Question

8. An engineer wanted to determine how the weight of a car (a) Determine which variable is the likely explanatory affects gas mileage. The following data represent the weight variable and which is the likely response variable. of various cars and their gas mileage. Complete parts (a) through (d). The explanatory variable is the miles Miles per per gallon and the response variable is Car Weight (pounds) Gallon the weight А 3310 19 The explanatory variable is the weight 3680 19 and the response variable is the miles 2770 24 per gallon D 3340 21 2690 26 (b) Draw a scatter diagram of the data. Choose the correct scatter plot. Click the icon to view the critical values table. A. B. colom С E 1 4000- 304 (som 2500- 15 15-1 30 2500 4000 Weight (lbs) mpe O c. D 30- 40004 Odu Weight(s) 2500 15- 2500 4000 Weight (lbs) 30 mo (c) Compute the linear correlation coefficient between the weight of a car and its miles per gallon. (Round to three decimal places as needed.) (d) Comment on the type of relation that appears to exist between the weight of a car and its miles per gallon based on the scatter diagram and the linear correlation coefficient. Because the correlation coefficient is (1) and the absolute value of the correlation coefficient _.is (2) than the critical value for this data set, (3) linear relation exists between the weight of a car and its miles per gallon. (Round to three decimal places as needed.) 1: Data Table Critical Values for Correlation Coefficient

Answer 1

a) -

Explanatory variable is weight & response variable is miles per gallon. Since, miles per depends on the weight of the car.

Option B is correct.

b) -

Scatter plot for the data -

Scatter Plot 30 25 20 Miles per Gallon 15 10 5 0 0 500 1000 1500 2500 3000 3500 4000 2000 Weight

Option C is correct.

c) -

Formula for correlation coefficient -

$r =\frac{\sum (x_{i} - \bar{x})(y_{i}-\bar{y})}{\sqrt{\sum (x_{i} - \bar{x})^{2}\sum (y_{i} - \bar{y})^{2}}}$

Observation table -

Car	weight(pounds)(x)	Miles per Gallon(y)	(x-x_bar)	(y-y_bar)	(x-x_bar)(y-y_bar)	(x-x_bar)^2	(y-y_bar)^2
A	3310	19	152	-2.8	-425.6	23104	7.84
B	3680	19	522	-2.8	-1461.6	272484	7.84
C	2770	24	-388	2.2	-853.6	150544	4.84
D	3340	21	182	-0.8	-145.6	33124	0.64
E	2690	26	-468	4.2	-1965.6	219024	17.64
Total	15790	109	-	-	-4852	698280	38.8

Calculations -

$r =\frac{\sum (x_{i} - \bar{x})(y_{i}-\bar{y})}{\sqrt{\sum (x_{i} - \bar{x})^{2}\sum (y_{i} - \bar{y})^{2}}}$

$= \frac{-4852}{\sqrt{(698280)(38.8)}}$

$= \frac{-4852}{\sqrt{27093264}}$

$= \frac{-4852}{5205.119}$

$= -0.9321 \approx 0.932$

Correlation coefficient between weight & mpg is 0.932.

d) -

Because the correlation coefficient is -0.932 & the absolute value of correlation coefficient is 0.932 is not greater than critical value for this data set 0.997. Negative linear relation exist between the weight of car & its miles per gallon.