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Chapter 4: Problem 19

A sample of automobiles traveling on a particular segment of a highway is selected. Each one travels at roughly a constant rate of speed, although speed does vary from auto to auto. Let \(x=\) speed and \(y=\) time needed to travel this segment. Would the sample correlation coefficient be closest to \(0.9,0.3,-0.3,\) or \(-0.9 ?\) Explain.

Short Answer

Expert verified
The sample correlation coefficient would most likely be closest to \(-0.9\), indicating a strong inverse relationship between speed and time.

Step by step solution

01

Understand the relationship between variables

Determine the relationship between the variables. The relationship between speed (x) and time (y) is inverse or negative. As one increases, the other decreases.
02

Correlation analysis

On analyzing the correlation options given, we can rule out the positive values \(0.9\) and \(0.3\) as they represent a positive correlation. We are left with two options, \(-0.9\) and \(-0.3\). A correlation of \(-0.9\) indicates a strong negative correlation while \(-0.3\) indicates a weak negative correlation.
03

Determine the most likely correlation

Since we know the relationship between speed and time is a strong inverse one, based on real-life experience, we can assume that the correlation coefficient would be closer to \(-0.9\) rather than \(-0.3\).

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Key Concepts

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

Inverse Relationship
An inverse relationship between two variables means that as one variable increases, the other decreases. This is a crucial concept in understanding how different data points relate to one another in a dataset. In the context of the exercise, this relationship is seen with speed and time.
When the speed of an automobile increases, the time taken to travel a set distance generally decreases. Conversely, as the speed decreases, the time taken increases. This kind of inverse relationship can be observed in various real-life scenarios.
Recognizing an inverse relationship helps in determining the type of correlation between variables. By analyzing this relationship, we can infer that the correlation between speed and time would likely be negative since one variable's increase results in another's decrease.
Negative Correlation
A negative correlation occurs when an increase in one variable leads to a decrease in another. This is encapsulated by a correlation coefficient that is less than zero.
In the given exercise, the negative correlation between speed and time suggests that as automobiles travel faster, the amount of time required to reach a destination decreases.
Correlation coefficients range from -1 to 1.
  • A value close to -1 indicates a strong negative correlation.
  • A value close to 0 suggests no correlation.
  • A value close to 1 represents a strong positive correlation.

In this exercise, understanding negative correlation is key to identifying the strength of this inverse relationship. The closer the correlation is to -1, the stronger the negative relationship, suggesting a more pronounced inverse relationship between the two variables in question.
Speed and Time Correlation
The correlation between speed and time on a highway segment is a practical example of an inverse, negative correlation. Given that speed and time often show a near-perfect inverse relationship when considered over a constant distance, it's reasonable to estimate this relationship with a correlation coefficient of around -0.9.
A strong negative correlation such as this implies:
  • High speeds result in less travel time.
  • Low speeds lead to more travel time.

The importance of understanding this concept lies in its application in real-world situations where speed limits, travel planning, and efficiency metrics are considered. Grasping speed and time correlation can aid in predictive modeling and optimizing travel routes, as it provides insight into how changes in speed can impact travel time significantly.

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It may seem odd, but biologists can tell how old a lobster is by measuring the concentration of pigment in the lobster's eye. The authors of the paper "Neurolipofuscin Is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster, in Florida" (Biological Bulletin [2007]: 55-66) wondered if it was sufficient to measure the pigment in just one eye, which would be the case if there is a strong relationship between the concentration in the right eye and the concentration in the left eye. Pigment concentration (as a percentage of tissue sample) was measured in both eyes for 39 lobsters, resulting in the following summary quantities (based on data from a graph in the paper): $$ \begin{array}{cll} n=39 & \sum_{x}=88.8 & \sum y=86.1 \\ \sum x y=281.1 & \sum x^{2}=288.0 & \sum y^{2}=286.6 \end{array} $$ An alternative formula for calculating the correlation coefficient that doesn't involve calculating the z-scores is $$ r=\frac{\sum_{x y}-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to calculate the value of the correlation coefficient, and interpret this value.
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