Used Car Mileage and Price. The Toyota Camry is one of the...

Used Car Mileage and Price. The Toyota Camry is one of the best-selling cars in North America. The cost of a previously owned Camry depends upon many factors, including the model year, mileage, and condition. To investigate the relationship between the car's mileage and the sales price for a 2007 model year Camry, the following data show the mileage and sale price for 19 sales (PriceHub website). a. Develop a scatter diagram with the car mileage on the horizontal axis and the price on the vertical axis. b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables? c. Develop the estimated regression equation that could be used to predict the price $(\$ 1000 \mathrm{~s})$ given the miles $(1000 \mathrm{~s})$ d. Test for a significant relationship at the .05 level of significance. e. Did the estimated regression equation provide a good fit? Explain. f. Provide an interpretation for the slope of the estimated regression equation. g. Suppose that you are considering purchasing a previously owned 2007 Camry that has been driven 60,000 miles. Using the estimated regression equation developed in part (c), predict the price for this car. Is this the price you would offer the seller?

Used Car Mileage and Price. The Toyota Camry is one of the best-selling cars in North America. The cost of a previously owned Camry depends upon many factors, including the model year, mileage, and condition. To investigate the relationship between the car's mileage and the sales price for a 2007 model year Camry, the following data show the mileage and sale price for 19 sales (PriceHub website).
a. Develop a scatter diagram with the car mileage on the horizontal axis and the price on the vertical axis.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Develop the estimated regression equation that could be used to predict the price $(\$ 1000 \mathrm{~s})$ given the miles $(1000 \mathrm{~s})$
d. Test for a significant relationship at the .05 level of significance.
e. Did the estimated regression equation provide a good fit? Explain.
f. Provide an interpretation for the slope of the estimated regression equation.
g. Suppose that you are considering purchasing a previously owned 2007 Camry that has been driven 60,000 miles. Using the estimated regression equation developed in part (c), predict the price for this car. Is this the price you would offer the seller?

Key Concepts

Prediction Using Regression

Prediction using regression involves estimating the value of the dependent variable for a given value of the independent variable using the regression equation. This is a practical application of the model, where the predicted value can be compared to real-world observations or used to inform decision-making based on the expected outcome.

Interpretation of the Slope

The slope in a regression equation represents the change in the dependent variable for a one-unit change in the independent variable. It provides a quantitative measure of the direction and strength of the relationship. A positive slope indicates that as the independent variable increases, the dependent variable tends to increase, and vice versa for a negative slope.

Goodness-of-fit

Goodness-of-fit refers to how well a regression model fits the observed data. Measures such as the coefficient of determination (R²) indicate the proportion of variance in the dependent variable that is explained by the independent variable in the model. A good fit means that the model explains a substantial portion of the variability, while a poor fit suggests that the model does not capture the underlying trends well.

Linear Regression

Linear regression is a statistical method used for modeling the relationship between a dependent variable and one (simple linear regression) or more (multiple linear regression) independent variables. It involves fitting a line that minimizes the differences between the observed values and the predicted values, which is essential for prediction and inference about the relationship between variables.

Regression Equation

The regression equation is the mathematical representation of the linear relationship between the independent and dependent variables. It is typically expressed in the form y = ?0 + ?1x, where ?0 is the intercept and ?1 is the slope. This equation is used to predict values of the dependent variable based on known values of the independent variable.

Hypothesis Testing in Regression

Hypothesis testing in regression involves testing whether the slope coefficient (?1) is statistically significantly different from zero, which would indicate a relationship between the independent and dependent variable. This is often done using a t-test at a specified level of significance, such as 0.05, to determine if the observed relationship is unlikely to have occurred by chance.

Scatter Diagram

A scatter diagram, or scatter plot, is a graphical representation of the relationship between two numerical variables. In the context of regression analysis, it is used to visually assess the pattern, direction, and strength of the association between the independent variable (predictor) and the dependent variable (response), helping to determine whether a linear model might be appropriate.

Transcript

00:01 First of all, i want to state that the data is coming from exercise 41, which is a data set of costs of toyotas.

00:09 And those were put into a list in a calculator, list 1 and 2.

00:13 And then we're given the slope intercept, slope and intercept for this.

00:17 So the equation, linear regression equation, will be the price of the car equals 14 ,286, that comes out as the intercept, minus 959 times the years.

00:36 The slope is minus 959.

00:41 Let me fix that up.

00:43 That's a five, nine, 959 times the years.

00:48 To interpret the slope, as the car gets one year older, the car is going to drop by 959 dollars in price.

00:57 So each year the car is depreciating or going down 959 dollars.

01:04 Interpreting the intercept, when the car is brand new, zero years old, it would cost 14 ,286.

01:12 That's what the cost of the car would be at the purchase price, at the very beginning, when it's brand new or zero years old.

01:20 For d, if i'm going to predict the price of a seven -year -old car here, i'm going to plug in seven into that equation that i wrote for part a and then just solve.

01:28 So 14 ,286 minus 959 times seven is 7 ,573 dollars.

01:37 So that's what a seven -year -old toyota would be.

01:42 Would you rather purchase a car for the negative or positive residual? you want it to be negative because that would mean that the actual price, the actual cost of the car, would be lower than predicted.

01:57 So we'd want to have a negative residual, lower than predicted, because that means we got a better deal than what was predicted on there.

02:09 So actual lower than predicted, that's the negative residual.

02:14 For f, they said that the actual price of this car was 3 ,500 and it's a ten -year -old car...

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