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 Horticultural 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. $$ \begin{array}{l|ccccccccc} x & 46 & 48 & 55 & 57 & 60 & 72 & 81 & 85 & 94 \\ \hline y & 2.18 & 2.10 & 2.13 & 2.28 & 2.34 & 2.53 & 2.28 & 2.62 & 2.63 \\ x & 109 & 121 & 132 & 137 & 148 & 149 & 184 & 185 & 187 \\ \hline y & 2.50 & 2.66 & 2.79 & 2.80 & 3.01 & 2.98 & 3.34 & 3.49 & 3.26 \end{array} $$ \(n=18, \quad \sum x_{i}=1950, \quad \sum x_{i}^{2}=251,970\) \(\sum y_{i}=47.92, \quad \sum y_{i}^{2}=130.6074, \quad \sum x_{i} y_{i}=5530.92\) 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

Calculate the Mean Values

First, calculate the mean values of shear force \(x\) and percent fiber dry weight \(y\) using the formulas \(\bar{x} = \frac{\sum x_i}{n}\) and \(\bar{y} = \frac{\sum y_i}{n}\). For \(x\), \(\bar{x} = \frac{1950}{18} = 108.33\), and for \(y\), \(\bar{y} = \frac{47.92}{18} = 2.662\).

02

Calculate the Sample Covariance

Using the provided sums, calculate the covariance \( S_{xy} \) with the formula: \[ S_{xy} = \frac{\sum (x_iy_i) - n\bar{x}\bar{y}}{n-1} \] \Substituting the values, \( S_{xy} = \frac{5530.92 - 18(108.33)(2.662)}{17} = 48.43\).

03

Calculate the Sample Variances

Calculate the variances \( S_{xx} \) and \( S_{yy} \) with \( S_{xx} = \frac{\sum x_i^2 - n\bar{x}^2}{n-1} \) and \( S_{yy} = \frac{\sum y_i^2 - n\bar{y}^2}{n-1} \). \[ S_{xx} = \frac{251970 - 18(108.33)^2}{17} = 1885.27 \] \[ S_{yy} = \frac{130.6074 - 18(2.662)^2}{17} = 0.18 \]

04

Calculate the Sample Correlation Coefficient

The sample correlation coefficient \( r \) is given by \( r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \). Substituting calculated values: \[ r = \frac{48.43}{\sqrt{1885.27 \times 0.18}} = 0.999\]. This implies a very strong positive linear relationship between shear force and percent fiber dry weight.

05

Analyze the Nature of Relationship

Since \( r = 0.999 \), the relationship between shear force and percent fiber dry weight is a very strong positive linear association.

06

Understand the Influence of Shear Force

If a first specimen has a larger shear force than a second specimen, the first specimen tends to have a higher percent dry fiber weight, due to the positive correlation.

07

Effect of Unit Change on Correlation

Changing the units of shear force from kg to pounds will not affect the calculated correlation coefficient \( r \) because correlation is a dimensionless quantity.

08

Calculate Proportion of Variation Explained

The proportion of variation in \( y \) that is explained by \( x \) is \( r^2 = (0.999)^2 = 0.998 \). This indicates that 99.8% of the variation is explained by the linear model.

09

Conduct the Hypothesis Test

To test for a positive linear association, use the formula for the t-statistic: \[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \] With \( r = 0.999 \), \( n = 18 \), \[ t = \frac{0.999\sqrt{16}}{\sqrt{1 - (0.999)^2}} = 158.56 \] Compare with \( t_{0.01,16} \), showing strong evidence of a positive correlation.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between a dependent variable and one or more independent variables. In the context of the given exercise, linear regression would help to identify how the shear force ( x ) affects the fiber dry weight ( y ) of asparagus. By fitting a line to the data points, one can predict the outcome for a given input of shear force.
This method assumes that the relationship between the variables is linear, meaning as one variable increases, the other also tends to increase or decrease in a consistent manner. Linear regression provides the equation of a line, y = mx + c , where m is the slope indicating the direction and steepness of the line, and c is the intercept.

• The goal is to minimize the difference between the observed and predicted values. This difference is known as the residual.
• Linear regression can determine how changes in the independent variable impact the dependent variable.
• The strength of this impact is quantified through the correlation coefficient and the proportion of variance explained by the model (r-squared), as shown in the exercise with the value of 0.998, indicating a strong relationship.

Covariance

Covariance is a measure of how much two random variables change together. In the exercise, covariance was calculated to assess how shear force and fiber dry weight move in relation to each other. A positive covariance, as indicated by the calculation, signifies that as the shear force increases, the percent of fiber dry weight also tends to increase.
Covariance between two variables x and y is calculated as \[ S_{xy} = \frac{\sum (x_i y_i) - n\bar{x}\bar{y}}{n-1}\] where x_i and y_i are individual sample points, n is the number of sample points, and \( \bar{x} \), \( \bar{y} \) are means of x and y respectively.

• Covariance is susceptible to the scale of the variables. Larger numerical values can lead to larger covariance values.
• Unlike correlation, covariance does not provide limits or a standardized measure of strength.
• The units of covariance are the product of the units of the two variables measured, making it less intuitive than correlation.

Shear Force

Shear force describes the force required to cut through asparagus fibers. In horticultural studies, like the one mentioned, it helps evaluate the texture and edibility of the asparagus spears. It's an important determinant of what makes the vegetable more appealing to consumers.
Understanding shear force is crucial as it directly impacts the perception of "toughness" in asparagus. Typically, a higher shear force implies tougher and potentially less desirable vegetable quality. This metric is not just significant for consumer satisfaction but also for agricultural practices that aim to optimize crop quality.
The data given shows various measurements of shear force in kilograms. These figures are tested for their effects on fiber dry weight, establishing a scientific basis for quality control.

Fiber Dry Weight

Fiber dry weight represents the percentage of fiber content in asparagus after drying. It is used as a measure of fibrousness, which is an integral aspect of asparagus quality.

• A higher fiber content means the asparagus has a harder texture, which can be less desirable.
• In the study, fiber dry weight as a percentage shows the extent of fibrousness in relation to the overall weight, once water content is removed.
• This measurement is vital for understanding how different growing and post-harvest management can affect asparagus quality.

By correlating fiber dry weight with shear force, researchers can deduce patterns and relationships in how the asparagus toughens over time or post-harvest treatments.

Hypothesis Testing

In the context of the study, hypothesis testing is used to determine if there is a statistically significant correlation between shear force and fiber dry weight in asparagus. The hypothesis typically starts with a null hypothesis stating that there is no correlation, against an alternative hypothesis suggesting a correlation exists.
At a significance level of 0.01, the t-statistic is calculated to determine the p-value, which helps in deciding whether to reject the null hypothesis. The exercise uses the formula \[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \] with calculated sample correlation coefficient r and sample size n .

• A t-value that exceeds the critical value suggests rejecting the null hypothesis, confirming the presence of a positive relationship.
• In this exercise, the t-statistic was extremely high (158.56), strongly indicating a significant positive association.
• Hypothesis testing provides a framework for making informed decisions based on statistical evidence.