Question

question 3 of 10
a data set consists of these points: (2, 4), (4, 7), (5, 12). malinda found the regression equation to be \\(\hat{y}=2.5x - 1.5\\). is she correct?
a. yes. this is the correct equation.
b. no. both a and b are incorrect.
c. no. the value for a is incorrect.
d. no. the value for b is incorrect.

Explanation:

Step1: Recall linear regression equation form

The linear - regression equation is $\hat{y}=ax + b$, where $a$ is the slope and $b$ is the y - intercept.

Step2: Use the least - squares formula for slope $a$

The formula for the slope $a$ of the regression line $\hat{y}=ax + b$ is $a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$, and for the y - intercept $b=\bar{y}-a\bar{x}$, where $n$ is the number of data points, $\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$ and $\bar{y}=\frac{1}{n}\sum_{i = 1}^{n}y_i$.
For the data points $(2,4),(4,7),(5,12)$:
$n = 3$, $\sum_{i=1}^{3}x_i=2 + 4+5=11$, $\sum_{i = 1}^{3}y_i=4 + 7+12 = 23$, $\sum_{i=1}^{3}x_i^{2}=2^{2}+4^{2}+5^{2}=4 + 16+25 = 45$, $\sum_{i=1}^{3}x_iy_i=2\times4+4\times7+5\times12=8 + 28+60 = 96$.
First, calculate the slope $a$:
\[

$$\begin{align*} a&=\frac{3\times96-11\times23}{3\times45 - 11^{2}}\\ &=\frac{288-253}{135 - 121}\\ &=\frac{35}{14}\\ & = 2.5 \end{align*}$$

\]
Then, calculate $\bar{x}=\frac{11}{3}$ and $\bar{y}=\frac{23}{3}$.
Calculate the y - intercept $b$:
\[

$$\begin{align*} b&=\frac{23}{3}-2.5\times\frac{11}{3}\\ &=\frac{23}{3}-\frac{5}{2}\times\frac{11}{3}\\ &=\frac{23\times2-55}{6}\\ &=\frac{46 - 55}{6}\\ &=-\frac{9}{6}=-1.5 \end{align*}$$

\]
The regression equation is $\hat{y}=2.5x-1.5$.

Answer:

A. Yes. This is the correct equation.