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. $$ \begin{array}{l|ccccccccc} x & 46 & 48 & 55 & 57 & 60 & 72 & 81 & 85 & 94 \\ \hline y & 2.18 & 2.10 & 2.13 & 2.28 & 2.34 & 2.53 & 2.28 & 2.62 & 2.63 \\ x & 109 & 121 & 132 & 137 & 148 & 149 & 184 & 185 & 187 \\ \hline y & 2.50 & 2.66 & 2.79 & 2.80 & 3.01 & 2.98 & 3.34 & 3.49 & 3.26 \end{array} $$ 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

Calculate the sample correlation coefficient (r)

First, calculate the means for both shear force (\(\bar{x}\)) and percent fiber dry weight (\(\bar{y}\)). Then, use the formula for the sample correlation coefficient:\[r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}}}\]Substitute the data and solve to find \(r\). Solving yields \(r \approx 0.922\).

02

Interpret the correlation coefficient

The calculated correlation coefficient \(r \approx 0.922\) indicates a strong positive linear relationship between shear force and percent fiber dry weight. This suggests that as the shear force increases, the percent fiber dry weight also tends to increase.

03

Analyze relationship between two specimens

When a specimen has a larger shear force value than another specimen, its percent dry fiber weight also tends to be higher, due to the strong positive linear relationship indicated by the correlation coefficient.

04

Effect of measuring shear force in pounds on r

Changing the units of shear force from kg to pounds does not affect the correlation coefficient \(r\) because \(r\) is a dimensionless measure that depends only on the relative distributions and not on units.

05

Determine explained variation by linear model

The proportion of variation in percent fiber dry weight that can be explained by the linear model is given by the square of the correlation coefficient, \(r^2\). Therefore, \(r^2 \approx (0.922)^2 \approx 0.850\).This means approximately 85% of the variability in percent fiber dry weight is explained by the shear force.

06

Test for positive linear association

Set up hypotheses: \(H_0: \rho = 0\) (no linear association) and \(H_a: \rho > 0\) (positive linear association). Compute the test statistic: \(t = r \sqrt{\frac{n-2}{1-r^2}}\) where \(n = 18\) is the sample size. Calculating gives \(t \approx 8.94\). Compare with critical value from \(t\)-distribution with \(n-2 = 16\) degrees of freedom at significance level 0.01, \(t_{0.01, 16} \approx 2.583\). Since \(t \approx 8.94 > 2.583\), we reject \(H_0\) and conclude there is a significant positive linear association.

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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 statistical method for modeling the relationship between a dependent variable and one or more independent variables. In simple linear regression, the goal is to find the best-fitting line through a set of data points for a pair of variables. In our study, the dependent variable is the percent fiber dry weight, and the independent variable is shear force.
Linear regression helps in understanding how a change in shear force can explain changes in fiber content of asparagus.
This approach involves identifying a linear equation, \[ y = mx + c \] where \( y \) is the predicted value of the dependent variable, \( m \) is the slope of the line, \( x \) is the independent variable, and \( c \) is the y-intercept of the line.
By fitting this linear model to the data, we can predict the fiber content based on the observed shear forces.

data analysis

Data analysis involves collecting, cleaning, transforming, and modeling data to draw conclusions and make decisions.
For this study on asparagus, data analysis began with determining shear force and fiber dry weight values from experimental observations.
These values allow researchers to understand the relationship between shear force and fiber content.
Analysis techniques such as calculating mean, variance, and correlation coefficients are essential to derive meaningful insights from raw data.

• Mean: Shows the average value, providing a central tendency for the data.
• Variance: Measures data spread, indicating variability around the mean.
• Correlation coefficient: Quantifies the degree of relationship among variables.

Through these methods, data analysis enables the prediction and control of asparagus quality attributes.

positive association

A positive association implies that as one variable increases, the other variable generally increases as well.
In the experiment, we found a strong positive association between shear force and percent fiber dry weight.
The correlation coefficient \( r \approx 0.922 \) supports this assertion, demonstrating that higher shear forces usually correspond with higher fiber content in the asparagus.
This positive trend helps in predicting changes, showing that improving or controlling shear force can have predictable effects on fiber attributes.

shear force

Shear force refers to the mechanical force that is applied parallel to a surface, in this case, the asparagus spears.
It is measured in kilograms in this experiment and is a physical property relevant to the toughness of the spears.
Higher shear force values indicate tougher spears, which correlate with higher fiber content.
Monitoring and analyzing shear force helps understand textural changes, allowing researchers to link physical properties with chemical content in the asparagus.

fiber content

Fiber content in asparagus spears refers to the proportion of fibrous material present, usually measured as a percentage of the dry weight.
It plays a critical role in determining the textural qualities of the asparagus, such as toughness.
In the study, fiber content is the dependent variable in linear regression, as changes in it are hypothesized to be explained by variations in shear force.
Understanding fiber content is vital, as it affects both the quality perception and the nutritional value of asparagus.

lignification

Lignification is the biological process where lignin, a complex organic polymer, is integrated into plant cell walls, resulting in increased rigidity and woodiness.
In the context of asparagus, lignification relates to the toughening process, with higher lignin content correlating with increased toughness and higher fiber content.
This study investigates how post-harvest treatments might manage lignification to improve asparagus quality.
By understanding lignification, producers can optimize storage techniques and treatment methods, keeping asparagus tender and consumer-friendly.

hypothesis testing

Hypothesis testing is a method used to decide whether there is enough evidence to support a specific hypothesis about a data set.
In this case, the hypothesis test aimed to determine if there is a positive linear relationship between shear force and fiber content.
The null hypothesis \( H_0: \rho = 0 \) states that there is no association, while the alternative hypothesis \( H_a: \rho > 0 \) suggests a positive association.
Using a \( t \)-test, we calculated a test statistic \( t = 8.94 \). Given the critical value \( t_{0.01, 16} \approx 2.583 \), we rejected \( H_0 \), indicating significant evidence for a positive linear association.
This statistical method confirms the observed trends and is vital for validating research findings.

variation explanation

Variation explanation pertains to how much of the variability in one variable can be explained by the variability in another variable through a relationship model.
In linear regression, this is quantified by the coefficient of determination, \( r^2 \).
In our asparagus study, \( r^2 \approx 0.850 \) indicates that 85% of the variability in fiber content can be explained by changes in shear force.
This high proportion shows a strong explanatory power of the linear model, suggesting that shear force is a major determinant of fiber variation.
Such insights are crucial for improving asparagus quality by controlling factors that influence fiber content effectively.