SOLUTION: Many people believe that there is a negative relationship between the amount of time students play video games each week and their GPA. x 10 3 0 2 5 4 7 y 1.5 2.4 3.2 3.5 2.7

Question 1176942: Many people believe that there is a negative relationship between the amount of time students play video games each week and their GPA.
x 10 3 0 2 5 4 7
y 1.5 2.4 3.2 3.5 2.7 3 2.1
Using the data shown above, predict a student's GPA when the student plays video games for 15 hours each week. Round your final answer to two decimal places.
What percentage of variation in GPAs can be explained by the number of hours students' play video games each week? Round your final answer to two decimal places.

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**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%

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