Open the file CAR Measurements using menu option Datasets and then Elementary Stats, 13th Edition in Statdisk. This file contains information, such as size, weight, length, braking distance, cylinders, displacement, city miles per gallon (MPG), highway MPG, and GHG (greenhouse gas emissions), for 21 cars.
Perform the following tasks to help you determine which car is right for you:
1. Scatterplots, Correlations, and the Correlation Coefficient
1A. Weight vs. Braking Columns
(i). Create a scatterplot for the data in the Weight and Braking columns. Paste it in your report.
(ii). Using Statdisk, calculate the linear correlation between the data in the Weight and Braking columns. Paste your results in your Word document.
(iii). Explain the mathematical relationship between weight and braking based on the linear correlation coefficient. Be certain to include comments about the magnitude (strength) and the direction (positive or negative) of the correlation. As weight increases, what happens to the braking distance?
B. Weight vs. City MPG
(i). Create a scatterplot for the data in the Weight and the City MPG columns. Paste it in your report.
(ii). Using Statdisk, calculate the linear correlation between the data in the Weight and City MPG columns. Paste your results in your Word document.
(iii). Explain the mathematical relationship between weight and city MPG based on the linear correlation coefficient. Be certain to include comments about the magnitude and the direction of the correlation. As weight increases, what happens to the city MPG?
(iv). Compare the correlations for weight and braking distance with that of weight and city MPG. How are they similar? How are they different?
2. Linear Regression and Prediction
A. Let’s say that we wanted to be able to predict the braking distance in feet for a car based on its weight in pounds.
(i). Using this sample data, perform a linear regression to determine the line-of-best fit. Use weight as your x (independent) variable and braking distance as your y (response) variable. Use four (4) places after the decimal in your answer. Paste it in your report.
(ii). What is the equation of the line-of-best fit (linear regression equation)? Present your answer in y = bo + b1x form.
(iii). What would you predict the braking distance would be for a car that weighs 2650 pounds? Show your calculation.
(iv). Let’s say you want to buy a muscle car that weighs 4250 pounds. What would you predict the braking distance would be for a muscle car that weighs 4250 pounds? Show your calculation.
(v). What effect would you predict weight would have on the braking distance of the car? Compare the breaking distance of the 2650-pound car to the 4250-pound car.
(vi). Calculate the coefficient of determination (R2 value) for this data. What does this tell you about this relationship?
3. Multiple Regression
A. Let’s say that we wanted to be able to predict the city MPG for a car using weight in pounds, length in inches, and cylinders. Using this sample data, perform a multiple-regression line-of-best-fit using weight, length, cylinder, and city MPG.
B. Select City MPG (Column 8) as your dependent variable. Paste it in your report.
(i). What is the equation of the line-of-best fit? The form of the equation is: Y = bo + b1X1 + b2X2 + b3X3 (fill in values for bo, b1, b2, and b3). Round coefficients to three (3) decimal places.
(ii). What would you predict for the city MPG of a car whose (1) Weight is 3410 pounds, (2) LENGTH is 130 inches, and (3) Cylinders is 6?
C. What is the R2 value for this regression? What does it tell you about the regression?
4. Making Decisions Based on Data
A. Based on the information gathered in this task on the relationship between weight and braking distance and weight and city MPG, which of the 21 cars listed would you choose to buy, and wh
- Scatterplots, Correlations, and the Correlation Coefficient
- Weight Vs. Braking Columns
1A (i): Weight Vs. Braking Scatterplot
Y values= Braking
X values= Weight
1A (ii): Linear Correlation
Correlation coeff, r: 0.3513217
Critical r: ±0.4328579
1A (iii): Mathematical Relationship
The scatterplot shows a weak positive correlation because r= 0.35 and is closer to zero. The critical r value is ±0.43 indicating a weak correlation and therefore the weight and braking values cannot be used for prediction (Bennett et al., 2018). Looking at the data points that appear to largely scatter from the straight line, it can be concluded that the relationship between weight and braking is weak. The direction of the line indicates that as the weight of the vehicle increases the braking time is also likely to increase.
- Weight Vs, City MPG
B(i): Weight Vs. City MPG Scatter Plot
Y values= City MPG
X values= Weight
B(ii): Linear Correlation
Correlation coeff, r: -0.8437604
Critical r: ±0.4328579
B(iii): Mathematical Relationship
The negative value observed after calculation indicates that car weight and city MPG have a negative correlation. Negative correlation values range between 0 and -1 and with a correlation coeff, r: -0.8437604 it indicates a strong negative correlation (Schober et al., 2018). The values in the scatter plot are close to the line due to the strong correlation of values. From the scatter plot, it can be observed that the line moves in a downward direction supporting the negative correlation. Additionally, the downward direction indicates that as the weight of the car gets heavier the fewer the MPG the car will have.
B(iii): Comparing Correlations
The Weight Vs. Braking values demonstrate a weak positive correlation while the Weight Vs. City MPG demonstrates a strong negative correlation. The values in the braking plot are scattered from the line compared to the values of the City MPG scatter plot. In both correlations, weight has a negative effect on braking and city MPG. When the weight increases, the car breaking distance gets worse, and also as the weight increases the car MPG decreases.
- Linear Regression and Prediction
2A(i): Regression Results
Y= b0 + b1x:
Y Intercept, b0: 125.308
Slope, b1: 0.0031873
2A(ii): Linear regression Equation
Form of the equation: y = bo + b1x
Equation based on the regression results is:
y= 125.308 + 0.0032x
2A(iii): Breaking Distance
For a 2650 pounds car, the breaking distance will be:
y= 125.308 + 0.0032x
y= 125.308 + 0.0032(2650)
y= 125.308 + 8.48
The prediction of braking for a car weighing 2650 pounds is 133.788 feet.
2A(iv): Breaking Distance
For a 4250 pounds muscle car, the breaking distance will be:
y= 125.308 + 0.0032x
y= 125.308 + 0.0032(4250)
y= 125.308 + 13.6
The prediction of braking for a muscle car weighing 4250 pounds is 138.908 feet.
The results above indicate that the more the weight of the car the further the braking distance. A 2650 pounds car has a braking distance of 133.788 feet while a 4250 pounds muscle car has a braking distance of 138.908. There is a significant braking distance of about 5.12 feet between the two cars.
2A(V): Coefficient of Determination
The Coeff of Det, R^2 is : 0.123427.
This value is not close to 1 indicating that correlation between the variables is weak. The variation in weight only explains about 12.3% of the braking distance while the rest remains unexplained.
- Multiple Regression
The multiple regression results are as follows:
Number of columns used: 4
Dependent column: 8
Coeff, b0: 46.01974
Coeff, b1: -0.0034893
Coeff, b2: -0.0578495
Coeff, b3: -0.4463389
Total Variation: 288.6667
Explained Variation: 212.7625
Unexplained Variation: 75.90414
Standard Error: 2.113043
Coeff of Det, R^2: 0.7370526
Adjusted R^2: 0.6906502
P Value: 0.0000352
Form: Y = bo + b1X1 + b2X2 + b3X3
Equation: Y= 46.02 – 0.003×1 – 0.058×2 – 0.446×3
The city MPG of a car weighing 3410 pounds, length 130 inches with 6 cylinders will be:
y= 46.02- 0.003(3410)-0.058(130)-0.446(6)
The car will have an MPG of 25.574 in the city.
3C: R^2 Value
Coeff of Det, R^2: 0.7370526
The r^2 value is closer to 1 indicating a positive correlation between the variables. From this value, it is likely that most values will be closer to the line of best fit.
- Decision Making
The information gathered indicates that the higher the weight of the car the longer the distance it will take to slow down when braking. However, the correlation is not strong indicating that other factors could contribute to the longer braking time (Bennett et al., 2018). Secondly, the weight and city MPG demonstrates a strong correlation where increased weight leads to a decreased city MPG. Based on this information, I would choose to buy the Kia Rio which has a braking distance of 132 feet and a city MPG of 27.
Bennett, J., Briggs, W. L., & Triola, M. F. (2018). Statistical reasoning for everyday life (5th ed.). Boston, MA: Pearson.
Schober, P., Boer, C., & Schwarte, L. A. (2018). Correlation coefficients: Appropriate use and interpretation. Anesthesia and Analgesia, 126(5), 1763–1768. https://doi.org/10.1213/ANE.0000000000002864
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