Linear Regression Models Statistics 4205/5205 — Fall 2020
统计作业代写 By Monday, November 9, read Chapters 1–7 (skip Sections 5.4–5.6) of Applied Linear Regression, fourth edition; by Sanford Weisberg
Assignment 7
Reading:
By Monday, November 9, read Chapters 1–7 (skip Sections 5.4–5.6) of Applied Linear Regression, fourth edition; by Sanford Weisberg, and the corresponding sections of Weisberg’s Computing Primer for Linear Regression Using R.
For Monday, November 16, read Section 8.1 of the textbook and the R primer.统计作业代写
For Wednesday, November 18, study for the second midterm exam! The exam will cover everything that was on the first midterm (see Assignment 4) as essential background, but the emphasis will be on material covered since then:
- Sections 4.1–4.5, 5.1–5.3 and 6.1–6.5 of thetextbook;
- Lectures 10 through 15, and the correspondingExamples;
- Homework assignments 4 through
The exam format will be similar to the first midterm; in particular, the exam is closed-book and closed-note, but you are permitted a single 8 1 × 11 cheat sheet, and a hand-held calculator.
For Monday, November 23, read Sections 8.2–8.3 of the textbook and the R primer. For Monday, November 30, read Chapter 9 of the textbook and the R primer.
Homework 7: 统计作业代写
The following problems are due by the end of the day on Wednesday, December 2.
- The data file stopping gives automobile stopping Distance in feet and Speed in mph for n = 62 trials of various automobiles.
(a)Draw a scatterplot of Distance versus Speed. Explain why this graph supports fitting a quadratic regression model.
(b)Fit the quadratic model assuming constant variance.Conduct a score test of constant variance against the alternative that the variance depends on Speed. State the null and alternative hypotheses, and summarize results.统计作业代写
(c)Still assuming a quadratic mean function, conduct a score test of the null hypothesis thatthe variance depends on Speed against the alternative that the variance depends on Speed and Speed2. Find the p-value and summarize results.
(d)Refitthe quadratic regression model assuming Var(Distance|Speed) = σ2Speed. Compare the estimates and their standard errors with the unweighted case.
(e)Based on the unweighted model, use a sandwich estimator of variance to correct for non- constant variance. Compare with the results of part(d).
2.In an 1868 study of coinage, W. Stanley Jevons weighed 274 gold sovereigns that he had collected from circulation in Manchester, England.
For each coin, Jevons recorded the weight after cleaning to the nearest 0.0001 g, and the date of issue. The data file jevons includes Age, the age of the coin in decades, n, the number of coins in the age class, Weight, the average weight of the coins in the age class, and SD, the standard deviation of the weights. The minimum Min and maximum Max of the weights are also given. The standard weight of a gold sovereign was 9876 g; the minimum legal weight was 7.9379g.统计作业代写
(a)Draw a scatterplot of Weight versus Age, and comment on the applicability of the usualassumptions of the linear regression model. Also draw a scatterplot of SD versus Age, and summarize the information in this
(b)To fit a simple linear regression model with Weight as the response, wls should be used with variance function Var(Weight|Age) = SD2/ It is reasonable here to pretend the SD are population values. Fit the wlsmodel.
(c)Is the fitted regression consistent with the known standard weight for a new coin?
(d)Forpreviously unsampled coins of Age = 1, 2, 3, 4, 5, estimate the probability that the weightof the coin is less than the legal minimum, assuming the population is normally distributed.
(e)Estimatethe Age at which the the mean weight of coins is equal to the legal minimum, and use the delta method to get a standard error for the estimate.
3.The data file mile gives the world record times for the one-mile run. 统计作业代写
For males, the records are for the period from 1861 to 2003, and for females, for the period 1967–2003. The variables in the file are Year, year of the record; Time, the record time, in seconds; Name, the name of the runner; Country, the runner’s home country; Place, the place where the record was run (missingformany of the early records); and Gender, either male or female.
(a)Draw a scatterplot of Time versus Year, using a different symbol for men and women. Comment on the graph.
(b)Fit a regression model with separate intercepts and slopes for each gender.Provide an interpretation of the slopes.统计作业代写
(c)Find the year in which the female record is expected to be 240 seconds, or 4 minutes; use the delta method to estimate the standard error.
(d)Using the model fit in part (b), estimate the year in which the female record will match the male record, and use the delta method to estimate the standard error of the year in which they will agree. Comment on whether you think using the point at which the fitted regression lines cross is a reasonable estimator of the crossing time.
4.The data in baeskel were collected for a study of the effect of dissolved sulfur on the surface tension of liquid copper. 统计作业代写
The predictor Sulfur is the weight percent sulfur, and the response Tension is the decrease in surface tension in dynes per centimeter. Two replicate observations were taken at each value of Sulfur.
(a)Drawthe plot of Tension versus Sulfur to verify that a transformation is required to achieve a straight-line mean function. 统计作业代写
(b)Set λ = −1, and fit the meanfunction
E (Tension|Sulfur) = β0 + β1Sulfurλ
using ols. Add a line for the fitted values from this fit to the plot you drew in part (a).
(c)Repeat part (b) for λ = 0, and λ = 1, so in the end you will have three lines on your plot.Which of these three choices of λ gives fitted values that match the data most closely?
(d)ReplaceSulfur by its logarithm, and consider transforming the response Draw the inverse fitted values plot, with the fitted values from the regression Tension ∼ log(Sulfur) on the vertical axis and Tension on the horizontal axis. Decide if transformation of the response will be helpful.
5.We return to the stopping distance data used in Problem 1. 统计作业代写
(a)Using Distance as the response, and Speed as the only regressor, find an appropriate trans- formation for Distance that can linearize this regression.
(b)Using Distance as the response, transform the predictor Speed using a power transformation with each λ ∈ {−1, 0, 1}, and show that none of these transformations is adequate. Then show that using λ = 2 to transform the predictor Speed does match the data well. This suggestsusing a quadratic polynomial, including both Speed and Speed2.统计作业代写
(c)In Problem 1 we fit a quadratic mean function for Distance given Speed, assuming Var(Distance|Speed) = Speed σ2. Draw the plot of Distance versus Speed, and add a line on the plot of this fitted curve. Then obtain the fitted values from regressingthe transformed Distance on Speed, using the transformation you found in part (a). Transformthose fitted values back to the original scale, and add to your plot the line corresponding to the back-transformed fitted values. Compare the fit of the two models.
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