Question

Data We are interested in exploring the relationship between the weight of a vehicle and its fuel efficiency (gasoline mileage). The data in the table show the weights, in pounds, and fuel efficiency, measured in miles per gallon, for a sample of vehicles. Note the units here are pounds and miles per gallon.

Weight (pounds)	Fuel Efficiency (miles per gallon)
2695	25
2510	27
2680	29
2730	38
3000	25
3410	23
3640	21
3700	27
3880	21
3900	19
4060	20
4710	15

Question

correlation coefficient

regression equation line (HINT: use LinRegTTest, the line is of the form yhat = a + bx)

p-value (HINT: use LinRegTTest)

s (the standard deviation of |y – y-hat|) (HINT: use LinRegTTest)

SSE (sum of squared errors – you can calculate this directly or use the formula for s)

Correlation coefficient

The percent of variation in fuel efficiency can be explained by the variation in weight using the regression line.([Hint: Coefficient of determination) Give answer as percentage to 2 decimal places.

For a car that has weight 2900 pounds, predict its fuel efficiency to 2 decimal places. Include units.

1. Is the correlation significant? Explain how you determined this in complete sentences.
2. Is the relationship a positive one or a negative one? Explain how you can tell and what this means in terms of weight and fuel efficiency.
3. In one or two complete sentences, what is the practical interpretation of the slope of the least squares line in terms of fuel efficiency and weight?
4. For a car that weighs 4,000 pounds, predict its fuel efficiency. Include units.
5. Can we predict the fuel efficiency of a car that weighs 10,000 pounds using the least squares line? Explain why or why not.
6. Answer each question in complete sentences.
   1. Does the line seem to fit the data? Why or why not?
   2. What does the correlation imply about the relationship between fuel efficiency and weight of a car? Is this what you expected?
7. Are there any outliers? If so, which point is an outlier?

Answer 1

a) Pearson correlation of Weight (x) and Fuel Efficiency (y) = -0.785

b) The regression equation is
Fuel Efficiency (y) = 47.120 - 0.0067 Weight (x)

y^=47.120 - 0.0067 (x)

c) The p-value for the significance linear regression test is  0.002.

d) s=  3.82383

e) SSE=  146.22

d) Correlation coefficient (r)= -0.785

f) The percent of variation in fuel efficiency can be explained by the variation in weight using the regression line=R^2= 61.69%

g)  For a car that has weight 2900 pounds, predict its fuel efficiency = 47.120 - 0.0067 *2900=27.60 miles per gallon

1. Is the correlation significant? Explain how you determined this in complete sentences.

Ans: Pearson correlation of Weight (x) and Fuel Efficiency (y) = -0.785, P-Value = 0.002. The p-value of the correlation is less than 0.05. Hence, the correlation is significant.

2.  Is the relationship a positive one or a negative one? Explain how you can tell and what this means in terms of weight and fuel efficiency.

Ans: The value of the Pearson correlation of Weight (x) and Fuel Efficiency (y) has a negative sign. Hence, the relationship is a negative one. So, increase the value of the weight, decreases the corresponding value of fuel efficiency and vice-versa.

3. In one or two complete sentences, what is the practical interpretation of the slope of the least-squares line in terms of fuel efficiency and weight?

Ans: The practical interpretation of the slope of the least-squares line in terms of fuel efficiency and weight is that when a pound is increased on weight, it decreases the mean fuel efficiency by  0.0067 miles.

4. For a car that weighs 4,000 pounds, predict its fuel efficiency. Include units.

Ans: The predicted fuel efficiency when weighs is equal to 4,000 pounds is  47.120 - 0.0067 *4000=20.32 miles per gallon.

5. Can we predict the fuel efficiency of a car that weighs 10,000 pounds using the least-squares line? Explain why or why not.

Ans: We can predict the fuel efficiency of a car that weighs 10,000 pounds using the least-squares line but the variance of the predicted value will be very large because the variance of the predicted value is directly proportional to the square of the difference between the mean of weight and 10,000.

6. Answer each question in complete sentences.

Does the line seem to fit the data? Why or why not?

No, the line does not seem to fit the data because the residual of the regression model does not follow a normal distribution.

What does the correlation imply about the relationship between fuel efficiency and weight of a car? Is this what you expected?

Ans:

Yes, the correlation implies about the relationship between fuel efficiency and weight of a car because both the variables follow the normal distribution.

7) Are there any outliers? If so, which point is an outlier?

From the above box-plots, we can say that the variable fuel efficiency has an outlier observation.