Question Help Most cars are fuel efficient when running at a steady speed of aro

Question

Question Help Most cars are fuel efficient when running at a steady speed of around 40 to 50 mph. A scatterplot relating fuel consumption (measured in mpg) and steady driving speed (measured in mph) for a mid-sized car is shown below. Complete parts a through c below click here to view the scatterplot a. The correlation equals 0.106. Comment on the use of the correlation coefficient as a measure for the association between fuel consumption and steady driving speed O A. The high correlation means that there is a strong relationship between the variables O B. The positive correlation means that there is a positive relationship between the variables O C. The low correlation means that there is not a strong relationship between the variables 0 D. The correlation coefficient is meaningless because the scatter plot shows a norinear or arved pattern b. Comment on the use of the regression equation as a tool for predicting fuel consumption from the velocity of the car A. Since there is a positive correlation, fuel consumption should always ncrease as velocity ncreases O B. Since there is a low correlation, tho regression equation can be used, howover may not be very accurate O c. snco there is a high correlation, the regression equation should yield farty accurate results O D. Since the relationship appears to be nonlinear, the regression equation cannot bo used to make predictions c. Over what subrange of steady driving speed might fitting a regrossion equation be appropriate? Select all that apply A. The range trom about 25 mph to 75 mph B. The range from about 15 mph to 40 mph ne he tanna from about 40 moh to 85 mph

Explanation / Answer

# Below is the R-code

> df <- data.frame(Velocity=seq(5,85,by=5),MPG=c(15.5,17,30.9,31.8,38.6,38.3,37.9,43.6,41.3,39.9,37.9,36.2,34.6,30.9,28,24.5,23.8))
> df
Velocity MPG
1 5 15.5
2 10 17.0
3 15 30.9
4 20 31.8
5 25 38.6
6 30 38.3
7 35 37.9
8 40 43.6
9 45 41.3
10 50 39.9
11 55 37.9
12 60 36.2
13 65 34.6
14 70 30.9
15 75 28.0
16 80 24.5
17 85 23.8
> cor(df)
Velocity MPG
Velocity 1.0000000 0.1061128
MPG 0.1061128 1.0000000
> # Fitting a regression Line
> model.lm <- lm(MPG~.,data=df)
> summary(model.lm)

Call:
lm(formula = MPG ~ ., data = df)

Residuals:
Min 1Q Median 3Q Max
-15.502 -5.438 1.510 6.428 11.380

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 30.82794 4.31440 7.145 3.36e-06 ***
Velocity 0.03480 0.08421 0.413 0.685   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.505 on 15 degrees of freedom
Multiple R-squared: 0.01126, Adjusted R-squared: -0.05466
F-statistic: 0.1708 on 1 and 15 DF, p-value: 0.6852

># R2 is very poor

> # For Range 25 to 75
> model.lm1 <- lm(MPG~.,data=df[4:15,])
> df[4:15,]
Velocity MPG
4 20 31.8
5 25 38.6
6 30 38.3
7 35 37.9
8 40 43.6
9 45 41.3
10 50 39.9
11 55 37.9
12 60 36.2
13 65 34.6
14 70 30.9
15 75 28.0
> df[5:15,]
Velocity MPG
5 25 38.6
6 30 38.3
7 35 37.9
8 40 43.6
9 45 41.3
10 50 39.9
11 55 37.9
12 60 36.2
13 65 34.6
14 70 30.9
15 75 28.0
> model.lm1 <- lm(MPG~.,data=df[5:15,])
> summary(model.lm1)

Call:
lm(formula = MPG ~ ., data = df[5:15, ])

Residuals:
Min 1Q Median 3Q Max
-3.9864 -2.4405 0.6009 2.3850 4.5691

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 47.08182 3.15404 14.927 1.18e-07 ***
Velocity -0.20127 0.06015 -3.346 0.00857 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.154 on 9 degrees of freedom
Multiple R-squared: 0.5544, Adjusted R-squared: 0.5049
F-statistic: 11.2 on 1 and 9 DF, p-value: 0.008572

> # For the range 15 to 40

> df[3:8,]
Velocity MPG
3 15 30.9
4 20 31.8
5 25 38.6
6 30 38.3
7 35 37.9
8 40 43.6
> model.lm2 <- lm(MPG~.,data=df[3:8,])
> summary(model.lm2)

Call:
lm(formula = MPG ~ ., data = df[3:8, ])

Residuals:
3 4 5 6 7 8
-0.1286 -1.5571 2.9143 0.2857 -2.4429 0.9286

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 24.0429 2.9078 8.268 0.00117 **
Velocity 0.4657 0.1010 4.612 0.00994 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.112 on 4 degrees of freedom
Multiple R-squared: 0.8417, Adjusted R-squared: 0.8021
F-statistic: 21.27 on 1 and 4 DF, p-value: 0.009942

> # For the range 40 to 85
> df[8:17,]
Velocity MPG
8 40 43.6
9 45 41.3
10 50 39.9
11 55 37.9
12 60 36.2
13 65 34.6
14 70 30.9
15 75 28.0
16 80 24.5
17 85 23.8
> model.lm3 <- lm(MPG~.,data=df[8:17,])
> summary(model.lm3)

Call:
lm(formula = MPG ~ ., data = df[8:17, ])

Residuals:
Min 1Q Median 3Q Max
-1.5539 -0.6683 0.0703 0.3623 1.6751

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 62.6988 1.3915 45.06 6.50e-11 ***
Velocity -0.4581 0.0217 -21.11 2.66e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9854 on 8 degrees of freedom
Multiple R-squared: 0.9824, Adjusted R-squared: 0.9802
F-statistic: 445.7 on 1 and 8 DF, p-value: 2.663e-08

> # For the range 5 to 80
> data=df[1:16,]
> df[1:16,]
Velocity MPG
1 5 15.5
2 10 17.0
3 15 30.9
4 20 31.8
5 25 38.6
6 30 38.3
7 35 37.9
8 40 43.6
9 45 41.3
10 50 39.9
11 55 37.9
12 60 36.2
13 65 34.6
14 70 30.9
15 75 28.0
16 80 24.5
> model.lm4 <- lm(MPG~.,data=df[1:16,])
> summary(model.lm4)

Call:
lm(formula = MPG ~ ., data = df[1:16, ])

Residuals:
Min 1Q Median 3Q Max
-14.254 -5.192 1.281 6.357 10.881

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 29.33000 4.33747 6.762 9.14e-06 ***
Velocity 0.08474 0.08971 0.945 0.361   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.271 on 14 degrees of freedom
Multiple R-squared: 0.0599, Adjusted R-squared: -0.007246
F-statistic: 0.8921 on 1 and 14 DF, p-value: 0.3609

# Answers for the questions

a. (D)

b. (B)

c. (B) & (C) # Using multiple r-square. Lower value means poor relationship.

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