Question

the data seems to have a linear trend. so, melissa wants to find a linear equation that could approximately model the data. select \compute\ to generate the values used for a linear regression model. write the values in your answer to 3 decimal places.
x: 2, 3, 4, 5, 6, 7, 9
y: 8, 7, 6, 4, 5, 2, 2
lin reg: $y = ax + b$
$a = $
$b = $
$r^2 = $
$r = $
the linear equation that best fits the data is $y = $
) use the equation from part (b) to help melissa predict the value of $y$ when $x = 1$. write your answer to 3 decimal places.
$y = $

Explanation:

Step1: Calculate sums for regression

First, compute necessary sums:
$n=7$
$\sum x = 2+3+4+5+6+7+9 = 36$
$\sum y = 8+7+6+4+5+2+2 = 34$
$\sum xy = (2*8)+(3*7)+(4*6)+(5*4)+(6*5)+(7*2)+(9*2) = 16+21+24+20+30+14+18 = 143$
$\sum x^2 = 2^2+3^2+4^2+5^2+6^2+7^2+9^2 = 4+9+16+25+36+49+81 = 220$
$\sum y^2 = 8^2+7^2+6^2+4^2+5^2+2^2+2^2 = 64+49+36+16+25+4+4 = 198$

Step2: Calculate slope $a$

Use slope formula:
$$a = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$
$$a = \frac{7*143 - 36*34}{7*220 - 36^2} = \frac{1001 - 1224}{1540 - 1296} = \frac{-223}{244} \approx -0.914$$

Step3: Calculate intercept $b$

Use intercept formula:
$$b = \frac{\sum y - a\sum x}{n}$$
$$b = \frac{34 - (-0.914)*36}{7} = \frac{34 + 32.904}{7} = \frac{66.904}{7} \approx 9.558$$

Step4: Calculate correlation $r$

Use correlation formula:
$$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$
$$n\sum y^2 - (\sum y)^2 = 7*198 - 34^2 = 1386 - 1156 = 230$$
$$r = \frac{-223}{\sqrt{244*230}} = \frac{-223}{\sqrt{56120}} \approx \frac{-223}{236.897} \approx -0.941$$

Step5: Calculate $r^2$

Square the correlation coefficient:
$r^2 = (-0.941)^2 \approx 0.885$

Step6: Write linear equation

Substitute $a$ and $b$ into $y=ax+b$:
$y = -0.914x + 9.558$

Step7: Predict $y$ when $x=1$

Substitute $x=1$ into the regression equation:
$y = -0.914(1) + 9.558 = 8.644$

Answer:

$a = -0.914$
$b = 9.558$
$r^2 = 0.885$
$r = -0.941$
The linear equation that best fits the data is $y = -0.914x + 9.558$
When $x=1$, $y = 8.644$