Solution

To find the coefficient of determination ($R^2$), we first need to calculate the correlation coefficient ($r$), which is related to $R^2$ by:

$R^2 = r^2$

Steps to Solve:

1. Regression Equation and Relationship: The regression equation is given as: $\hat{y} = 562.102x - 1960.998$ This predicts sales ($y$) based on the total square footage ($x$).

2. Data Summary:

   • $x$: Total Square Footage (in billions)
   • $y$: Sales (in billions of dollars)

   We have: $x = [5.1, 5.2, 5.3, 5.2, 5.4, 5.8, 5.8, 5.9, 5.9, 6.2]$ $y = [850.7, 939.3, 980.4, 1073.5, 1102.8, 1218.7, 1287.1, 1329.4, 1447.5, 1525.9]$

3. Calculate $r$: Using the formula for correlation coefficient: [ r = \frac{n \sum(xy) - \sum x \sum y}{\sqrt{\left[ n \sum x^2 - (\sum x)^2 \right] \left[ n \sum y^2 - (\sum y)^2 \right]}} ] Here:

   • $n = 10$ (number of data points)
   • $\sum x$: Sum of $x$
   • $\sum y$: Sum of $y$
   • $\sum xy$: Sum of products of $x$ and $y$
   • $\sum x^2$: Sum of squares of $x$
   • $\sum y^2$: Sum of squares of $y$

4. Interpret $R^2$: $R^2$ indicates the proportion of variability in $y$ (sales) that is explained by $x$ (square footage). A higher $R^2$ implies a stronger relationship.

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Calculation:

Let me compute the required values. numpy as np

Data points

x = np.array([5.1, 5.2, 5.3, 5.2, 5.4, 5.8, 5.8, 5.9, 5.9, 6.2]) # Square footage y = np.array([850.7, 939.3, 980.4, 1073.5, 1102.8, 1218.7, 1287.1, 1329.4, 1447.5, 1525.9]) # Sales

Number of data points

n = len(x)

Summations

sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x2 = np.sum(x ** 2) sum_y2 = np.sum(y ** 2)

Correlation coefficient (r)

numerator = n * sum_xy - sum_x * sum_y denominator = np.sqrt((n * sum_x2 - sum_x ** 2) * (n * sum_y2 - sum_y ** 2)) r = numerator / denominator

Coefficient of determination (R^2)

R2 = r ** 2

Results

r, R2### Results:

1. Correlation Coefficient ($r$): $0.961$
2. Coefficient of Determination ($R^2$): $0.924$ (rounded to three decimal places)

Interpretation:

The coefficient of determination ($R^2 = 0.924$) implies that 92.4% of the variability in sales ($y$) is explained by the total square footage ($x$) of retailing space. This indicates a very strong positive linear relationship between square footage and sales.

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Would you like further clarification or additional calculations?

Related Questions:

1. How is the regression line used for prediction in this context?
2. What would be the predicted sales for a total square footage of $6.0$ billion?
3. How would you calculate residuals for the given data points?
4. What does the correlation coefficient ($r$) indicate in practical terms?
5. How could outliers affect the $R^2$ value?

Tip:

When interpreting $R^2$, remember it does not imply causation—just the proportion of variability explained by the regression model.