Question

determine the regression equation for the following data set. then, use the regression equation to determine y when x = 15.

x	1	2	3	4	5	6	7	8	9	10
y	73	26	22	5	6	13	13	36	57	96

711.39
823.05
373.05
776.18

Explanation:

Step1: Calculate sums

Let \(n = 10\). Calculate \(\sum_{i = 1}^{n}x_i=1 + 2+\cdots+10=\frac{10\times(10 + 1)}{2}=55\), \(\sum_{i = 1}^{n}y_i=73+26+\cdots+96 = 347\), \(\sum_{i = 1}^{n}x_i^2=1^2+2^2+\cdots+10^2=\frac{10\times(10 + 1)\times(2\times10 + 1)}{6}=385\), \(\sum_{i = 1}^{n}x_iy_i=1\times73+2\times26+\cdots+10\times96 = 2107\).

Step2: Calculate slope \(b_1\)

The formula for the slope \(b_1\) of the regression - line \(y=b_0 + b_1x\) is \(b_1=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\). Substitute the values: \(b_1=\frac{10\times2107-55\times347}{10\times385 - 55^2}=\frac{21070-19085}{3850 - 3025}=\frac{1985}{825}\approx2.406\).

Step3: Calculate intercept \(b_0\)

The formula for the intercept \(b_0\) is \(b_0=\overline{y}-b_1\overline{x}\), where \(\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{55}{10}=5.5\) and \(\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{347}{10}=34.7\). Then \(b_0 = 34.7-2.406\times5.5=34.7 - 13.233=21.467\).
The regression equation is \(y = 21.467+2.406x\).

Step4: Predict \(y\) for \(x = 15\)

Substitute \(x = 15\) into the regression equation: \(y=21.467+2.406\times15=21.467 + 36.09=776.18\).

Answer:

776.18