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Chapter 12: Problem 3

What value does \(r\) assume if all the data points fall on the same straight line in these cases? a. The line has positive slope. b. The line has negative slope

Short Answer

Expert verified
Answer: When data points fall exactly on a straight line with a positive slope, the Pearson's correlation coefficient (r) is +1. When data points fall exactly on a straight line with a negative slope, the Pearson's correlation coefficient (r) is -1.

Step by step solution

01

Case A: Line with positive slope

In this case, the data points form a straight line with a positive slope, meaning that as one variable increases, the other variable also increases. This represents the strongest possible positive correlation between the two variables. Therefore, in this case, \(r\) assumes the maximum positive value of +1.
02

Case B: Line with negative slope

In this case, the data points form a straight line with a negative slope, meaning that as one variable increases, the other variable decreases. This represents the strongest possible negative correlation between the two variables. Therefore, in this case, \(r\) assumes the maximum negative value of -1.

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Key Concepts

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

Positive Correlation
Positive correlation refers to a relationship between two variables where they both move in the same direction. Imagine plotting two variables on a graph. As one variable increases, the other also goes up. This is what a positive correlation looks like.
When all data points fall perfectly on a straight line with a positive slope, it means that we have a perfect positive correlation. In numerical terms, this is denoted by a correlation coefficient, denoted as \( r \), with a value of \( +1 \).
This value of \( +1 \) represents the strongest possible agreement between the two variables when one goes up, the other follows in perfect harmony.
Negative Correlation
Negative correlation, on the other hand, occurs when two variables move in opposite directions. Visualize two variables on a graph where an increase in one variable leads to a decrease in the other.
In the instance where all data points fall exactly on a straight line with a negative slope, this indicates a perfect negative correlation. Here, the correlation coefficient \( r \) will be \( -1 \).
This value of \( -1 \) portrays the strongest possible disagreement between the variables, meaning as one variable rises, the other falls in precise alignment.
Slope of a Line
The slope of a line is a crucial concept when discussing correlation. It describes the steepness and direction of the line formed by our data points. The slope can be either positive or negative.
A positive slope signifies that as one variable on the graph increases, the other variable increases as well. Conversely, a negative slope indicates that as one variable rises, the other decreases.
Understanding the slope helps in predicting the relationship between variables and is key to knowing whether the correlation should be positive or negative.
Data Points
Data points are individual pairs of values that represent the relationship between two variables. Each point on a graph corresponds to one pair of data values.
When these points are laid out on a graph, they can reveal patterns or relationships between the variables.
If all points fall along a single straight line, it captures a perfect correlation, either positive or negative, depending on the direction of the slope.
  • Positive correlation: all points align on an upward-sloping line
  • Negative correlation: all points align on a downward-sloping line
Mathematical Relationship
A mathematical relationship involves the interaction between variables that can often be represented as an equation or a graph. Such relationships can be linear, where the points form a straight line.
In our scenario, linear relationships exactly capture the essence of correlations 鈥� through the slope of that line. A line with a positive slope indicates a positive mathematical relationship and vice versa for a negative slope.
These relationships allow us to summarize and predict how one variable affects another, and are measurable through the correlation coefficient.

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Most popular questions from this chapter

A social skills training program was implemented with seven mildly challenged students in a study to determine whether the program caused improvement in pre/post measures and behavior ratings. For one such test, the pre- and post test scores for the seven students are given in the table. $$ \begin{array}{lrr} \text { Subject } & \text { Pretest } & \text { Posttest } \\ \hline \text { Earl } & 101 & 113 \\ \text { Ned } & 89 & 89 \\ \text { Jasper } & 112 & 121 \\ \text { Charlie } & 105 & 99 \\ \text { Tom } & 90 & 104 \\ \text { Susie } & 91 & 94 \\ \text { Lori } & 89 & 99 \end{array} $$ a. What type of correlation, if any, do you expect to see between the pre- and posttest scores? Plot the data. Does the correlation appear to be positive or negative? b. Calculate the correlation coefficient, \(r\). Is there a significant positive correlation?

How does the coefficient of correlation measure the strength of the linear relationship between two variables \(y\) and \(x ?\)

Give the equation and graph for a line with \(y\) -intercept equal to 3 and slope equal to -1 .

Some varieties of nematodes, roundworms that live in the soil and feed on the roots of lawn grasses and other plants, can be treated by the application of nematicides. Data collected on the percent kill of nematodes for various rates of application (dosages given in pounds per acre of active ingredient) are as follows: $$ \begin{array}{l|l|l|l|l} \text { Rate of Application, } x & 2 & 3 & 4 & 5 \\\ \hline \text { Percent Kill, } y & \mid 50,56,48 & 63,69,71 & 86,82,76 & 94,99,97 \end{array} $$ Use an appropriate computer printout to answer these questions: a. Calculate the coefficient of correlation \(r\) between rates of application \(x\) and percent kill \(y .\) b. Calculate the coefficient of determination \(r^{2}\) and interpret. c. Fit a least-squares line to the data. per acre. What do the diagnostic plots tell you about the validity of the regression assumptions? Which assumptions may have been violated? Can you explain why? d. Suppose you wish to estimate the mean percent kill for an application of 4 pounds of the nematicide

In Exercise 3.18 , \(\mathrm{EX}_{1223}\) we found that male crickets chirp by rubbing their front wings together, and their chirping is temperature dependent. The table below shows the number of chirps per second for a cricket, recorded at 10 different temperatures: $$ \begin{array}{l|llllllllll} \text { Chirps per Second } & 20 & 16 & 19 & 18 & 18 & 16 & 14 & 17 & 15 & 16 \\ \hline \text { Temperature } & 88 & 73 & 91 & 85 & 82 & 75 & 69 & 82 & 69 & 83 \end{array} $$ a. Use the formulas given in this chapter to find the least-squares regression line relating the number of chirps to temperature. Compare to the results obtained in Exercise 3.18 b. Do the data provide sufficient evidence to indicate that there is a linear relationship between number of chirps and temperature? c. Calculate \(r^{2}\). What does this value tell you about the effectiveness of the linear regression analysis?
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