study variability of a blood pH measurements among animals from different dams as well as from different sires described in Sokal and Rohlf (1995), ten female mice (dams) were successfully mated over a period of time to two males (sires). Different sires were employed for the
A manufacturer of diet foods suspects that the batches of raw material furnished by her supplier differ significantly in sodium content. There is a large number of batches currently in the warehouse, and the variability of sodium content among these batches is of interest. Five of these are
A sugar manufacturer wants to determine whether there is significant variability in purity of batches of raw cane among batches obtained from different suppliers as well as among different batches obtained from the same supplier (Montgomery 1991). Four batches of raw cane are obtained at random
A textile mill weaves a fabric on a large number of looms (Montgomery 1991). To investigate whether there is an appreciable variation among the output of cloth per minute by the looms, the process engineer selects five looms at random and measured their output on five randomly chosen days. The
The following experiment is described in (Montgomery 2013). A resin(PFTE) is used to produce artificial vascular grafts by extruding into tubes. A study is performed to determine the cause of hard protrusions called flicks. The extrusion pressure is suspected to be a factor in the occurrence of
A consumer product-testing organization wished to compare the annual power consumption of five different brands of dehumidifier (Devore 1982). Because power consumption depends on the prevailing humidity level, it was decided to monitor each brand at four different areas of humidity, ranging from
Four machines are compared to assess the differences in the rate of production of a certain part (Part No. Z-15) (Ostle 1963). The data were collected over 5 days. All four machines were run each day (in random order), thus using a randomized complete block design with “Day” as a blocking
Four brands of airplane tires are compared to assess the differences in the rate of tread wear. The data were collected on eight planes, with two tires used under each wing. The researcher uses each airplane as a block, mounting four test tires, one of each brand, in random order on each airplane.
Kutner et al. (2005) discussed an example in which human growth hormone was administered at a clinical research center to short prepubescent children. The investigator was studying the effects of gender and bone development levels on the rate of growth induced by hormone administration. Three
Use the following data set, which is similar to that used in SAS Example E8 (Fig. 5.49) except that different data values are missing.Drug Poison ABCD I0.310.82. 0.450.45 0.45 0.71 0.880.63 0.66 0.43 0.72. 0.62 I0.36 0.92 0.44 0.56. . 0.35.0.40 0.490.31 0.71. .0.40 0.38 III0.22 0.300.23 0.30.0.0.18
The yield (grams per plant) of beetroots grown in pots in response to two crossed factors, wood chips from three different sources and nitrogen at three levels, in a completely randomized design with three replications, is given below (Bliss 1970). The rate of application of the wood chips was 10
Ostle (1963) presented an example of an agronomic experiment to assess the effects of date of planting and type of fertilizer on the yield of soybeans. Thirty-two experimental plots were randomly assigned to the treatment combinations so that each combination was replicated four times. The yield
A mechanical engineer is comparing the thrust force produced by a drill press under various combinations of drill speed and feed rate. A two-way factorial with four feed rates (0.015, 0.030, 0.045, and 0.060 in./min) and two drill speeds (125 and 200 rpm) is used in a completely randomized design
A medical experiment is run to determine the side effects on children when they take various dosages of a drug administered by different methods. A two-way factorial with four dosages (0.5, 1.0, 1.5, and 2.0 milligrams) and three methods of administering (oral, extended release, intravenous) is
In an experiment described in Kirk (1982), four methods for teaching arithmetic were being evaluated. Thirty-two students were randomly assigned to four classrooms, each with eight students. An intelligence test was administered to each student at the beginning of the experiment.The resulting
An experiment conducted to study the friction properties of lubricants is described in Mason et al. (1989). A key constituent of lubricants that is of interest to the researchers is the additive that is mixed with the base lubricant. In order to ascertain whether two competing additives produce a
The data displayed below are results from an experiment described in Snedecor and Cochran (1989) on the use of drugs in the treatment of leprosy. The drugs were A and D, which were antibiotics, and F is an inert drug used as a control. The dependent variable Y was a score of leprosy bacilli
A process engineer is interested in determining if there is a difference in the breaking strength of a monofilament fiber produced using three different machines for a textile company (Montgomery 1991). However, the strength of the fiber is also related to its diameter, with thicker fibers being
A researcher wants to evaluate the difference in mean film thickness of a coating placed on silicon wafers using three different processes(Ott and Longnecker 2001). Six wafers are randomly assigned to each of the processes. The film thickness and the temperature in the lab during the coating
An experiment, carried out in a completely randomized design (Devore 1982), to compare the effects of five different plate lengths of 4, 6, 8, 10, and 12 inches on axial stiffness of metal plate-connected trusses used for roof support, yielded the following observations on axial stiffness
Researchers conducted an experiment to compare the effectiveness of four new weight-reducing agents to that of an existing agent (Ott and Longnecker 2001). The researchers randomly divided a random sample of 50 males into 5 equal groups, with preparation A assigned to the first group, B to the
In an experiment conducted to compare the effects of sleep deprivation on reaction time to onset of light (Kirk 1982), 32 subjects were randomly divided into 4 groups of 8 subjects each. Four levels of sleep deprivation(12, 24, 36, and 48 hours) were randomly assigned to the four groups.The
A marketing consultant conducted an experiment to compare four different package designs for a new breakfast cereal (this problem is described in Kutner et al. (2005); however, the data given are artificial). Twentyfour stores with approximately similar sales volumes were selected, and each store
The following are clotting times of plasma (in minutes) for four different levels of a drug compared in a completely randomized design (modified from a different experiment reported in Armitage and Berry 1994). Blood samples were taken from each of eight subjects randomly assigned to each of the
Six samples of each of four types of cereal grain grown in a certain region were randomly selected and analyzed to determine the thiamin content(mcg/gm) in an experiment reported in Devore (1982). The data were input to the following SAS program:Assuming the one-way fixed effects model yij = μ +
An experiment was carried out in a completely randomized design to compare the differences in the levels of physiologically active polyunsaturated fatty acids (PAPUFA, in percentages) of six different brands of diet margarine, resulting in the following data (Devore 1982):Imperial 14.1 13.6 14.4
In this problem use the adjusted R2 criterion with the stepwise method model selection and the cv method for selecting the best model (in terms of the smallest cross-validation prediction error, called CV PRESS) in each step. Specify that fivefold cross-validation method to be used. Also specify
In this problem use the significance level criterion with the stepwise method model selection and the validate method for selecting the best model (in terms of smallest validation ASE) in each step. Partition the data randomly so that 35% of the data are used for validation. Request plots of the
This problem may be considered a complete SAS project. Import an Excel data set named air pollution.xls using proc import to create a SAS data set. The data shown in Table B.7 consists of air pollution measured as SO2 content in the air and related other variables for 41 U.S. cities. SO2 is the
Since the previous analyses indicated that hospital 14 might have been inefficiently run, use SAS procedures to obtain the “best” model resulting from using possible combinations of the five explanatory variables after hospital 14 is removed from the data set.a. Use a SAS procedure to do all
Consider relating y (HOURS) to x1 (LOAD), x2 (XRAY), x3 (BEDDAYS), x4(POP), and x5(LENGTH) by fitting the model y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x5 +using all 17 hospitals.a. Use the scatter plot matrix and correlation matrices to carry out a preliminary assessment of the relationship
For two hospitals not used in the above analysis whose efficiency the Navy questions, the values of XRAY, BEDDAYS, and LENGTH are 56,194, 14,077.88, and 6.89 and 6021, 1651.42, and 5.41, respectively.Use SAS to obtain predictions of the number of monthly labor hours used for these hospitals. Use

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Statistical Data Analysis Using SAS Intermediate Statistical Methods 2nd Edition Mervyn G. Marasinghe, Kenneth J. Koehler - Solutions

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