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Chapter 12: Problem 44

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

Short Answer

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Question: Explain the coefficient of correlation and its interpretation in measuring the strength of the linear relationship between two variables, y and x. Answer: The coefficient of correlation, denoted by the symbol 'r', measures the strength and direction of the linear relationship between two variables, y and x. Its value ranges between -1 and 1. To interpret the coefficient of correlation, consider the following: 1. \(r = 1\) represents a perfect positive linear relationship, meaning as x increases, y also increases. 2. \(r = -1\) represents a perfect negative linear relationship, meaning as x increases, y decreases. 3. \(r = 0\) indicates no linear relationship between the variables x and y. 4. \(0<r<1\): A higher value of r indicates a stronger positive linear relationship between the variables. 5. \(-1<r<0\): A more negative value of r indicates a stronger negative linear relationship between the variables. To analyze the strength of the linear relationship between y and x, calculate the correlation coefficient using the associated formula and interpret the resulting value based on the guidelines above.

Step by step solution

01

Define the Coefficient of Correlation

The coefficient of correlation, also known as the Pearson correlation coefficient or simply correlation coefficient, is a measure of the strength and direction of the linear relationship between two variables. The correlation coefficient is denoted by the symbol 'r' and its value ranges between -1 and 1.
02

Calculate the Coefficient of Correlation

To calculate the correlation coefficient (r) between two variables x and y, use the following formula: \[r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \sum_{i=1}^{n} (y_i - \bar{y})^2}}\] Where \(x_i\) and \(y_i\) are individual data points, \(\bar{x}\) and \(\bar{y}\) are the means of the x and y variables, and n is the number of data points for both x and y.
03

Interpret the Coefficient of Correlation

The value of the correlation coefficient (r) allows you to interpret the strength and direction of the linear relationship between the two variables: 1. \(r = 1\) implies a perfect positive linear relationship, meaning as x increases, y increases. 2. \(r = -1\) implies a perfect negative linear relationship, meaning as x increases, y decreases. 3. \(r = 0\) implies no linear relationship between the variables x and y. 4. \(0<r<1\): With increase in the values of r, the strength of positive linear relationship increases. 5. \(-1<r<0\): As(r) becomes more negative, the strength of negative linear relationship increases. To measure the strength of the linear relationship between two variables y and x, calculate the correlation coefficient using the given formula and interpret the resulting value as described above.

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Most popular questions from this chapter

G. W. Marino investigated the variables related to a hockey player's ability to make a fast start from a stopped position. \({ }^{11}\) In the experiment, each skater started from a stopped position and attempted to move as rapidly as possible over a 6-meter distance. The correlation coefficient \(r\) between a skater's stride rate (number of strides per second) and the length of time to cover the 6 -meter distance for the sample of 69 skaters was -.37 . a. Do the data provide sufficient evidence to indicate a correlation between stride rate and time to cover the distance? Test using \(\alpha=.05 .\) b. Find the approximate \(p\) -value for the test. c. What are the practical implications of the test in part a?

Graph the line corresponding to the equation \(y=-2 x+1\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line. How is this line related to the line \(y=2 x+1\) of Exercise \(12.1 ?\)

How is the cost of a plane flight related to the length of the trip? The table shows the average round-trip coach airfare paid by customers of American Airlines on each of 18 heavily traveled U.S. air routes. $$ \begin{array}{lrr} & \text { Distance } & \\ \text { Route } & \text { (miles) } & \text { Cost } \\ \hline \text { Dallas-Austin } & 178 & \$ 125 \\ \text { Houston-Dallas } & 232 & 123 \\ \text { Chicago-Detroit } & 238 & 148 \\ \text { Chicago-St. Louis } & 262 & 136 \\ \text { Chicago-Cleveland } & 301 & 129 \\ \text { Chicago-Atlanta } & 593 & 162 \\ \text { New York-Miami } & 1092 & 224 \\ \text { New York-San Juan } & 1608 & 264 \\ \text { New York-Chicago } & 714 & 287 \\ \text { Chicago-Denver } & 901 & 256 \\ \text { Dallas-Salt Lake } & 1005 & 365 \\ \text { New York-Dallas } & 1374 & 459 \\ \text { Chicago-Seattle } & 1736 & 424 \\ \text { Los Angeles-Chicago } & 1757 & 361 \\ \text { Los Angeles-Atlanta } & 1946 & 309 \\ \text { New York-Los Angeles } & 2463 & 444 \\ \text { Los Angeles-Honolulu } & 2556 & 323 \\ \text { New York-San Francisco } & 2574 & 513 \end{array} $$ a. If you want to estimate the cost of a flight based on the distance traveled, which variable is the response variable and which is the independent predictor variable? b. Assume that there is a linear relationship between cost and distance. Calculate the least-squares regression line describing cost as a linear function of distance. c. Plot the data points and the regression line. Does it appear that the line fits the data? d. Use the appropriate statistical tests and measures to explain the usefulness of the regression model for predicting cost.

The number of passes EX1242 completed and the total number of passing yards for Tom Brady, quarterback for the New England Patriots, were recorded for the 16 regular games in the 2006 football season. \({ }^{8}\) Week 6 was a bye and no data was reported. $$ \begin{array}{ccc} \text { Week } & \text { Completions } & \text { Total Yards } \\ \hline 1 & 11 & 163 \\ 2 & 15 & 220 \\ 3 & 31 & 320 \\ 4 & 15 & 188 \\ 5 & 16 & 140 \\ 6 & * & * \\ 7 & 18 & 195 \\ 8 & 29 & 372 \\ 9 & 20 & 201 \\ 10 & 24 & 253 \\ 11 & 20 & 244 \\ 12 & 22 & 267 \\ 13 & 27 & 305 \\ 14 & 12 & 78 \\ 15 & 16 & 109 \\ 16 & 28 & 249 \\ 17 & 15 & 225 \end{array} $$ a. What is the least-squares line relating the total passing yards to the number of pass completions for Tom Brady? b. What proportion of the total variation is explained by the regression of total passing yards \((y)\) on the number of pass completions \((x) ?\) c. If they are available, examine the diagnostic plots to check the validity of the regression assumptions.

An experiment was conducted to investigate the effect of a training program on the length of time for a typical male college student to complete the 100 -yard dash. Nine students were placed in the program. The reduction \(y\) in time to complete the 100 -yard dash was measured for three students at the end of 2 weeks, for three at the end of 4 weeks, and for three at the end of 6 weeks of training. The data are given in the table. $$ \begin{array}{l|l|l|l} \text { Reduction in Time, } y(\mathrm{sec}) & 1.6, .8,1.0 & 2.1,1.6,2.5 & 3.8,2.7,3.1 \\ \hline \text { Length of Training, } x(\mathrm{wk}) & 2 & 4 & 6 \end{array} $$ Use an appropriate computer software package to analyze these data. State any conclusions you can draw.
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