Question 1176842
Let's solve this step-by-step.

**1. Calculate the Regression Equation**

We need to find the equation of the form ŷ = a + bx, where:

* x = hours playing video games
* y = GPA
* b = slope
* a = y-intercept

First, calculate the necessary sums:

* Σx = 10 + 3 + 0 + 2 + 5 + 4 + 7 = 31
* Σy = 1.5 + 2.4 + 3.2 + 3.5 + 2.7 + 3 + 2.1 = 18.4
* Σx² = 100 + 9 + 0 + 4 + 25 + 16 + 49 = 203
* Σy² = 2.25 + 5.76 + 10.24 + 12.25 + 7.29 + 9 + 4.41 = 51.2
* Σxy = 15 + 7.2 + 0 + 7 + 13.5 + 12 + 14.7 = 69.4
* n = 7 (number of data points)

Now, calculate b (slope):

* b = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)
* b = (7 * 69.4 - 31 * 18.4) / (7 * 203 - 31²)
* b = (485.8 - 570.4) / (1421 - 961)
* b = -84.6 / 460
* b ≈ -0.1839

Next, calculate a (y-intercept):

* a = (Σy - bΣx) / n
* a = (18.4 - (-0.1839) * 31) / 7
* a = (18.4 + 5.6999) / 7
* a = 24.0999 / 7
* a ≈ 3.4428

Therefore, the regression equation is:

* ŷ = 3.4428 - 0.1839x

**2. Predict GPA for 15 Hours of Video Games**

* x = 15
* ŷ = 3.4428 - 0.1839 * 15
* ŷ = 3.4428 - 2.7585
* ŷ ≈ 0.6843

Rounded to two decimal places, the predicted GPA is 0.68.

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

R² represents the percentage of variation in GPAs explained by the number of hours playing video games.

First, calculate the correlation coefficient (r):

* r = (nΣxy - ΣxΣy) / √((nΣx² - (Σx)²)(nΣy² - (Σy)²))
* r = -84.6 / √((460)(7 * 51.2 - 18.4²))
* r = -84.6 / √((460)(358.4 - 338.56))
* r = -84.6 / √((460)(19.84))
* r = -84.6 / √9126.4
* r = -84.6 / 95.5322
* r ≈ -0.8856

Now, calculate R²:

* R² = r²
* R² = (-0.8856)²
* R² ≈ 0.7843

Expressed as a percentage and rounded to two decimal places:

* R² ≈ 78.43%

**Answers**

* **Regression equation:** ŷ = 3.4428 - 0.1839x
* **Predicted GPA (15 hours):** 0.68
* **Percentage of variation explained:** 78.43%