Question

New Car Sales

New car sales have increased at a roughly linear rate. Sales, in millions of vehicles, from 1992 to 2005, are given in the table below. Let x represent the number of years since 1990.
Year Sales Year Sales
1992 12.8 1999 17.0
1993 13.9 2000 17.4
1994 15.0 2001 17.2
1995 14.7 2002 16.8
1996 15.1 2003 16.7
1997 15.2 2004 16.9
1998 15.6 2005 17.0

A. Find the equation of the least squares line and the coefficient of correaltion

B. Find the equation of the least squares line using only the data for every other year starting with 1993, 1995, aand so on. Find the coefficient of correlation.

C. Compare your results for parts a and b. What do you find? Why do you think this happens?

Answer 1

this is what i have so far and i am stuck..:
let x represent years x=1 corresponding to the year 1992.

let y represent sales

value for x is 105 (1+2+3+4+5+6+7+8+9+10+11+12+13+14

value for y is 145.2 (12.8+13.9+15.0+14.7+15.1+15.2+15.6+17.0+17.4+17.2+16.8+16.7+16.9+17.0)

Answer 2

They say
Let x represent the number of years since 1990.

that means 1992 corresponds to x=2

there are a couple different ways to find the least squares line
Can you post the details on the approach they are teaching you ?

Answer 3

i can post what i came up with... and what the instructor replied to

Answer 4

what i originally came up with:
page 43 #10

New Car Sales

New car sales have increased at a roughly linear rate. Sales, in millions of vehicles, from 1992 to 2005, are given in the table below. Let x represent the number of years since 1990.
Year Sales Year Sales
1992 12.8 1999 17.0
1993 13.9 2000 17.4
1994 15.0 2001 17.2
1995 14.7 2002 16.8
1996 15.1 2003 16.7
1997 15.2 2004 16.9
1998 15.6 2005 17.0

A. Find the equation of the least squares line and the coefficient of correaltion

see below for equation of the least squares line and the coefficient of correaltion is 1.35

let x represent years x=1 corresponding to the year 1992.

let y represent sales

value for Σx is 105 (1+2+3+4+5+6+7+8+9+10+11+12+13+14

value for Σy is 145.2 (12.8+13.9+15.0+14.7+15.1+15.2+15.6+17.0+17.4+17.2+16.8+16.7+16.9+17.0)

value for Σxy is 1722.2

value for Σx2 is 1015

value for Σy2 is 3519.89



m=n(∑xy)−(∑x )(∑x )→÷n(∑x2) − (∑x)2 <<formula for m

1(1722.2)-(105)(145.2)

1(1015)-(105)2 <<sub from the table


1722.2-15246

1015-11025 <<multiply


-13523.8

-10010 <<subtract


m=1.35


B. Find the equation of the least squares line using only the data for every other year starting with 1993, 1995, and so on. Find the coefficient of correlation.

see equation below for least squares line coefficient of correlation is 2.07

value for Σx is 56

value for Σy is 111.7

value for Σxy is 918.6

value for Σx2 is 560

value for Σy2 is 1789.07

1(918.6)-(56)(111.7)

1(560)-(56)2 <<sub from the table


918.6-6255.2

560-3136<<multiply


-5336.6

-2576 <<subtract


m=2.07

C. Compare your results for parts a and b. What do you find? Why do you think this happens?

My results for part a is 1.34, and for part b is 2.07. I find that the new car sales is higher on the odd year cars. I feel this happens due to sales being higher in the odd years than what they were in the even years.


what the instructor replied back
your answers are incorrect. First, since x represent years since 1900, I subtracted 1900 from 1992 and got 92. you can do this for all of the other years. then, if you solve for the sums, you should be getting

∑x = 1,379

∑y = 221.3

∑xy = 21,866.1

∑x2 = 136,059

∑y2 = 3,524

then you solve for teh slope of the least squares line, m.

Once you have that you can use it to determine the b (the y-intercept) of the least squares line.

You shoudl then be able to write your equation of yoru line in y=mx+b format.

Then you need to solve for the coefficient of the correlation, r.

You might find it easier to use the calculator, it will take less time and give you the same result.

Answer 5

I am confused. The question says
Let x represent the number of years since 1990.

Your teacher says
since x represent years since 1900

which is it ?

Answer 6

For part A, you said

A. Find the equation of the least squares line and the coefficient of correaltion

see below for equation of the least squares line and the coefficient of correaltion is 1.35



You got a coeff of correlation of 1.35 That should make you very suspicious.


Remember this fact:
Pearson correlation coefficient is between +1 and −1 inclusive.

This tells you that any estimate outside this range -1 to +1 is not correct.

for 2 variables x and y
+1 means completely correlated, which means if you know x you know y. It also means that when x gets bigger, y gets bigger. In other words, there is a formula y= mx+b that you can figure out. It will have a positive m. This formula will perfectly predict y when you plug in x. Or, if you know y, you can figure out x using this formula.

-1 means completely anti-correlated. One could say the two variables are completely correlated, except that as x gets bigger, y gets smaller. So there will be a line y= mx +b. But in this case, m will be a negative number.

0 for a correlation between x and y means the two variables are not correlated. Example, the last digit of your telephone number and your weight. If you get a set of numbers (last digit, weight) you should see a coefficient near 0, because knowing someone's weight does not give any information on knowing the last digit of their telephone number.

On the other hand, if you have a correlation of +0.5 between x and y, that means that (on average) if x is bigger then y is bigger, but the rule is a bit "sloppy". An example in real life is height and weight. If you know someone's height, you can guess their weight better than if you did not know their height. But even if you know their height, you still don't know their exact weight. Height and weight are correlated, but not 100% (not with coeff=1)

Answer 7

For part A, you said

value for Σy is 145.2 (12.8+13.9+15.0+14.7+15.1+15.2+15.6+17.0+17.4+17.2+16.8+16.7+16.9+17.0)

If I copy and paste the line of numbers into google's search window I get
221.3
So you might want to use that as a check. Another way is use a programmable calculator which computes statistics (though it is a bother figuring out how to use them!)

Answer 8

I don't recognize your formula for m . Can you post it?

Meanwhile, here is a formula that I know works:

\[m = \frac{ \frac{Σxy}{n}-\bar{x}\bar{y}}{\frac{Σ(x^2)}{n} - (\bar{x})^2} \]

\[ \text{ average x value is }\bar{x}= \frac{Σx}{n} \]
\[ \text{ average y value is }\bar{y}= \frac{Σy}{n} \]

n is the number of terms. n =14 for this problem

Answer 9

so if i replace my n with the 14 on what i had will it work out?

Answer 10

m=n(∑xy)−(∑x )(∑x )
÷n(∑x2) − (∑x)2 is the formula that is in the book..

Answer 11

OK, your formula makes sense. But what is x? is it year - 1990 (like in the question) or year - 1900 (like what your teacher said)

Answer 12

But to answer your question, yes you should get the correct m