The purpose of this project is to run statistical analyses on AutosRUs's prototype MechaCar, which is experiencing some production challenges. This project performs multiple linear regression on the MechaCar's dataset to determine which variables impact the vehicle's miles per gallon (MPG), produces summary statistics of the pressure exerted (in PSI) on the suspension coils of each manufacturing lot, and runs t-tests on the suspension coil data to determine if the lots have any significant statistical variation from the mean population.
A linear regression was performed to examine which of the five variables -- vehicle length, vehicle weight, spoiler angle, ground clearance, and all wheel drive (AWD) -- may have a significant impact on the MechaCar's miles per gallon (MPG) measurement.
Equation:
y = -1.040e+02 = -1.040e+02 + 6.267e+00*(vehicle_length) + 1.245e-03*(vehicle_weight) + 6.877e-02*(spoiler_angle) + 3.546e+00*(ground_clearance) - 3.411e+00*(AWD)
Equation Coefficients:
| Estimate | Std. | t value | Pr(>|t|) | Significance | |
|---|---|---|---|---|---|
| (Intercept) | -1.040e+02 | 1.585e+01 | -6.559 | 5.08e-08 | *** |
| vehicle_length | 6.267e+00 | 6.553e-01 | 9.563 | 2.60e-12 | *** |
| vehicle_weight | 1.245e-03 | 6.890e-04 | 1.807 | 1.807 | . |
| spoiler_angle | 6.877e-02 | 6.653e-02 | 1.034 | 0.3069 | |
| ground_clearance | 3.546e+00 | 5.412e-01 | 6.551 | 5.21e-08 | *** |
| AWD | -3.411e+00 | 2.535e+00 | -1.346 | 0.1852 |
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Multiple R-squared: 0.7149
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p-value: 5.35e-11
Which variables/coefficients provided a non-random amount of variance to the mpg values in the dataset?
Based on the Pr(>|t|) values of the results, the vehicle length and the ground clearance (as well as the intercept) of each vehicle is unlikely to provide a random amount of variance to the MPG values in a linear regression model. This means that vehicle length, and ground clearance both have a clear positive correlation with the MechaCar's MPG.
Is the slope of the linear model considered to be zero? Why or why not?
The slope of the linear model cannot be considered zero, because of the two variables that have a positive correlation with the MPG. A zero slope would indicate that there is no significant linear correlation between the MPG and each of the five variables.
Does this linear model predict mpg of MechaCar prototypes effectively? Why or why not?
Based on the R-squared value and the p-value of the linear model, the model appears to fit the data rather well, though it is not a perfect fit.
The R-squared value, also known as the coefficient of determination, describes the amount of variance in the response variable (y) in relation to the five predictor variables (vehicle length, vehicle weight, spoiler angle, ground clearance, and AWD). The closer to 1 the R-squared value is, the better the predictor variables are able to determine the response variable. In this case, the R-squared value is 0.7149, which means that the linear model fits the data with roughly 70% accuracy.
The p-value shows whether or not the values of the predictor variables have any significant correlation the values of the response variable. If the p-value is 0.05 or greater, then there is insufficient evidence to conclude that the predictor variables have any significant correlation to the response variable. The p-value of the linear regression as a whole for the MechaCar is extremely small, so it is reasonable to conclude that there is a strong correlation between the predictor variables and the response variable.
The mean, median, variance, and standard deviation of the pressure exerted on the MechaCar prototypes' suspension coils were determined from the raw data provided. Pressure was measured in units of pounds per square inch (PSI).
Summary of All Lots
Summary of Individual Lots
The design specifications of the MechaCar note that the variance of the suspension coils must not exceed 100 PSI. Although Lots 1 and 2 meet this requirement, Lot 3 greatly exceeds the limit. This means that there is a great deal of inconsistency in the prototypes from Lot 3 that is affecting the amount of pressure being placed on the suspension coils.
One-sample t-tests were run on the PSI values of the MechaCar prototypes' suspension coils in order to determine whether the mean of the sample data deviates from the ideal mean, 1500 PSI. T-tests were run on all the data together, and individually on each lot.
The null hypothesis being tested is that the mean PSI is equal to the ideal mean. The results of the t-tests are as follows.
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All Lots Together
- t = -1.8931
- degrees of freedom (df) = 149
- p-value = 0.06028
- 95 percent confidence interval: 1497.507, 1500.053
- mean of x: 1498.78
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Lot 1
- t = 0
- degrees of freedom (df) = 49
- p-value = 1
- 95 percent confidence interval: 1499.719, 1500.281
- mean of x: 1500
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Lot 2
- t = 0.51745
- degrees of freedom (df) = 49
- p-value = 0.6072
- 95 percent confidence interval: 1499.423, 1500.977
- mean of x: 1500.2
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Lot 3
- t = -2.0916
- degrees of freedom (df) = 49
- p-value = 0.04168
- 95 percent confidence interval: 1492.431, 1499.849
- mean of x: 1496.14
The t-tests for lot 1, lot 2, and all the lots together all have p-values well above 0.05, which means that the null hypothesis cannot be rejected, and it would be reasonable to assume that their mean PSI values are equal to the ideal mean of 1500 PSI. The p-value for the t-test for lot 3, however is below 0.05, though it is still relatively close. While the null hypothesis cannot be completely rejected in this instance, when compared to the other tests, it is still a strong indication of manufacturing defects in lot 3, and should be investigated.
In a hypothetical analysis to compare the MechaCar to its potential competitors, a few factors that may be worth investigating include fuel efficiency, comfort, and price.
Fuel efficiency would be tested by measuring a vehicle's average miles per gallon (MPG). A simple two-sample t-test could be used to test the MechaCar against each competitor, using a null hypothesis such as -- the mean MPG of the MechaCar when traveling at 50 mph on a windless, level surface is the same as the mean MPG of its competitor under the same conditions. Or, an ANOVA test could be used to compare the mean MPG of multiple brands of vehicles all together. The necessary data to run this test would be a list of at least 50 vehicles of each type being compared.
Comfort can be measured through several different factors such as indoor air quality (IAQ), which is a measure of the concentration of pollutants like volatile organic compounds (VOC), carbon monoxide, and particulate matter (PM), or thermal comfort, by measuring the humidity and temperature within the vehicle to calculate the heat index. The null hypothesis would be to assume that none of these measures of comfort have any meaningful impact on a passenger's reported level of comfort, and an ANOVA test could be run to test that hypothesis. When comparing how the MechaCar's comfort performs against its competition, each measure of comfort can be taken individually and run through a two-factor t-test or ANOVA, using a null hypothesis such as -- the mean concentration of particulate matter inside the MechaCar with the ventilation moving the air at a constant velocity is the same as the mean concentration of particulate matter in the competitor vehicles under the same conditions.
After performing a linear regression, it appears that the variables that most impact a MechaCar's MPG are the vehicle's length and its grounds clearance. Other factors such as vehicle weight, spoiler angle, and all wheel drive (AWD) appeared to have little correlation to the MechaCar's MPG, at least when evaluating them in a linear relationship with the MPG.
While the 1st and 2nd lots of the MechaCar prototypes had very consistent PSI measurements on its suspension coils, the 3rd lot of prototypes had a large amount of variance, which may indicate that there were defects during production either in the suspension coils or in other parts of the vehicles that may apply pressure to the suspension coils.
Further areas to consider in making the MechaCar competitive in the market would be to compare how its fuel efficiency and user comfort measure up to its current competition, and in the case of user comfort, which environmental factors withing the enclosed space have an impact on reported comfort levels.