Question

If the values of $x$ are 1, 2, and 3 and the corresponding values of $y$ are 9, 8, and 10 respectively, then the slope of the line of regression equation of $y$ on $x$ is (up to 1 decimal place) __________________.

Hint:

In regression problems, always compute means first, then deviations to find covariance and variance. For small data sets, manual calculation is straightforward.

Answer 1

Correct Answer: 0.5

Step 1: Recall regression slope formula
The slope $b_{yx}$ of regression of $y$ on $x$ is: \[ b_{yx} = \frac{\text{Cov}(x,y)}{\text{Var}(x)} \] Step 2: Compute means
\[ \bar{x} = \frac{1+2+3}{3} = 2, \bar{y} = \frac{9+8+10}{3} = 9 \] Step 3: Compute deviations and products
\[ (x, y): (1,9), (2,8), (3,10) \] Deviations from mean: \[ (1-2, 9-9) = (-1,0), (2-2, 8-9) = (0,-1), (3-2, 10-9) = (1,1) \] Products: \[ (-1)(0)=0, (0)(-1)=0, (1)(1)=1 \] \[ \text{Cov}(x,y) = \frac{0+0+1}{3} = \frac{1}{3} \] Step 4: Compute variance of $x$
\[ (x- \bar{x})^2: (-1)^2=1, (0)^2=0, (1)^2=1 \] \[ \text{Var}(x) = \frac{1+0+1}{3} = \frac{2}{3} \] Step 5: Slope calculation
\[ b_{yx} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} = 0.5 \] Final Answer: \[ \text{Slope of regression line} = 0.5 \]