Screen Time Marks Fe6C0D

1. **Problem Statement:** We are given data on daily screen time and marks of BMS students and asked to identify predictor and response variables, calculate correlation coefficient, regression coefficients, and write the regression equation. 2. **Identify Predictor and Response Variables:** - Predictor variable (independent): Screen Time (hours) - Response variable (dependent): Marks (out of 100) 3. **Calculate the Correlation Coefficient $r$:** Given data: \begin{align*} \text{Name} & \text{Screen Time }(x) & \text{Marks }(y) \\ \text{Mei} & 4.3 & 84 \\ \text{Bao} & 6.1 & 78 \\ \text{Fang} & 6.9 & 75 \\ \text{Lihua} & 7.5 & 65 \\ \text{Ming-Yue} & 9.3 & 52 \end{align*} Calculate sums: $$\sum x = 4.3 + 6.1 + 6.9 + 7.5 + 9.3 = 34.1$$ $$\sum y = 84 + 78 + 75 + 65 + 52 = 354$$ $$\sum xy = (4.3)(84) + (6.1)(78) + (6.9)(75) + (7.5)(65) + (9.3)(52) = 361.2 + 475.8 + 517.5 + 487.5 + 483.6 = 2325.6$$ $$\sum x^2 = 4.3^2 + 6.1^2 + 6.9^2 + 7.5^2 + 9.3^2 = 18.49 + 37.21 + 47.61 + 56.25 + 86.49 = 246.05$$ $$\sum y^2 = 84^2 + 78^2 + 75^2 + 65^2 + 52^2 = 7056 + 6084 + 5625 + 4225 + 2704 = 25694$$ Number of data points $n=5$. Formula for correlation coefficient: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$$ Calculate numerator: $$5 \times 2325.6 - 34.1 \times 354 = 11628 - 12071.4 = -443.4$$ Calculate denominator: $$\sqrt{(5 \times 246.05 - 34.1^2)(5 \times 25694 - 354^2)} = \sqrt{(1230.25 - 1162.81)(128470 - 125316)} = \sqrt{67.44 \times 3154} = \sqrt{212726.76} = 461.1$$ Therefore, $$r = \frac{-443.4}{461.1} \approx -0.9619$$ Interpretation: Strong negative correlation between screen time and marks. 4. **Calculate Regression Coefficients $b_0$ and $b_1$:** Formulas: $$b_1 = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b_0 = \frac{\sum y - b_1 \sum x}{n}$$ Calculate denominator for $b_1$: $$5 \times 246.05 - 34.1^2 = 1230.25 - 1162.81 = 67.44$$ Calculate numerator for $b_1$ (already found): $$-443.4$$ So, $$b_1 = \frac{-443.4}{67.44} \approx -6.58$$ Calculate $b_0$: $$b_0 = \frac{354 - (-6.58)(34.1)}{5} = \frac{354 + 224.38}{5} = \frac{578.38}{5} = 115.68$$ 5. **Write the Regression Equation:** $$\hat{y} = b_0 + b_1 x = 115.68 - 6.58 x$$ This equation predicts marks based on screen time. --- **Final answers:** - Predictor variable: Screen Time - Response variable: Marks - Correlation coefficient $r \approx -0.962$ - Regression coefficients: $b_0 \approx 115.68$, $b_1 \approx -6.58$ - Regression equation: $$\hat{y} = 115.68 - 6.58 x$$