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Chapter 4: Problem 21

The points on a scatterplot lie very close to a straight line. The correlation between \(x\) and \(y\) is close to (a) \(-1 .\) (b) \(1 .\) (c) either \(-1\) or 1 , we can't say which.

Short Answer

Expert verified
(c) either -1 or 1, we can't say which.

Step by step solution

01

Understanding Correlation

Correlation measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1. A correlation close to 1 indicates a strong positive linear relationship, while a correlation close to -1 signifies a strong negative linear relationship.
02

Analyzing the Scatterplot Description

The problem states that the points on the scatterplot lie very close to a straight line. This implies a very strong linear relationship between the variables, indicating that the correlation is close to either 1 or -1.
03

Determining Sign of Correlation

Since the scatterplot points are close to a straight line but the specific direction (positive or negative slope) isn't specified, we cannot definitively determine if the correlation is exactly -1 or 1. Both are possible depending on the slope of the line.
04

Selecting an Answer

Given that both options -1 and 1 represent points lying close to a straight line, and the direction of the line isn't provided, the most accurate answer is (c) either \(-1\) or 1, we can't say which.

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Key Concepts

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

Scatterplot
A scatterplot is a type of graph used in statistics to represent the relationship between two numerical variables. Each point on the scatterplot corresponds to a pair of values from the dataset. By plotting these on the graph, we can visually assess any potential patterns or relationships between the variables.

When we look at a scatterplot, we're trying to determine if there's any noticeable trend or pattern in how the points are distributed:
  • If the points show a rising trend from left to right, it might indicate a positive relationship.
  • Conversely, if they fall, it might indicate a negative relationship.
  • If the points don't follow any clear pattern, this suggests no strong relationship.
Understanding scatterplots is a crucial skill as they provide an intuitive way to comprehend complex data, especially when we want to examine relationships visually before diving into more sophisticated statistical analyses.
Linear Relationship
In statistics, a linear relationship between two variables is one where the value of one variable generally increases or decreases in direct proportion to the other. Imagine plotting a straight line on a graph 鈥� this line represents the direction of the relationship.

To determine if a linear relationship exists, look for certain characteristics in the scatterplot:
  • A straight line pattern suggests a linear relationship.
  • The line can either slope upward (positive linear relationship) or downward (negative linear relationship).
For example, consider how study time (x-axis) might relate to test scores (y-axis). An increase in study time leading to higher test scores would represent a positive linear relationship. However, without a discernable straight line, there is likely no linear relationship.
Strength of Correlation
The strength of a correlation tells us how closely the data points in a scatterplot follow a linear pattern. This is quantitatively measured by the correlation coefficient, represented by the symbol "r," which ranges from -1 to 1.

Let鈥檚 break down what different correlation values mean:
  • r = 1: Perfect positive correlation, where all points lie exactly on a line with a positive slope.
  • r = -1: Perfect negative correlation, where all points lie exactly on a line with a negative slope.
  • r = 0: No linear correlation, suggesting randomness with no discernible linear trend.
  • 0 < r < 1: Positive correlation, with varying degrees of linearity.
  • -1 < r < 0: Negative correlation, with varying degrees of linearity.
Observing the strength of correlation in a scatterplot helps in predicting how changes in one variable might affect the other, which can be crucial in many scientific and practical applications.

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Most popular questions from this chapter

Toucan's beak . The toco toucan, the largest member of the toucan family, possesses the largest beak relative to body size of all birds. This exaggerated feature has received various interpretations, such as being a refined adaptation for feeding. However, the large surface area may also be an important mechanism for radiating heat (and hence cooling the bird) as outdoor temperature increases. Here are data for beak heat loss, as a percent of total body heat loss, at various temperatures in degrees Celsius: \({ }^{25}\) III ToucAN $$ \begin{array}{l|cccccccc} \hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 \\ \text { Percent heat loss from beak } & 32 & 34 & 35 & 33 & 37 & 46 & 55 & 51 \\\ \hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \text { Percent heat loss from beak } & 43 & 52 & 45 & 53 & 58 & 60 & 62 & 62 \\\ \hline \end{array} $$ Investigate the relationship between outdoor temperature and beak heat loss, as a percentage of total body heat loss.

If the correlation between two variables is close to 0 , you can conclude that a scatterplot would show (a) a strong straight-line pattern. (b) a cloud of points with no visible pattern. (c) no straight-line pattern, but there might be a strong pattern of another form.

Teacher salaries. For each of the 50 states and the District of Columbia, average Mathematics SAT scores and average high school teacher salaries for 2015 are available. \({ }^{28}\) Discuss whether the data support the idea that higher teacher salaries lead to higher Mathematics SAT scores.

Sloppy writing about correlation. Each of the following statements contains a blunder. Explain in each case what is wrong. (a) "There is a high correlation between the sex of American workers and their income." (b) "We found a high correlation \((r=1.09)\) between students' ratings of faculty teaching and ratings made by other faculty members." (c) "The correlation between height and weight of the subjects was \(r=0.63\) centimeter per kilogram."

Predicting Life Expectancy. Identifying variables that can be used to predict life expectancy is important for insurance companies, economists, and policymakers. Several researchers have investigated the extent to which poverty level can be used to predict life expectancy. Name two other variables that could be used to predict life expectancy.
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