Need help putting the equation in the form of ŷ = a + bx

Question

"The maximum discount value of the Entertainment® card for the "Fine Dining" section for various pages is given in the table below"

Page numberMaximum value ($)
4	17
14	18
25	15
32	18
43	20
57	15
72	15
85	16
90	16

Let page number be the independent variable and maximum value be the dependent variable.

1. I need help calculating the least-squares line, I keep getting it wrong so if you can show me how to solve step by step that would be great (I must put the equation in the form of: ŷ = a + bx and need to round my answers to three decimal places)

ŷ = ? + ? x

Answer 1

First off, you need, at most, a SEVEN (7) column table***

the first column is X

the 2nd column is Y

the 3rd column is X^2

the 4th column is XY

the 5th column is the Line Estimate = M*x+B

the 6th column is the ERROR = y - Line Estimate

the 7th column is Y^2 <--- ***this is for the calculation of

the R-coefficient; more on that

later; if you are not interested

in the R-coefficient, then this

column is not needed, which means

you only need 6 columns

Next, you put the stats for the independent variable

(the page number) in column X and the price of each

corresponding in column Y...

so the ordered pairs (x,y) in each row are (4,17),(14,18),

(25,15), (32,18), ... (90,16); so yes, there are N=9 rows of data

Now, Here are the formulas that I TEACH !!!

That is, when I TEACH STATISTICS, THESE are the formulae I use

for linear , least squares regression, so as to calculate the

SLOPE and the INTERCEPT of the trend line...

Denominator D = (# of rows) * (total of X^2) - (total of X)^2

note here the location and setting of the parenthesis....

in the first term (total of X^2) is the total of the 3rd

column labeled X^2... while in the second term of this

formula (total of X)^2 is the total of the first column

labeled X, AND THEN SQUARES THAT TOTAL !!! Do not confuse

these two values!!!

As you can see in the following formulae, this same value appears

in the denominator of both of the formulae for slope M and intercept B.

slope M = [(# of rows) * (total of XY) - (total of X)*(total of Y)] / D

intercept B = [ (total of Y)*(total of X^2) - (total of X)*(total of XY)]/ D

Once found, you WRITE the equation of the trend line y = mx+b, as you have asked.

This in turn, gets you the LINE ESTIMATE, that is where the LINE

is for each value of X.

The ERROR then is the vertical distance from the data point to the line for each x,

hence Y - Line Estimate = Y - (mx+b)

Yes, the total of the errors should be some value CLOSE to ZER0, at least within

order 10^(-N) for some HIGH degree of accuracy N digits; in other words, the total

of Y should BALANCE against the error and the line estimates

I have uploaded a spreadsheet that I REGULARLY use to perform all

of these calculations, and it includes a picture/graph as well as

the aforementioned R-coefficient which gives you a RATING of how

well the data fits a linear model. The more that R-coefficient

is closer to 1 or -1, the better. On the other hand,

if the R-coefficient is closer to 1/2 or -1/2, it is NOT GOOD LINEAR FIT!

The formula for the R-coefficient is MUCH more unfriendly, and for now

deferred. You can google it, look it up online or in wikipedia, and/or

even check it out from the formula in the spreadsheet. (Not for the weak at heart)

Here are the results, as shown in the spreadsheet for this data set:

N = 9 <--- there are 9 data points

D = 69128

M = -0.021785673 <--- the slope is practically zero,

so the trend line is almost horizontal

B = 17.68817267

the equation of the best fit trend line is then y = -0.021785673 x + 17.68817267

the R-Coefficient is -0.389736518, so the linear model is NOT a good fit

for this data. This is consistent with the picture, as the price fluctuates

between 15 and 18 with a spike in the middle at 20. Its more of a polynomial

or perhaps trigonometric/sinusoidal curve.

Finally, I would recommend that you study how these formulae

work with a SMALLER data set and simpler stats, so you can

see a little better how they work. For that reason, I will

conclude this mini-lesson with a VERY SIMPLE example

that runs just a few data points through these formulae, and then let you explore

your particular posted problem via the spreadsheet...sometimes LESS is MORE

Given the data set of N=4 data points:

(2,5); (5,11); (7,15); (10,21) <-- y = 2x+1

line estimate and error columns are deferred until we get

the slope and the intercept...

X Y X^2 XY

-----------------------------

2 5 4 10

5 11 25 55

7 15 49 105

10 21 100 210

-------------------------------

24 52 178 380 <---- totals

plugs these totals into the formulae above....

D = 4*178 - 24^2 = 136

M = [ 4 * 380 - 24 * 52 ] / 136

= [ 1520 - 1248]/126

= 272/136= 2

B = [ 52 * 178 - 24 * 380]/136

= 136/136 = 1

so yeah, y = 2x+1

extending the table for the line estimates and errors...

X Y X^2 XY line est = mx+b error=y - line est.

---------------------------------------------------------------------

2 5 4 10 5 0

5 11 25 55 11 0

7 15 49 105 15 0

10 21 100 210 21 0

---------------------------------------------------------------------

24 52 178 380 52 0