### Part A: Interpretation of Strength and Direction of Association
When interpreting the data in the table provided, we are interested in understanding the relationship between horsepower of the engine and fuel usage (in gallons per hour).
In a scatter plot, if most of the points closely follow a straight line, this indicates a strong linear association. If the points are more spread out, this indicates a weaker association. The direction of the association can also be identified by the general trend of the data points:
1. Strength of Association:
- The strength of the association between horsepower and fuel usage can be observed by how tightly the data points cluster around a hypothetical line of best fit. Here, the residuals (the differences between observed and predicted values) will tell us about the deviations from the line of best fit.
- In examining the residuals, a majority of them are relatively small except for a few larger values. This suggests that, while there is some deviation, many of the data points do align reasonably well with the assumed linear trend.
2. Direction of Association:
- The data shows that as horsepower increases, fuel usage generally increases as well, indicating a positive association.
- Despite some residuals being larger, the positive values of residuals, particularly as horsepower increases, indicate that the general direction of association is positive. This means as the engines have more horsepower, they tend to consume more fuel.
Overall, there is a positive direction in the association between horsepower and fuel usage. The strength of this association is moderate, indicated by the general alignment of data points to the trend with some notable outliers.
### Part B: Identifying the Outlier and Its Impact
To identify the outlier, we look for the residual with the largest absolute value. This residual represents the data point that deviates the most from the expected trend.
1. Outlier Identification:
- The residual with the largest absolute value is -0.605. This corresponds to the engine with 16 horsepower and a fuel usage of 0.5 gallons per hour.
- Therefore, the outlier ordered pair is (16, 0.5) with a residual of -0.605.
2. Impact of Removing the Outlier:
- Removing this outlier would likely result in the residuals for the remaining data points being smaller overall. This reduction in variance of residuals would cause the data points to cluster more closely around the line of best fit.
- As a result, the strength of the correlation (association) would increase as the data points would show a more consistent pattern of behavior without the distortion caused by the outlier.
- The direction of the association would still remain positive, as the overall trend of increasing horsepower leading to increasing fuel usage wouldn't change, but the relationship would appear stronger and clearer.
Thus, the outlier is the ordered pair (16, 0.5), and removing it would enhance the strength of the association while maintaining the positive direction of the relationship between horsepower and fuel usage.
