Question

performance assessment #1 the table below shows the median home price (in 1000s) given its distance from new york city (in miles).

distance from new city (mi.)	median home price (in $1000s)
15	400
28	310
20	290
5	410
9	400
25	300
2	490
13	370
10	350
18	320
8	400

a. find the linear regression model to represent the data.

b. find the correlation coefficient for the regression model.

c. explain the meaning of the slope of the regression model in the context of the situation.

d. explain the meaning of the y - intercept of the regression model in the context of the situation.

e. use your linear regression model to predict what the median home price would be for homes that are 50 miles from new york city.

Explanation:

Step1: Calculate necessary sums

Let $x$ be the distance from New - York City and $y$ be the median home price.
We have $n = 12$ data points.
Calculate $\sum x$, $\sum y$, $\sum x^{2}$, $\sum xy$.
$\sum x=12 + 15+28+20+5+9+25+2+13+10+18+8=165$
$\sum y=390 + 400+310+290+410+400+300+490+370+350+320+400 = 4330$
$\sum x^{2}=12^{2}+15^{2}+28^{2}+20^{2}+5^{2}+9^{2}+25^{2}+2^{2}+13^{2}+10^{2}+18^{2}+8^{2}$
$=144 + 225+784+400+25+81+625+4+169+100+324+64 = 2941$
$\sum xy=12\times390+15\times400+28\times310+20\times290+5\times410+9\times400+25\times300+2\times490+13\times370+10\times350+18\times320+8\times400$
$=4680+6000+8680+5800+2050+3600+7500+980+4810+3500+5760+3200 = 56560$

Step2: Find the slope $m$ and y - intercept $b$ of the regression line

The slope $m$ is given by the formula:
$m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}$
$m=\frac{12\times56560 - 165\times4330}{12\times2941-165^{2}}$
$=\frac{678720-714450}{35292 - 27225}=\frac{-35730}{8067}\approx - 4.43$
The y - intercept $b$ is given by the formula:
$b=\frac{\sum y-m\sum x}{n}$
$b=\frac{4330-(-4.43)\times165}{12}$
$=\frac{4330 + 730.95}{12}=\frac{5060.95}{12}\approx421.75$
The linear regression model is $y=-4.43x + 421.75$

Step3: Calculate the correlation coefficient $r$

The formula for the correlation coefficient $r$ is:
$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}}$
First, calculate $\sum y^{2}=390^{2}+400^{2}+310^{2}+290^{2}+410^{2}+400^{2}+300^{2}+490^{2}+370^{2}+350^{2}+320^{2}+400^{2}$
$=152100+160000+96100+84100+168100+160000+90000+240100+136900+122500+102400+160000 = 1672400$
$r=\frac{12\times56560-165\times4330}{\sqrt{(12\times2941 - 165^{2})(12\times1672400-4330^{2})}}$
$r=\frac{-35730}{\sqrt{8067\times(20068800 - 18748900)}}$
$r=\frac{-35730}{\sqrt{8067\times1319900}}\approx - 0.93$

Step4: Interpret the slope

The slope of the regression model $m=-4.43$ means that for every one - mile increase in the distance from New York City, the median home price decreases by approximately $\$4430$ (since the price is in thousands of dollars).

Step5: Interpret the y - intercept

The y - intercept $b = 421.75$ means that when the distance from New York City is $0$ miles (in the context of the model), the median home price is approximately $\$421750$ (since the price is in thousands of dollars).

Step6: Make a prediction

To predict the median home price when $x = 50$
$y=-4.43\times50+421.75$
$y=-221.5+421.75=200.25$
The median home price for homes that are 50 miles from New York City is approximately $\$200250$

Answer:

a. $y=-4.43x + 421.75$
b. $r\approx - 0.93$
c. For every one - mile increase in the distance from New York City, the median home price decreases by approximately $\$4430$.
d. When the distance from New York City is $0$ miles, the median home price is approximately $\$421750$.
e. $\$200250$