Miami University Statistics Homework Problems

OverviewThis assignment includes two regression problems. The first problem is straightforward in terms of coding, but covers the new material related to linear regression (note, you will need both module 05 and module 06 to complete this homework). We step you through the process! The second question combines elements of our design of experiment unit with regression and has you compare a fitted model to some plots. There is also the opportunity for some bonus points.Problem 1 (20pts)A data file, homePrices2020.csv, containing information on 76 single-family homes in Corvallis, Oregon during 2020 was provided by Victor Whitman, a realtor. We will model single-family home sale prices (Price, in thousands of dollars), using these predictor variables:Floor = floor size (thousands of square feet)Bed = number of bedrooms (between 2 and 6)Age = age (standardized: (year built – 1970)/10—explained below)Garage = garage size (0, 1, 2, or 3 cars)From the realtor’s experience, both very old homes and very new homes tend to command a premium price relative to “middle age” homes in this market, so a quadratic (curvature) effect might be expected for an age variable in a multiple linear regression model to predict price. To facilitate this, the Age variable was rescaled subtracting 1970 from “year built” and dividing by 10.Read in the data, reorder the variables (use select() ) so that the response variable Price is the last selected variable, and construct a scatterplot matrix of all variables. Describe any discernible patterns you see. (3pts)Fit a multiple regression modeling the home price as a function of floor size, number of bedrooms, the rescaled age and garage size. (1pt)Perform a residual analysis of the fitted model, in particular, do you see any concerns regarding the assumptions we make in linear regression? (2pts)Construct a Box-Cox transformation plot to the fitted model in Part 2, what is the optimal LaTeX: lambda? suggested? What type of transformation would that suggest? What other transformations are viable? (2pts)You should note that LaTeX: lambda=0? = 0 is well within the supplied confidence bands of the Box-Cox plot. So, fit a multiple regression modeling the log of the home price as a function of the floor size, number of bedrooms, rescaled age, and garage size. (1pt)Perform a residual analysis of the fitted model in part 5, does it appear that the logarithm has improved the residuals plots? (1pt)Interpret the intercept term for the fitted model from part 2. Does it contextually make any sense (2pts)Interpret each of the four slope coefficients in the fitted model from part 2. (3pts)Does the model from part 2 significantly predict the price of homes in Corvallis, Oregon? If so, what percentage of the variability of the price of homes is explained by the fitted model? (3pts)Use your fitted model from part 5 to predict, with 95% confidence, the price of a 3 bedroom home with a 2 car garage that has 1.9 thousand square feet of floor size and was built in 1975 (hint: make sure to construct a prediction interval). Make sure to interpret the interval in context. (2pts)Problem 2 (15pts + 4pts bonus)A study was conducted to assess the fuel efficiency as a function of car type (a sedan versus a van versus a sports utility vehicle), gasoline octane levels, and the odometer readings (mileage) on a vehicle. The file carEfficiency.csv???? contains the odometer reading, car type, octane of the gasoline (treat as a numeric measured outcome) used and the efficiency (in miles per gallon, MPG) for 25 sample vehicles.Construct two scatterplots, one showing the relationship between the response MPG and the Odometer reading and the other showing the relationship between the response MPG and the octane used. Describe/discuss any relationships you see – in particular, does it appear the Odometer reading or Octane level is predictive of MPG? (3pts)Fit a linear regression modeling the MPG of the vehicles based on the Odometer reading and the Octane used. Does this model significantly predict the mean MPG of vehicles? Discuss how this result agrees/disagrees with the findings in part 1. (2pts)Remake the two scatterplots from part 1, this time let the color of each point be determined by the `CarType` variable in the dataset. Describe/discuss any relationships you see. (2pts)Fit a linear regression modeling the MPG of the vehicles based on the Odometer reading, the Octane used and the categorical variable `CarType`. Does this model significantly predict the mean MPG of vehicles? Discuss how this result agrees/disagrees with the findings in part 3. (2pts)Interpret the intercept from the model in part 4. (2pts)At the 1% significance level, perform the appropriate hypothesis test on the Odometer reading, make sure to write a conclusion statement that properly interprets the results of the hypothesis test in context. (2pts)Fit a linear regression modeling the MPG of the vehicles where you only include significant predictors (based on the 1% significance level). Write a statement interpreting the adjusted R-squared value of this model in context. (2pts)Bonus:Construct a plot of the model you fit in part 7 including confidence bands. Hint: the resulting plot should include multiple lines plotted on a scatterplot with included confidence bands. (2pts)Discuss the plot and how it relates to the coefficients from the model in part 7. Specifically, how the MPG varies for the van and SUV categories of vehicles. What does this imply about the associated t-test for these variables for the model fit in part 7. (2pts)