Question

Chapter Three Exercises
Exercise 3.7
At a used dealership, let X be an independent variable representing the age in years of a motorcycle and Y be the dependent variable representing the selling price of used motorcycle. The data is now given to you.

X = {5, 10, 12, 14, 15}

Y = {500, 400, 300, 200, 100}

A.) Find the coefficient of correlation and interpret its value.
B.) Find the coefficient of determination and interpret its value.

Answer 1

Answer 2

To find the coefficient of correlation (r) and the coefficient of determination (r²) for the given data, we first need to calculate the means of X and Y, as well as the sums of squares and cross-products. Let's proceed step by step:

Step 1: Calculate the means of X and Y Mean of X (X̄) = (5 + 10 + 12 + 14 + 15) / 5 = 11.2 Mean of Y (Ȳ) = (500 + 400 + 300 + 200 + 100) / 5 = 300

Step 2: Calculate the sums of squares and cross-products Sum of squares of X (SSX) = Σ(Xi - X̄)² SSX = (5 - 11.2)² + (10 - 11.2)² + (12 - 11.2)² + (14 - 11.2)² + (15 - 11.2)² SSX = 38.4

Sum of squares of Y (SSY) = Σ(Yi - Ȳ)² SSY = (500 - 300)² + (400 - 300)² + (300 - 300)² + (200 - 300)² + (100 - 300)² SSY = 40000

Sum of cross-products (SP) = Σ(Xi - X̄)(Yi - Ȳ) SP = (5 - 11.2)(500 - 300) + (10 - 11.2)(400 - 300) + (12 - 11.2)(300 - 300) + (14 - 11.2)(200 - 300) + (15 - 11.2)(100 - 300) SP = -720

Step 3: Calculate the coefficient of correlation (r) r = SP / √(SSX * SSY) r = -720 / √(38.4 * 40000) r ≈ -0.447

Step 4: Calculate the coefficient of determination (r²) r² = r² ≈ (-0.447)² ≈ 0.20

A.) Interpretation of the coefficient of correlation (r): The coefficient of correlation (r) measures the strength and direction of the linear relationship between the variables X (age of motorcycle) and Y (selling price). In this case, the calculated r value is approximately -0.447. Since the coefficient is negative, it indicates a negative correlation, which means that as the age of the motorcycle increases, the selling price tends to decrease.

The magnitude of the correlation coefficient (0.447) suggests a moderate negative correlation. It's not very strong, but there is still a tendency for older motorcycles to be sold at lower prices.

B.) Interpretation of the coefficient of determination (r²): The coefficient of determination (r²) represents the proportion of the variance in the dependent variable (Y) that can be explained by the independent variable (X). In this case, the calculated r² value is approximately 0.20.

This means that about 20% of the variability in the selling prices of used motorcycles can be explained by their ages. The remaining 80% of the variability is attributed to other factors not accounted for in this simple linear relationship.