d Fit a more appropriate model and give the associated least

(d) Fit a more appropriate model and give the associated least-squares regression equation. (e) Create the residual plot corresponding to the new model and draw a rough sketch of the residual plot below. Does the new model appear to be appropriate? Explain.

Solution

15(A)

Solve this problem by making a scatter plot for the given data.

(1)Take the graph paper.

(2)Plot the values of Age (in years) on X axis and values of Prices(in Dollars) on Y-axis.

Here the Explanatory variable is Age(X) and

Response variable is Prices(in Dollars) which is Y.

The required equation of best fit is

Y=mX+C

That is Price Advertised =Slope(Age)+Y- intercept

Scale:

on x axis 1 cm=0.1 units

on y axis 1cm =50 units

After plotting draw a line

y = 11992.25 - 908.332 x

Here Slope(m) is -908.332 and Y intercept C is 11992.25

we get direction as negative,form is linear and strength is strong.

Explanation:

We will find an equation of the regression line in 4 steps.

Step 1: Find XY and X2 as it was done in the table below.

Step 2: Find the sum of every column:

X=113 , Y=113219 , XY=489484 , X2=953

Step 3: Use the following equations to find a and b:

ab=YX2XXYnX2(X)2=11321995311348948418953113211992.25=nXYXYnX2(X)2=1848948411311321918953(113)2908.332

Step 4: Substitute a and b in regression equation formula

y y = a + bx= 11992.25 908.332x

Answer15(C):If we study the residual plots and diagnostic tests more carefully, we notice that the errors are severely autocorrelated: there are long runs of negative errors alternating with long runs of positive errors.

That is

The linear trend model obviously fails the autocorrelation test in this case.

Answer15(d):

Least square regression is a method for finding a line that summarizes the relationship between the two variables, at least within the domain of the explanatory variable x.

Formula :

Another formula for Slope: Slope = (NXY - (X)(Y)) / (NX² - (X)²)

Where,

b = The slope of the regression line a = The intercept point of the regression line and the y axis. X = Mean of x values Y = Mean of y values SD_x = Standard Deviation of x SD_y = Standard Deviation of y

To Find,

Least Square Regression Line Equation

Solution :

Step 1 :

Count the number of given x values.

N = 18

Step 2 :

Find XY, X² for the given values. See the below table

Step 3 :

Now, Find X, Y, XY, X² for the values

Step 4 :

Substitute the values in the above slope formula given. Slope(b) = (NXY - (X)(Y)) / (NX² - (X)²)

Step 5 :

Now, again substitute in the above intercept formula given. Intercept(a) = (Y - b(X)) / N

Step 6 :

Then substitute these values in regression equation formula Regression Equation(y) = a + bx

Slope:-908.332

Y intercept is 11992.25

Least Square Regression Line Equation Y=-908.332x + 11992.25.

Helpful link is

http://www.stat.purdue.edu/~xuanyaoh/stat350/xyApr6Lec26.pdf

http://docs.statwing.com/interpreting-residual-plots-to-improve-your-regression/

X	Y	XY	XX
1	12995	12995	1
1	11950	11950	1
2	11495	22990	4
3	9800	29400	9
3	9700	29100	9
4	6995	27980	16
4	7990	31960	16
5	6000	30000	25
5	6200	31000	25
6	5990	35940	36
6	4995	29970	36
9	3200	28800	81
9	2250	20250	81
9	3300	29700	81
11	2859	31449	121
11	2850	31350	121
11	2900	31900	121
13	1750	22750	169