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Chapter 12: Problem 1

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 how the coefficient of correlation measures the strength of a linear relationship between two variables, x and y. Answer: The coefficient of correlation, also known as Pearson's correlation coefficient (r), is a statistical measure that quantifies the strength and direction of a linear association between two variables, x and y. It ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation), with 0 indicating no correlation. By calculating r using the formula, we can determine the strength and nature of the linear relationship between x and y. If the r value is close to -1 or 1, it signifies a strong linear relationship, while an r value close to 0 suggests little to no linear relationship. However, it's important to remember that the coefficient of correlation only assesses linear relationships, and non-linear relationships may not be accurately depicted by its values.

Step by step solution

01

Understanding the coefficient of correlation

Pearson's correlation coefficient, commonly denoted as \(r\), is a statistical measure that indicates the strength and direction of the linear association between two variables. The coefficient ranges from -1 to 1, where -1 signifies a perfect negative correlation, 0 means no correlation, and 1 represents a perfect positive correlation.
02

Calculating Pearson's correlation coefficient

To calculate the correlation coefficient between variable \(x\) and variable \(y\), use the following formula: $$ r = \frac{n(\sum_{i=1}^n x_i y_i) - (\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{\sqrt{[n\sum_{i=1}^n x_i^2 - (\sum_{i=1}^n x_i)^2][n\sum_{i=1}^n y_i^2 - (\sum_{i=1}^n y_i)^2]}}$$ Where: - \(n\) is the number of pairs of observations - \(x_i\) and \(y_i\) represent the \(i\)th observed value of variables \(x\) and \(y\) respectively - \(\sum_{i=1}^n x_i y_i \) is the sum of the product of corresponding \(x_i\) and \(y_i\) values
03

Interpreting the coefficient of correlation

The value obtained for \(r\) can be used to understand the strength and nature of the relationship between the two variables: - When \(r\) is close to -1 or 1, there is a strong linear relationship between \(x\) and \(y\). A value of -1 indicates a perfect negative correlation, meaning that as one variable increases, the other decreases, while a value of 1 signifies a perfect positive correlation, meaning that as one variable increases, the other also increases. - When \(r\) is close to 0, there is little to no linear relationship between \(x\) and \(y\), indicating that the variables are not strongly related to one another. It is important to note that the coefficient of correlation only assesses linear relationships and may not adequately depict the relationship between variables if it is non-linear.

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Most popular questions from this chapter

Why is it that one person may tend to gain weight, even if he eats no more and exercises no less than a slim friend? Recent studies suggest that the factors that control metabolism may depend on your genetic makeup. One study involved 11 pairs of identical twins fed about 1000 calories per day more than needed to maintain initial weight. Activities were kept constant, and exercise was minimal. At the end of 100 days, the changes in body weight (in kilograms) were recorded for the 22 twins. \({ }^{16}\) Is there a significant positive correlation between the changes in body weight for the twins? Can you conclude that this similarity is caused by genetic similarities? Explain. $$ \begin{array}{rrr} \text { Pair } & \text { Twin A } & \text { Twin B } \\\ \hline 1 & 4.2 & 7.3 \\ 2 & 5.5 & 6.5 \\ 3 & 7.1 & 5.7 \\ 4 & 7.0 & 7.2 \\\ 5 & 7.8 & 7.9 \\ 6 & 8.2 & 6.4 \\ 7 & 8.2 & 6.5 \\ 8 & 9.1 & 8.2 \\ 9 & 11.5 & 6.0 \\ 10 & 11.2 & 13.7 \\ 11 & 13.0 & 11.0 \end{array} $$

You are given five points with these coordinates:$$ \begin{array}{c|rrrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 1 & 1 & 3 & 5 & 5 \end{array} $$ a. Use the data entry method on your scientific or graphing calculator to enter the \(n=5\) observations. Find the sums of squares and cross-products, \(S_{x x} S_{x y},\) and \(S_{y y}\) b. Find the least-squares line for the data. c. Plot the five points and graph the line in part \(b\). Does the line appear to provide a good fit to the data points? d. Construct the ANOVA table for the linear regression.

An agricultural experimenter, investigating the effect of the amount of nitrogen \(x\) applied in 100 pounds per acre on the yield of oats \(y\) measured in bushels per acre, collected the following data: $$\begin{array}{l|lllll} x & 1 & 2 & 3 & 4 \\ \hline y & 22 & 38 & 57 & 68 \\\ & 19 & 41 & 54 & 65 \end{array} $$ a. Find the least-squares line for the data. b. Construct the ANOVA table. c. Is there sufficient evidence to indicate that the yield of oats is linearly related to the amount of nitrogen applied? Use \(\alpha=.05 .\) d. Predict the expected yield of oats with \(95 \%\) confidence if 250 pounds of nitrogen per acre are applied. e. Estimate the average increase in yield for an increase of 100 pounds of nitrogen per acre with \(99 \%\) confidence f. Calculate \(r^{2}\) and explain its significance in terms of predicting \(y,\) the yield of oats.

What is the difference between deterministic and probabilistic mathematical models?

Leonardo \(\mathrm{EXI} 217\) da Vinci ( \(1452-1519\) ) drew a sketch of a man. indicating that a person's arm span (measuring across the back with your arms outstretched to make a "T") is roughly equal to the person's height. To test this claim, we measured eight people with the following results: $$ \begin{aligned} &\begin{array}{l|clll} \text { Person } & 1 & 2 & 3 & 4 \\\ \hline \text { Armspan (inches) } & 68 & 62.25 & 65 & 69.5 \\ \text { Height (inches) } & 69 & 62 & 65 & 70\end{array}\\\ &\begin{array}{l|cccc} \text { Person } & 5 & 6 & 7 & 8 \\ \hline \text { Armspan (inches) } & 68 & 69 & 62 & 60.25 \\ \text { Height (inches) } & 67 & 67 & 63 & 62 \end{array} \end{aligned} $$ a. Draw a scatte rplot for arm span and height. Use the same scale on both the horizontal and vertical axes. Describe the relationship between the two variables. b. If da Vinci is correct, and a person's arm span is roughly the same as the person's height, what should the slope of the regression line be? c. Calculate the regression line for predicting height based on a person's arm span. Does the value of the slope \(b\) confirm your conclusions in part b? d. If a person has an arm span of 62 inches, what would you predict the person's height to be?
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