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Chapter 6: Problem 36

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

Expert verified
a. Partly justified, correlation is negative but not strongly. b. Partially justified, as the correlation is weak. c. Not justified, cannot identify outliers from correlation coefficient. d. Not justified, scale of measurement won't affect the correlation.

Step by step solution

01

Clarification Of Conclusion A

The correlation coefficient \(r=-0.34\) indeed signifies a negative relationship between price and fuel efficiency, but it's a moderate correlation, not a strong one. This implies that it's not always true that the more you pay for a car, the lower its fuel efficiency will be. Therefore, this conclusion is only partially justified. There may be a trend, but there's also a fair amount of variability.
02

Clarification Of Conclusion B

Although the correlation is negative, it is also weak, thus the relationship between Fuel Efficiency and Price isn't strongly linear. Therefore, describing the relationship as 'moderately straight' would not be completely fitting. It only reflects a general trend.
03

Clarification Of Conclusion C

We cannot conclusively identify the existence of several outliers based solely on the given correlation coefficient. Outliers can influence the correlation coefficient significantly, but to identify outliers, one needs more information (like a scatter plot or the raw data). Hence, this conclusion is not justified.
04

Clarification Of Conclusion D

The correlation between two variables does not depend on the scale of measurement. The correlation will remain the same, whether miles per gallon or kilometers per liter are used to measure fuel efficiency. Therefore, this assumption is not justified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient
The correlation coefficient is a statistical measure that describes the strength and direction of a relationship between two variables. It is represented by the symbol \( r \). The value of \( r \) ranges from -1 to +1.

Here are what these numbers imply:
  • \( r = +1 \): A perfect positive linear relationship.
  • \( r = -1 \): A perfect negative linear relationship.
  • \( r = 0 \): No linear relationship between the variables.
A correlation of \( r = -0.34 \) means there is a weak negative linear relationship between two variables. This suggests that as one variable increases, the other tends to decrease slightly.
In our exercise, the correlation indicates that higher car prices tend to come with slightly lower fuel efficiency. However, the association is not strong enough to make exact predictions or conclusions.
Fuel Efficiency
Fuel efficiency describes how far a vehicle can travel using a specific amount of fuel. It's commonly measured in miles per gallon (MPG) or kilometers per liter (km/L).

When assessing fuel efficiency:
  • Higher values indicate better fuel economy, meaning fewer resources consumed over greater distances.
  • Conversely, lower values mean the vehicle consumes more fuel for the same distance traveled.
Fuel efficiency is an important factor for car buyers, especially those cost-sensitive or environmentally conscious. A car's fuel efficiency can vary based on engine type, vehicle size, and even the driving conditions.
In the case of our exercise, cars with high prices might not always be the most fuel-efficient due to a negative correlation, but it's a moderate one, indicating variations.
Linear Relationship
A linear relationship between two variables means that as one variable changes, the other changes at a constant rate. In a graphical sense, this relationship will form a straight line on a plot.

Here are some features of linear relationships:
  • If the correlation is positive, both variables increase together.
  • If the correlation is negative, as one variable increases, the other decreases.
  • The exactness of this relationship is described by the correlation coefficient \( r \).
In our scenario, the given correlation of \( r = -0.34 \) implies a negative linear tendency, but it's weak enough that it can't confidently predict the form of the relationship as moderately straight.
While there might be a general downward trend, only with additional data, such as a scatter plot, can we better identify how linear or scattered the relationship is between car price and fuel efficiency.

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Most popular questions from this chapter

A researcher investigating the association between two variables collected some data and was surprised when he calculated the correlation. He had expected to find a fairly strong association, yet the correlation was near 0 . Discouraged, he didn't bother making a scatterplot. Explain to him how the scatterplot could still reveal the strong association he anticipated.

Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a. When climbing mountains: altitude, temperature b. For each week: ice cream cone sales, air-conditioner sales c. People: age, grip strength d. Drivers: blood alcohol level, reaction time

A researcher studies children in elementary school and finds a strong positive linear association between height and reading scores. a. Does this mean that taller children are generally better readers? b. What might explain the strong correlation?

If we assume that the conditions for correlation are met, which of the following are true? If false, explain briefly. a. A correlation of -0.98 indicates a strong, negative association. b. Multiplying every value of \(x\) by 2 will double the correlation. C. The units of the correlation are the same as the units of \(y\).

Smartphones and life expectancy A survey of the world's nations in 2014 shows a strong positive correlation between percentage of the country using smartphones and life expectancy in years at birth. a. Does this mean that smartphones are good for your health? b. What might explain the strong correlation?
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