91Ó°ÊÓ

Emily Smith decides to buy a fuel-efficient used car. Here are several vehicles she is considering, with the estimated cost to purchase and the age of the vehicle. $$ \begin{array}{|lrr|} \hline \text { Vehicle } & \text { Estimated Cost } & \text { Age } \\ \hline \text { Honda Insight } & \$ 5,555 & 8 \\ \text { Toyota Prius } & \$ 17,888 & 3 \\ \text { Toyota Prius } & \$ 9,963 & 6 \\ \text { Toyota Echo } & \$ 6,793 & 5 \\ \text { Honda Civic Hybrid } & \$ 10,774 & 5 \\ \text { Honda Civic Hybrid } & \$ 16,310 & 2 \\ \text { Chevrolet Cruz } & \$ 2,475 & 8 \\ \text { Mazda3 } & \$ 2,808 & 10 \\ \text { Toyota Corolla } & \$ 7,073 & 9 \\ \text { Acura Integra } & \$ 8,978 & 8 \\ \text { Scion xB } & \$ 11,213 & 2 \\ \text { Scion xA } & \$ 9,463 & 3 \\ \text { Mazda3 } & \$ 15,055 & 2 \\ \text { Mini Cooper } & \$ 20,705 & 2 \\ \hline \end{array} $$ a. Plot these data on a scatter diagram with estimated cost as the dependent variable. b. Find the correlation coefficient. c. A regression analysis was performed and the resulting regression equation is Estimated Cost \(=18358-1534\) Age. Interpret the meaning of the slope. d. Estimate the cost of a five-year-old car. e. Here is a portion of the regression software output. What does it tell you? $$ \begin{array}{lrrrr} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 18358 & 1817 & 10.10 & 0.000 \\ \text { Age } & -1533.6 & 306.3 & -5.01 & 0.000 \end{array} $$ f. Using the 10 significance level, test the significance of the slope. Interpret the result. Is there a significant relationship between the two variables?

Step by step solution

01

Plot Data on Scatter Diagram

To plot the data on a scatter diagram, place the 'Age' of each vehicle on the horizontal axis (independent variable) and the 'Estimated Cost' on the vertical axis (dependent variable). Each vehicle is represented by a point in this plot.

02

Calculate Correlation Coefficient

To find the correlation coefficient, use the formula to calculate the Pearson correlation: \( r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} \), where \(x\) is the age, \(y\) is the cost, and \(n\) is the number of data points. The result reflects the strength and direction of a linear relationship between two variables.

03

Interpret the Slope of Regression Equation

The regression equation provided is Estimated Cost = 18358 - 1534*Age. The slope is -1534, which means for each additional year of the car's age, the estimated cost decreases by $1534, assuming a linear relationship.

04

Estimate Cost of a Five-Year-Old Car

Use the regression equation Estimated Cost = 18358 - 1534*Age to estimate the cost of a 5-year-old car: Estimated Cost = 18358 - 1534*5 = $10,688.

05

Analyze Regression Output

The regression software output shows the coefficients for the constant and Age. It indicates that the coefficient for Age is -1533.6 with a standard error of 306.3. The p-value for the Age is 0.000, suggesting the slope is statistically significant.

06

Test Significance of Slope at 10% Level

Test the null hypothesis that the slope is zero using a t-test: the t-statistic is -5.01 (given) and compares with the critical value from the t-distribution at a 10% significance level. Since the p-value (0.000) is less than 0.10, we reject the null hypothesis. This indicates a significant relationship between Age and Estimated Cost.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Diagram

A scatter diagram is an essential tool used in regression analysis. It visually represents the relationship between two numerical variables. In our scenario, we plot each vehicle's age on the horizontal axis and its estimated cost on the vertical axis. Each point on the scatter diagram corresponds to a vehicle, showing how its age pairs with its cost.

By examining the scatter diagram, we can identify trends, patterns, or any form of correlation. For instance, if the points tend to slope downwards from left to right, it might suggest a negative relationship, meaning older cars generally cost less. This visual approach helps establish the groundwork for further statistical analysis, making it easier to understand the potential relationship between the age and the cost of a vehicle.

Correlation Coefficient

The correlation coefficient is a statistical measure that calculates the strength and direction of a linear relationship between two variables, typically denoted by 'r'. To find 'r', we use the Pearson correlation formula:

\[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} \]

Here, 'x' represents the age, 'y' is the cost, and 'n' is the number of observations. The resulting value of 'r' ranges from -1 to 1.

• A value close to 1 indicates a strong positive correlation, where increases in age correspond to increases in cost.
• A value close to -1 suggests a strong negative correlation, where as the age increases, the cost typically decreases.
• A value around 0 implies no linear correlation.

Understanding the correlation coefficient allows us to gauge how tightly the costs of the cars align with their ages, which is crucial for prediction and analysis.

Statistical Significance

Statistical significance in regression analysis helps us determine whether the relationship observed in our sample can be generalized to the larger population. It is often assessed by looking at the p-value associated with the predictor in a regression model.

In our analysis, the p-value for the coefficient of Age is 0.000. This indicates the result is statistically significant when we conduct a test at the 10% significance level. Since our p-value is much lower than 0.10, we have strong evidence to reject the null hypothesis. This means that there is indeed a significant linear relationship between the age of the vehicles and their costs.

• This significance implies that as a car gets older, its estimated cost reliably decreases.

By analyzing statistical significance, we confirm that the observed trend is not just due to random chance and provides grounded insights into the data.

Regression Equation

The regression equation is a mathematical expression representing the relationship between the independent and dependent variables. In this case study, the regression equation is given by:

Estimated Cost = 18358 - 1534 * Age

Here, the slope of the regression line is -1534. This indicates that every one-year increase in a vehicle's age results in an average cost decrease of $1,534, assuming a linear relationship. The intercept (18358) represents the estimated cost of a vehicle with an age of zero, which serves as the starting point of our equation, though not practical in real-world terms.

This equation allows us to predict the expected cost of a car based on its age. For example, to estimate a 5-year-old car's cost, we substitute Age with 5:

• Estimated Cost = 18358 - 1534 * 5 = 10688

The regression equation is a powerful tool that extends descriptive analysis to predictive analytics, offering tangible insights and decisions based on the observed data.