Question

A sample of 20 cars, including measurements of fuel consumption (city mi/gal and highway mi/gal), weight (pounds), number of cylinders, engine displacement (in liters), amount of greenhouse gases emitted (in tons/year), and amount of tailpipe emissions of NO_x (in lb/yr).

CAR	CITY	HWY	WEIGHT	CYLINDERS	DISPLACEMENT	MAN/AUTO	GHG	NOX
Chev. Camaro	19	30	3545	6	3.8	M	12	34.4
Chev. Cavalier	23	31	2795	4	2.2	A	10	25.1
Dodge Neon	23	32	2600	4	2	A	10	25.1
Ford Taurus	19	27	3515	6	3	A	12	25.1
Honda Accord	23	30	3245	4	2.3	A	11	25.1
Lincoln Cont.	17	24	3930	8	4.6	A	14	25.1
Mercury Mystique	20	29	3115	6	2.5	A	12	34.4
Mitsubishi Eclipse	22	33	3235	4	2	M	10	25.1
Olds. Aurora	17	26	3995	8	4	A	13	34.4
Pontiac Grand Am	22	30	3115	4	2.4	A	11	25.1
Toyota Camry	23	32	3240	4	2.2	M	10	25.1
Cadillac DeVille	17	26	4020	8	4.6	A	13	34.4
Chev. Corvette	18	28	3220	8	5.7	M	12	34.4
Chrysler Sebring	19	27	3175	6	2.5	A	12	25.1
Ford Mustang	20	29	3450	6	3.8	M	12	34.4
BMW 3-Series	19	27	3225	6	2.8	A	12	34.4
Ford Crown Victoria	17	24	3985	8	4.6	A	14	25.1
Honda Civic	32	37	2440	4	1.6	M	8	25.1
Mazda Protege	29	34	2500	4	1.6	A	9	25.1
Hyundai Accent	28	37	2290	4	1.5	A	9	34.4

To determine whether there is any linear relationship between the number of cylinders (CYLINDERS) a car has and the greenhouse emission gasses (GHG) , first we make a scatterplot for the data, then we calculate the linear correlation coefficient. If there is strong linear correlation then we do regression.Answer the following questions:

1. Make a scatterplot for CYLINDERS and GHG. Use your independent variable as CYLINDERS and dependent variable as GHG.

i. Describe the type of linear correlation- positive, negative, no correlation. Is it nonlinear?

2. Find the linear correlation coefficient between CLYLINERS and GHG.

i. Describe the linear correlation coefficient. Is it positive or negative? Is it strong, moderate or week?

ii. Use Table A6 and a = 0.05  to determine whether there is correlation between CYLINDER and GHG in the population.

3. Find the regression line between CYLINDERS and GHG.

i. What is the meaning of the slope for your regression equation?

ii. What is the meaning of y-intercept for your regression equation?

iii. Estimate the greenhouse emission gasses amount if the number of cylinders for cars could be 5.

Answer 1

1) SCATTER PLOT:

I have used excel to construct scatter plot.

The scatter plot for Cylinders and GHG, with cylinders as independent variable and greenhouse emission gasses (GHG) as dependent variable is given below:

(i) From the scatter plot, it is evident that there is positive linear pattern. Thus there is positive linear correlation between the variables Cylinders and GHG.

2) LINEAR CORRELATION COEFFICIENT:

I have used R code to find linear correlation coefficient between Cylinders and GHG.

(i) The linear correlation coefficient between Cylinders and GHG is . Since the coefficient is positive and closer to , there is a strong positive linear correlation between Cylinders and GHG.

(ii) TEST FOR SIGNIFICANCE OF CORRELATION:

HYPOTHESIS:

(That is, there is no statistically significant linear correlation between Cylinders and GHG in population)

(That is, there is statistically significant linear correlation between Cylinders and GHG in population)

R OUTPUT:

Since the p value is less than the significance level , we reject the null hypothesis and conclude that there is statistically significant linear correlation between Cylinders and GHG in population.

3) SIMPLE LINEAR REGRESSION OUTPUT:

I have used R code to build simple linear regression model to data with cylinders as independent variable and greenhouse emission gasses (GHG) as dependent variable.

ESTIMATED LINEAR REGRESSION EQUATION:

The estimated simple linear regression equation is,

where

is the predicted dependent variables "greenhouse emission gasses (GHG) "

is the intercept

is the slope coefficient of independent variable "cylinders"

is the independent variable "cylinders"

(i)  The slope coefficient of regression equation is . That is, the mean amount of greenhouse gases emitted increases by 0.8788 (in tons/year) for increase in number of cylinders by one.

(ii) The  y-intercept for regression equation is . That is, the mean amount of greenhouse gases emitted is 6.3788 (in tons/year) when the number cylinders is 0 (or without involving number of cylinders).

(iii) PREDICTED VALUE OF GHG:

If the number of cylinders for cars is 5, the predicted value of amount of greenhouse gases emitted is,

  

If the number of cylinders for cars is 5, the predicted value of amount of greenhouse gases emitted is 10.7728 (in tons/year).