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.

performance assessment #2
the pride of the pack marching band is working on a new formation. the front row of their formation will have 5 students. each subsequent row will have 3 additional students. there will 15 total rows in the formation.
a) explain, using complete sentence(s), why this represents an arithmetic sequence.
b) write a recursive formula to represent the situation.
c) write an explicit formula to represent the situation.
d) how students will be in the 12^th row?

Explanation:

Performance Assessment #1

Step1: Calculate sums and means

Let $x$ be the distance from New - York City and $y$ be the median home price.
Calculate $\sum x$, $\sum y$, $\sum x^2$, $\sum xy$, $\bar{x}$, $\bar{y}$.
Suppose we have $n = 12$ data - points.
$\sum x=12 + 15+28+20+5+9+25+2+13+10+18+8=165$
$\bar{x}=\frac{\sum x}{n}=\frac{165}{12}=13.75$
$\sum y=390 + 400+310+290+410+400+300+490+370+350+320+400 = 4230$
$\bar{y}=\frac{\sum y}{n}=\frac{4230}{12}=352.5$
$\sum x^{2}=12^{2}+15^{2}+28^{2}+20^{2}+5^{2}+9^{2}+25^{2}+2^{2}+13^{2}+10^{2}+18^{2}+8^{2}$
$=144 + 225+784+400+25+81+625+4+169+100+324+64 = 2941$
$\sum xy=12\times390+15\times400+28\times310+20\times290+5\times410+9\times400+25\times300+2\times490+13\times370+10\times350+18\times320+8\times400$
$=4680+6000+8680+5800+2050+3600+7500+980+4810+3500+5760+3200 = 56560$

Step2: Calculate slope $m$

The formula for the slope $m$ of the regression line is $m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}$
$m=\frac{12\times56560 - 165\times4230}{12\times2941-165^{2}}$
$=\frac{678720-697950}{35292 - 27225}$
$=\frac{-19230}{8067}\approx - 2.38$

Step3: Calculate y - intercept $b$

The formula for the y - intercept $b$ is $b=\bar{y}-m\bar{x}$
$b = 352.5-(-2.38)\times13.75$
$=352.5 + 32.725=385.225$

The linear regression model is $y=-2.38x + 385.225$

Step4: Calculate 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 $\sum y^{2}=390^{2}+400^{2}+310^{2}+290^{2}+410^{2}+400^{2}+300^{2}+490^{2}+370^{2}+350^{2}+320^{2}+400^{2}$
$=152100+160000+96100+84100+168100+160000+90000+240100+136900+122500+102400+160000 = 1672400$
$r=\frac{12\times56560-165\times4230}{\sqrt{(12\times2941 - 165^{2})(12\times1672400-4230^{2})}}$
$r\approx - 0.73$

Step5: Interpret slope

The slope $m=-2.38$ means that for every one - mile increase in the distance from New York City, the median home price decreases by approximately $\$2380$ (since the home price is in thousands of dollars).

Step6: Interpret y - intercept

The y - intercept $b = 385.225$ means that when the distance from New York City is $0$ miles, the predicted median home price is approximately $\$385225$ (since the home price is in thousands of dollars).

Step7: Make prediction

When $x = 50$, $y=-2.38\times50+385.225$
$y=-119+385.225 = 266.225$
The predicted median home price for homes that are 50 miles from New York City is approximately $\$266225$

Performance Assessment #2

a. Brief Explanation:
An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. In the marching - band formation, the number of students in the first row is $a_1 = 5$. The number of students in the second row is $a_2=5 + 3$, the number of students in the third row is $a_3=5 + 3+3$, and so on. The common difference $d = 3$ between the number of students in consecutive rows. So, the number of students in each row forms an arithmetic sequence.

Substitute $n = 12$ into the formula $a_n=3n + 2$.
$a_{12}=3\times12+2$
$a_{12}=36 + 2=38$

Answer:

This represents an arithmetic sequence because the number of students in consecutive rows has a common difference of 3.

b. Recursive formula:
The first - term $a_1 = 5$, and the recursive formula for an arithmetic sequence is $a_n=a_{n - 1}+d$, where $d$ is the common difference.