Absolutely, let's tackle these questions step-by-step.
### Part A: Real-world Interpretation of the Strength and Direction of Association
To determine the strength and direction of the association between horsepower of the engine and fuel usage, we compute the correlation coefficient. The correlation coefficient is a measure that determines the degree to which two variables' movements are associated. The value of the correlation coefficient ranges from -1 to +1, inclusive, where:
- `+1` indicates a perfect positive linear relationship,
- `-1` indicates a perfect negative linear relationship,
- `0` indicates no linear relationship.
Given the correlation strength of approximately `-0.578`, we can derive the following interpretation:
Interpretation:
The correlation coefficient of `-0.5778450107445933` suggests a moderate negative linear relationship between horsepower and fuel usage. This means that, generally speaking, as the horsepower of the engine increases, the fuel usage tends to decrease. The negative sign indicates the inverse relationship, and the magnitude of 0.578 implies it is a moderate (neither very weak nor very strong) relationship.
### Part B: Identifying the Outlier and its Effects on the Correlation
An outlier is an observation point that is distant from other observations. In a residual plot, an outlier will have a residual that is much larger (positive or negative) than other residuals.
Identifying the Outlier:
To identify the outlier, we look at the residuals—specifically, the one with the highest absolute value. From the data provided:
[tex]\[ \text{Residuals} = [0.165, -0.054, -0.210, 0.071, -0.055, -0.274, 0.601, 0.244] \][/tex]
The largest residual in absolute terms is `0.601` which corresponds to the data point with:
[tex]\[ \text{Horsepower} = 20 \text{ and Fuel Usage} = 1.25 \][/tex]
Hence, the ordered pair [tex]\( (20, 1.25) \)[/tex] is identified as the outlier.
Effect of Removing the Outlier:
To understand the effect of removing the outlier on the correlation:
Before removing the outlier, the correlation coefficient was `-0.578`, indicating a moderate negative linear relationship.
If we remove the outlier [tex]\( (20, 1.25) \)[/tex]:
The new correlation coefficient is approximately `-0.929`.
Interpretation of the New Correlation Coefficient:
After removing the outlier, the new correlation coefficient of `-0.9287993117110809` suggests a much stronger negative linear relationship than before. This significant change indicates that the outlier had a substantial effect on the linear relationship between horsepower and fuel usage. By removing the outlier, we see that the remaining data shows a stronger inverse relationship, where higher horsepower corresponds to lower fuel usage more consistently.
Summary:
Part A: The original correlation coefficient of `-0.578` indicates a moderate negative linear relationship, suggesting that as horsepower increases, fuel usage typically decreases.
Part B: The outlier is the data point [tex]\( (20, 1.25) \)[/tex]. Removing this outlier changes the correlation coefficient to `-0.929`, indicating a much stronger negative linear relationship. This shows that the outlier was diminishing the apparent strength of the relationship.
