91Ó°ÊÓ

Toughness and fibrousness of asparagus are major determinants of quality. This was the focus of a study reported in "Post-Harvest Glyphosphate Application Reduces Toughening, Fiber Content, and Lignification of Stored Asparagus Spears" (J. of the Amer. Soc. of Hort. Science, 1988: 569-572). The article reported the accompanying data (read from a graph) on \(x=\) shear force \((\mathrm{kg})\) and \(y=\) percent fiber dry weight. $$ \begin{array}{c|ccccccccc} x & 46 & 48 & 55 & 57 & 60 & 72 & 81 & 85 & 94 \\ \hline y & 2.18 & 2.10 & 2.13 & 2.28 & 2.34 & 2.53 & 2.28 & 2.62 & 2.63 \\ x & 109 & 121 & 132 & 137 & 148 & 149 & 184 & 185 & 187 \\ \hline y & 2.50 & 2.66 & 2.79 & 2.80 & 3.01 & 2.98 & 3.34 & 3.49 & 3.26 \end{array} $$ \(n=18, \Sigma x_{i}=1950, \Sigma x_{i}^{2}=251,970\) \(\Sigma y_{i}=47.92, \Sigma y_{i}^{2}=130.6074, \Sigma x_{i} y_{i}=5530.92\) a. Calculate the value of the sample correlation coefficient. Based on this value, how would you describe the nature of the relationship between the two variables? b. If a first specimen has a larger value of shear force than does a second specimen, what tends to be true of percent dry fiber weight for the two specimens? c. If shear force is expressed in pounds, what happens to the value of \(r\) ? Why? d. If the simple linear regression model were fit to this data, what proportion of observed variation in percent fiber dry weight could be explained by the model relationship? e. Carry out a test at significance level .01 to decide whether there is a positive linear association between the two variables.

Step by step solution

01

Understand the Variables

We have two variables: \(x\), the shear force in kilograms, and \(y\), the percent fiber dry weight. We need to analyze their relationship using statistical methods.

02

Calculate the Correlation Coefficient

The formula for the sample correlation coefficient \(r\) is \[ r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{(n \Sigma x^2 - (\Sigma x)^2)(n \Sigma y^2 - (\Sigma y)^2)}} \] Plug the given values into the formula: \( n = 18 \), \( \Sigma x = 1950 \), \( \Sigma x^2 = 251970 \), \( \Sigma y = 47.92 \), \( \Sigma y^2 = 130.6074 \), \( \Sigma xy = 5530.92 \). Calculate the result.

03

Interpret the Correlation

Once the correlation coefficient \(r\) is calculated, assess the strength and direction of the linear relationship. Positive \(r\) indicates a direct relationship, while negative \(r\) suggests an inverse relationship. The closer \(r\) is to \(1\) or \(-1\), the stronger the relationship.

04

Assess the Effect of Unit Conversion

If shear force is converted from kilograms to pounds, the correlation coefficient \(r\) remains unchanged because \(r\) is a dimensionless measurement. Changing units does not affect the strength or direction of a linear relationship.

05

Calculate Proportion of Variance Explained

The coefficient of determination \( r^2 \) represents the proportion of variance in \( y \) explained by the linear relationship. Square the correlation coefficient \( r \) to determine \( r^2 \), which indicates the explanatory power of the regression model.

06

Hypothesis Test for Significance of Correlation

For a hypothesis test at a significance level of \( \alpha = 0.01 \), the null hypothesis \( H_0: \rho = 0 \) (no correlation) is tested against the alternative hypothesis \( H_a: \rho > 0 \) (positive correlation). Use the test statistic \[ t = r \sqrt{\frac{n-2}{1-r^2}} \] and compare with the critical value from the \( t \)-distribution table with \( n-2 \) degrees of freedom.

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Key Concepts

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

Linear Regression

Linear regression is a method used to model the relationship between two variables by fitting a linear equation to observed data. In our asparagus study, these variables are the shear force, denoted as \(x\), and the percent fiber dry weight, denoted as \(y\). A simple linear regression will often appear as a straight line on a graph where \(x\) values are plotted on the horizontal axis and \(y\) values on the vertical axis. This line is called the "best fit" line.

The equation of this line is typically written as \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) is the y-intercept. The slope \(m\) shows how much \(y\) changes for a one-unit change in \(x\).

This method is particularly useful for making predictions. If you have a data point for \(x\), you can plug it into the equation to predict \(y\). In the case of asparagus, having a known shear force allows estimation of the percent fiber dry weight. This predictive power is why linear regression is a go-to tool for analysts and researchers.

Hypothesis Testing

Hypothesis testing in the context of correlation involves evaluating whether a relationship between two variables is statistically significant. In the asparagus example, you might want to test if the observed correlation between shear force and percent fiber dry weight is due to a true association, rather than mere chance.

Hypothesis testing starts with forming a null hypothesis, \(H_0\), and an alternative hypothesis, \(H_a\). For these variables, \(H_0\) might state that there is no correlation \((\rho = 0)\), whereas \(H_a\) suggests there is a positive correlation \((\rho > 0)\).

Using a significance level, such as \(\alpha = 0.01\), which reflects the risk of accepting \(H_a\) when \(H_0\) is true, you calculate a test statistic from the sample data. This statistic is then compared against a critical value derived from the \(t\)-distribution with \(n-2\) degrees of freedom, where \(n\) is the number of data pairs.

If the test statistic exceeds this critical value, the null hypothesis is rejected, supporting the alternative hypothesis. Thus, you conclude that a significant positive correlation exists between shear force and percent fiber dry weight, confirming it's not just by chance.

Variance Explained

Variance explained in a dataset refers to how much of the variability in one variable is accounted for by its relationship with another variable, often through a linear model. The measure used here is the coefficient of determination, \( r^2 \).

In the asparagus study, once you have found the correlation coefficient \(r\), squaring it yields \( r^2 \). This value provides insight into the proportion of variance in percent fiber dry weight that can be explained by the changes in shear force.

A higher \( r^2 \) value indicates a stronger relationship, meaning that a greater percentage of \( y \) (percent fiber dry weight) variation can be predicted from \( x \) (shear force) variations. For example, if \( r^2 = 0.58 \), it implies that 58% of the variability in percent fiber dry weight is explained by the linear relationship with shear force.

This aspect of analysis is crucial because it shows the effectiveness of your linear model in explaining the variability, guiding decisions and insights drawn from the data.