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Question 1176842: 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
Write the regression equation below. Round all numbers to four decimal places.
y hat =
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! 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%
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