Question

Correlation conclusions II. The correlation between Fuel Efficiency (as measured by miles per gallon) and Price of 150 cars at a large dealership is \(r=-0.34\). Explain whether or not each of these possible conclusions is justified: a) The more you pay, the lower the fuel efficiency of your car will be. b) The form of the relationship between Fuel Efficiency and Price is moderately straight. c) There are several outliers that explain the low correlation. d) If we measure Fuel Efficiency in kilometers per liter instead of miles per gallon, the correlation will increase.

Short answer

a) It's not strongly justified, b) exaggerated claim, c) can't conclude without data, d) incorrect, unit change doesn't affect correlation.

Step-by-step solution

Understand the Correlation Value

The correlation coefficient \( r = -0.34 \) suggests a weak negative linear relationship between Fuel Efficiency and Price. This indicates a tendency, though not a strong one, that as the fuel efficiency increases, the price tends to decrease, and vice versa.

Evaluate Conclusion A

Statement a) suggests a direct relationship between price and fuel efficiency. While \( r = -0.34 \) does indicate an inverse relationship, the weak nature of the correlation means it is not strong enough to definitively support the conclusion that paying more will definitively lead to lower fuel efficiency.

Assess Conclusion B

Statement b) refers to the shape of the relationship as moderately straight. Since \( r = -0.34 \) is not close to -1, the assumption of a moderately straight linear relationship is exaggerated. A stronger correlation is needed to confirm a distinctly linear relationship.

Consider Conclusion C

Statement c) infers the presence of outliers that cause the low correlation. While outliers can affect correlation, without additional information such as a scatterplot, we cannot conclusively attribute the low correlation to outliers alone. Thus, further investigation is needed.

Examine Conclusion D

Statement d) involves changing units for Fuel Efficiency from miles per gallon to kilometers per liter. Since correlation is unitless, such a change will not affect the correlation coefficient. Thus, the correlation value will remain the same regardless of unit conversion.

Key concepts

Fuel Efficiency

Fuel efficiency is a crucial aspect when considering the overall performance of a vehicle. It is often quantified by how many miles a car can travel per gallon of fuel, commonly known as miles per gallon (MPG). A higher MPG indicates a vehicle that is more efficient in its fuel use, thus offering better fuel economy.
In the study of correlations between different vehicle parameters, fuel efficiency plays a significant role. It signifies how economically a vehicle runs on the fuel it consumes. This is important for both environmental reasons and cost-effectiveness for the owner.

• Higher fuel efficiency can lead to less environmental pollution.
• Fuel-efficient vehicles often help reduce the cost of travel.
• It portrays a vehicle's technological advancement in terms of engine performance.

Understanding how fuel efficiency relates to other factors, such as price, helps consumers make better choices when purchasing a car.

Linear Relationship

In statistics, a linear relationship is a connection between two variables that can be represented by a straight line on a graph. The strength of this relationship is often measured by a correlation coefficient, denoted as \( r \).
In the exercise, a correlation coefficient of \( r = -0.34 \) indicates a weak negative linear relationship between fuel efficiency and price. This means that, generally, as one variable increases, the other tends to decrease, but this relationship is not strong.
Here’s why a linear relationship might be important:

• Predictability: A strong linear relationship allows for more accurate predictions.
• Consistency: With a linear model, the rate of change is uniform.
• Ease of interpretation: Linear relationships are straightforward to interpret and analyze.

In the context of our example, the weak correlation suggests the relationship isn't fully linear, and other factors may be influencing the variables.

Outliers

Outliers are data points that differ significantly from other observations in a dataset. They can drastically affect statistical measurements, including mean values and correlation coefficients.
This is especially relevant when analyzing the correlation between fuel efficiency and price. Outliers might be unusually priced vehicles or those with atypically high or low fuel efficiency, which could skew the perception of a relationship between these variables.
Here’s how outliers can impact data analysis:

• Distortion of correlation: Outliers can either inflate or deflate the perceived strength of a correlation.
• Influence on mean and variance: Outliers can alter basic statistical measures, making data appear more varied than it truly is.
• Sparking further investigation: Discovering outliers often leads analysts to delve deeper into data to understand underlying causes.

In this case, without a visual scatterplot or additional analysis, it’s hard to definitively assess the role of outliers.

Unit Conversion

Unit conversion is the process of changing a measure to a different unit, for example from miles per gallon to kilometers per liter for fuel efficiency. While the numeric value of the measure changes, the physical quantity remains the same.
In statistical terms, a correlation is dimensionless, meaning it is unaffected by unit changes. Therefore, converting fuel efficiency from miles per gallon to kilometers per liter will not change the correlation between fuel efficiency and price.
Benefits of understanding unit conversion include:

• Global standardization: Facilitates understanding across different regions and measurement systems.
• Consistent analysis: Ensures that comparisons and analyses remain valid across various units.
• Avoids misinterpretation: Helps prevent confusion over differences that are only due to units, not actual values.

Thus, while unit conversion is important for standardization, it has no effect on the correlation coefficient in this context.