Question

problem 3 the table to the right shows the length in inches and weight in pounds of several different dogs. regression equation:
if a dog is 75 inches in length what is its predicted weight?
if a dog is 65 inches in length what is its predicted weight?

Explanation:

Step1: Calculate slope ($m$) and y - intercept ($b$)

Let $x$ be the length and $y$ be the weight. Using the formula for the slope of the regression line $m=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}$ and $b = \bar{y}-m\bar{x}$, where $n = 7$.
First, calculate the necessary sums:

$x$	$y$	$xy$	$x^{2}$
62	114	7068	3844
64	124	7936	4096
66	131	8646	4356
68	139	9452	4624
70	149	10430	4900
72	159	11448	5184

$\sum x=60 + 62+64+66+68+70+72 = 462$
$\sum y=105 + 114+124+131+139+149+159 = 921$
$\sum(xy)=6300+7068+7936+8646+9452+10430+11448 = 61280$
$\sum(x^{2})=3600+3844+4096+4356+4624+4900+5184 = 30604$
$m=\frac{7\times61280-462\times921}{7\times30604 - 462^{2}}$
$=\frac{428960-425502}{214228 - 213444}$
$=\frac{3458}{784}\approx4.41$
$\bar{x}=\frac{462}{7}=66$
$\bar{y}=\frac{921}{7}\approx131.57$
$b = 131.57-4.41\times66$
$b=131.57 - 291.06=- 159.49$
The regression equation is $y = 4.41x-159.49$

Step2: Predict weight for $x = 75$

Substitute $x = 75$ into the regression equation:
$y=4.41\times75-159.49$
$y = 330.75-159.49=171.26$

Step3: Predict weight for $x = 65$

Substitute $x = 65$ into the regression equation:
$y=4.41\times65-159.49$
$y = 286.65-159.49 = 127.16$

Answer:

Regression Equation: $y = 4.41x-159.49$
If a dog is 75 inches in length, its predicted weight is 171.26 pounds.
If a dog is 65 inches in length, its predicted weight is 127.16 pounds.