We have data on a number of varibles on AutosRUs' newest prototype, MechaCar. We have a:
- table containing data on weight, length, ground clearance, spoiler angle, AWD (binary), and mpg of vehicles; and
- another table on pounds per square inch (PSI) of the suspension coils of vehicles from different manufacturing lots.
The purposes of the analyses are:
- to use regression to identify which factors can predict mpg of MechaCar prototypes;
- to produce summary statistics on the PSI of the suspension coils from the manufacturing lots;
- to conduct t-tests to determine if the manufacturing lots are statistically different from the mean population.
An initial visual inspection using scatter plots suggest some positive correlation between mpg and vehicle length as well as ground clearnce.
Vehicle length vs MPG
Ground clearance vs MPG
Multiple linear regression model was fitted to predict the mpg of Mechacar prototypes. The independent variables can be seen in the image below.
The summary() function returns below statistics which measure the overall fit (R-squared) as well as the level of significance (pr(>|t|)) of each of the explanatory variable.
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According to our data, vehicle_length and ground_clearance are statistically significant in providing non-random variance to mpg as the corresponding p-values are both less than even 0.001. In other words, these two variables are statistically significant predictors of mpg.
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On the other hand, vehicle weight, spoiler angle, and AWD are not siginificant predictors of mpg as can be concluded from their large p-values (>0.05).
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It should be noted that the intercept of the fitted model is also statistically significant. However, considering practical scenario, we can ignore it (the intercept estimates mpg of a vehicle with zero length, weight, etc. and is of no practical use therefore).
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The R-squared statistic for the fitted model is 0.71 meaning that roughly 71% of the variability of the mpg can be explained by the model which is overall statistically significant (p-value of 5.35e-11).
From the above results, it can be said that the model did a fairly good job in predicting the mpg of MechaCar prototypes.
- The data indicate that the design specification that the variance of the suspension coils must not exceed 100 pounds PSI across all manfucaturing lots is met (62.3).
Total summary table
- However, when calculated by individual manufacturing lots, lot-3 fails to meet the criteria (170.29) although lot-1 and lot-2 do meet it (0.98 and 7.47, respectively).
Lot summary table
- Results of t-test on suspension coils PSI with null hypothesis that the true mean is equal to 1500 indicate that we don't have any evidence from the data to reject this hypothesis at a level of significance of 0.05. In other words, our data suggest that the true mean is indeed 1500 PSI.
- However, when the test is conducted by lot, we see wide variability in the result in terms of level of sigificance. While the null hypothesis (true mean = 1500 PSI) cannot be rejected for lot-1 and lot-2, we can reject it for lot-3 at 0.05 level of significance. In other words, this suggest that the true mean PSI for suspension coils from lot-3 is not equal to 1500.
Lot-1
Lot-2
Lot-3
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MechaCar needs to meet consumers' expectation on number of factors compared to its competitiors to win the market. These will include objective criteria such as price, MPG, horse power, safety ratings; future performance criteria such as maintenance cost; and more subjective criteria such as aesthetics and design. We have data on vehicle length, weight, ground clearance and whether they are AWD but these things are individual preferences rather than universally sought qualities such as MPG.
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Since buyers are likely to decide first what type of vehicle they want to purchase before choosing a brand, we need to compare MechaCar varieties with their peers. This means that we need to create categories of vehicles with more or less same length, rows and design classes (sedan, SUV, van, etc.), AWD (binary), horse power and then compare how MechaCar fares in terms of price, MPG, safety ratings etc. compared to its competitors.
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Our null hypotheses in these cases will be of the form H0: MechaCar's true mean of performance criterion is equal to or worse than competitor's mean. For example, H0: MechaCar's mean price is same or higher than paired competitor's mean. If our data allows us to reject this hypothesis, we can say that MechaCar is performing better in that particular criterion compared to its competitor.
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The ideal statistic for this task is a series of paired t-test where we test how MechaCar performs compared to its paired competitors. For example, for a MechaCar-Competitor paired sample on length, rows and design classes (sedan, SUV, van, etc.), AWD (binary), horse power, etc., are MechaCars cheaper than the mean price of competitor's cars?
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To conduct this analyses, we will need a dataset containing all variables for pairing (length, rows and design classes, AWD, horse power), and those for comparing (price, MPG, safety ratings). Also, our dataset needs to have enough records for each paired sample.