MODEL 1: (A) Use a simple linear regression of y=Price against X=Odometer. State your NULL & ALTERNATE hypotheses (for each of these models) and your conclusions. Use a=.05 throughout this project (for each of these models). Plot the graph of price against odometer and add it to your report. Decide how much the value of a car is lowered if it is driven an extra 1000 miles. MODEL 1:(B) Use a simple linear regression of y=Price against X=AGE. State your NULL & ALTERNATE hypotheses (for each of these models) and your conclusions. Use a=.05 throughout this project (for each of these models). Decide how much the value of a car is lowered if it is one year older. MODEL 1:(C) Use a simple linear regression of y=Price against X=AUCTIONNUM. State your NULL & ALTERNATE hypotheses (for each of these models) and your conclusions. Discuss why this model DOES NOT MAKE ANY SENSE! What can we learn from this?
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- ALTERNATE hypothesis: There is a relationship between the odometer reading and the price of a car. - Conclusion: Perform a simple linear regression analysis to determine if there is a significant relationship between the odometer reading and the price of a car. Show more…
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1. Data has been gathered from 11 used cars. If you look at this car sheet in the picture file, you will be able to see data on the 11 used cars. Create a scatterplot (Excel or Sheets) with price as the response variable (variable on the y-axis) and either age or mileage as the explanatory variable (on the x-axis). b) Based on your scatterplot, describe the relationship between the two variables. c)Using technology, create the least-squares regression line for your scatterplot. Write the equation of the least-squares regression line. Be sure to describe the variables in your equation. d) Interpret the slope of the least-squares regression line in context. e) Interpret the y-intercept of the least-squares regression line in context. f) Find the value of the correlation coefficient and the value of the coefficient of determination for your least-squares regression line. Interpret both values in context. g) If you chose to create a least-squares regression line based on age, use your least-squares regression equation to predict the price for a car that is 4 years old. How does your prediction compare to the actual price of the car that was 4 years old? If you chose to create a least-squares regression line based on mileage, use your least-squares regression equation to predict the price for a car with 19,860 miles. How does your prediction compare to the car that had a mileage of 19,860?
Jerelyn N.
(b) Develop an estimated regression equation showing how price is related to miles. What is the estimated regression model? Let x represent the miles. If required, round your answers to four decimal places. For subtractive or negative numbers, use a minus sign even if there is a + sign before the blank (Example: -300) ŷ = 16.47 + -0.059 x (c) Test whether each of the regression parameters β0 and β1 is equal to zero at a 0.01 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable? (d) How much of the variation in the sample values of price does the model estimated in part (b) explain? If required, round your answer to two decimal places. (e) For the model estimated in part (b), calculate the predicted price and residual for each automobile in the data. Identify the two automobiles that were the biggest bargains. If required, round your answer to the nearest whole number. The best bargain is the Camry #_____ in the data set, which has _____ miles and sells for $_____ less than its predicted price. The second best bargain is the Camry #_____ in the data set, which has _____ miles and sells for $_____ less than its predicted price. (f) Suppose that you are considering purchasing a previously owned Camry that has been driven 40,000 miles. Use the estimated regression equation developed in part (b) to predict the price for this car. If required, round your answer to one decimal place. Do not round intermediate calculations. Predicted price: $_____ Is this the price you would offer the seller? Explain.
4. Consider the "sale.txt" data posted (at onQ, below this file). A company conducted a survey of a new product to develop a marketing strategy. The response Y is the amount (in thousand dollars) that each individual can spend on the new product. The possible explanatory variable x is the yearly income (in hundred dollars). There are 21 participants in the study. (a) Fit a simple linear regression for Y and x. Define this model clearly in mathematical form. Assess the model fit by the 3 types of residual plots introduced in Section 5.2. Do you find any problems with the constant variance assumption? (b) Use the Box-Cox transformation on Y to improve the model. What transformation do you choose? Define a new model in mathematical form based on this transformation. Fit the model, and assess the model performance using suitable residual plot or plots. Is it a good remedy for the problem identified in (a)? (c) Now consider a model without intercept, Yi = ̢̢xi + ̢i where Yi and xi are the amount individual i can spend on the new product, and his/her yearly income. The error terms ̢i are assumed to be i.i.d. N(0, ̢2). Fit this model to the data. Does it fit the data better than the model in (a)? Repeat residual analysis (and plots) for this model as in (a) and comment. (d) Suppose the true model is Yi = ̢xi + ̢i where error terms ̢i are independent from N(0, ̢2xi) distribution. That is, Var(Yi) = Var(̢i) = ̢2xi. Notice this is a model without an intercept, and for heteroscedastic data! A possible remedy for heteroscedasticity is to consider the model Yi* = ̢xi* + ̢i*, where Yi* = Yi/√xi, xi* = √xi and ̢i* = ̢i/√xi. In theory, do ̢i*'s have equal variances now? Fit this model to the data, and assess the model performance through relevant residual plot or plots. Does it work in fixing the non-constant variance problem for the model in (c)? Remark: Part (d) is really applying a weighted least squares method to the heteroscedastic data. We can also apply it directly to the original data through the lm() function with weights. Try this to see if you get exactly the same output as from the model you fit for (d). > fitwt=lm(Y~X-1,weights=1/X) # Original data attached.
Dominador T.
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