Question: 41-45. Seven students from B.S. in Criminology class of Arellano University were given dummy homework that needs to be completed. The table below shows the number of hours of studies and performance in the examination for Mathematics in the Modern World.
Study Hours (X) Exam Score (Y)
4 70
12 98
9 85
6 80
2 65
3 75
10 92
1. **State the problem:** We are given data of study hours ($X$) and exam scores ($Y$) for seven students. We want to analyze the relationship between $X$ and $Y$, typically by finding the linear regression line $Y = a + bX$.
2. **Formula used:** The formulas for the slope $b$ and intercept $a$ of the regression line are:
$$b = \frac{n\sum XY - \sum X \sum Y}{n\sum X^2 - (\sum X)^2}$$
$$a = \frac{\sum Y - b \sum X}{n}$$
where $n$ is the number of data points.
3. **Calculate sums:**
- $n = 7$
- $\sum X = 4 + 12 + 9 + 6 + 2 + 3 + 10 = 46$
- $\sum Y = 70 + 98 + 85 + 80 + 65 + 75 + 92 = 565$
- $\sum XY = (4)(70) + (12)(98) + (9)(85) + (6)(80) + (2)(65) + (3)(75) + (10)(92) = 280 + 1176 + 765 + 480 + 130 + 225 + 920 = 3976$
- $\sum X^2 = 4^2 + 12^2 + 9^2 + 6^2 + 2^2 + 3^2 + 10^2 = 16 + 144 + 81 + 36 + 4 + 9 + 100 = 390$
4. **Calculate slope $b$:**
$$b = \frac{7 \times 3976 - 46 \times 565}{7 \times 390 - 46^2} = \frac{27832 - 25990}{2730 - 2116} = \frac{1842}{614}$$
Show cancellation:
$$b = \frac{\cancel{1842}}{\cancel{614}} = 3$$
Actually, $1842 \div 614 = 3$ exactly.
5. **Calculate intercept $a$:**
$$a = \frac{565 - 3 \times 46}{7} = \frac{565 - 138}{7} = \frac{427}{7} = 61$$
6. **Regression line:**
$$Y = 61 + 3X$$
7. **Interpretation:** For each additional hour studied, the exam score increases by 3 points on average. The base score with zero study hours is 61.
**Final answer:** The linear regression equation relating study hours to exam score is $$Y = 61 + 3X$$.