Question 1176942
**1. Linear Regression**

We'll use linear regression to model the relationship between video game hours (x) and GPA (y).

* **Calculate the means:**
    * Mean of x (x̄): (10 + 3 + 0 + 2 + 5 + 4 + 7) / 7 = 31 / 7 ≈ 4.43
    * Mean of y (ȳ): (1.5 + 2.4 + 3.2 + 3.5 + 2.7 + 3 + 2.1) / 7 = 18.4 / 7 ≈ 2.63

* **Calculate the sums:**
    * Σ(x - x̄)(y - ȳ):
        * (10 - 4.43)(1.5 - 2.63) + (3 - 4.43)(2.4 - 2.63) + (0 - 4.43)(3.2 - 2.63) + (2 - 4.43)(3.5 - 2.63) + (5 - 4.43)(2.7 - 2.63) + (4 - 4.43)(3 - 2.63) + (7 - 4.43)(2.1 - 2.63) ≈ 6.26 + 0.33 + 2.59 + 2.11 + 0.06 - 0.16 - 1.48 ≈ 9.71
    * Σ(x - x̄)²:
        * (10 - 4.43)² + (3 - 4.43)² + (0 - 4.43)² + (2 - 4.43)² + (5 - 4.43)² + (4 - 4.43)² + (7 - 4.43)² ≈ 30.91 + 2.04 + 19.62 + 5.90 + 0.33 + 0.18 + 6.59 ≈ 65.57

* **Calculate the slope (b):**
    * b = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)² ≈ 9.71 / 65.57 ≈ -0.148

* **Calculate the y-intercept (a):**
    * a = ȳ - b * x̄ ≈ 2.63 - (-0.148) * 4.43 ≈ 2.63 + 0.655 ≈ 3.29

* **Regression equation:**
    * y = a + bx ≈ 3.29 - 0.148x

**2. Predict GPA for 15 Hours**

* Plug x = 15 into the regression equation:
    * y ≈ 3.29 - 0.148 * 15 ≈ 3.29 - 2.22 ≈ 1.07

* Therefore, the predicted GPA for a student playing video games for 15 hours is approximately 1.07.

**3. Coefficient of Determination (R²)**

* **Calculate Σ(y - ȳ)²:**
    * (1.5 - 2.63)² + (2.4 - 2.63)² + (3.2 - 2.63)² + (3.5 - 2.63)² + (2.7 - 2.63)² + (3 - 2.63)² + (2.1 - 2.63)² ≈ 1.28 + 0.05 + 0.33 + 0.76 + 0.005 + 0.14 + 0.28 ≈ 2.855

* **Calculate R²:**
    * R² = 1 - (Σ(y - ŷ)² / Σ(y - ȳ)²)
    * We need to find Σ(y-ŷ)²
    * ŷ values are found by inputting all the original x values into the regression equation: y = 3.29 - 0.148x.
    * ŷ = [1.81, 2.84, 3.29, 3.00, 2.55, 2.69, 2.25]
    * Σ(y-ŷ)² = (1.5-1.81)² + (2.4-2.84)² + (3.2-3.29)² + (3.5-3.00)² + (2.7-2.55)² + (3-2.69)² + (2.1-2.25)² = 0.0961 + 0.1936 + 0.0081 + 0.25 + 0.0225 + 0.0961 + 0.0225 = 0.689
    * R² = 1 - (0.689 / 2.855) ≈ 1 - 0.241 ≈ 0.759

* **Percentage of variation:**
    * R² * 100% ≈ 75.9%

**Answers**

* **Predicted GPA for 15 hours:** 1.07
* **Percentage of variation explained:** 75.90%