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Whole crop weight versus rice weight: A study by Horie \({ }^{24}\) compared the dry weight \(B\) of brown rice with the dry weight \(W\) of the whole crop (including stems and roots). These data, in tons per hectare, \({ }^{25}\) for the variety Nipponbare grown in various environmental conditions are partially presented in the table below. $$ \begin{array}{|c|c|} \hline \begin{array}{|c|} \hline \text { Whole crop } \\ \text { weight } W \end{array} & \text { Rice weight } B \\ \hline 6 & 1.8 \\ \hline 11.1 & 3.2 \\ \hline 13.7 & 3.7 \\ \hline 14.9 & 4.3 \\ \hline 17.6 & 5.2 \\ \hline \end{array} $$ a. Find an approximate linear model for \(B\) as a function of \(W\). b. Which sample might be considered to have a lower rice weight than expected from the whole crop weight? c. How much additional rice weight can be expected from 1 ton per hectare of additional whole crop weight?

Step by step solution

01

Identify the Data Points

From the table, we extract the data points as coordinate pairs \((W, B)\): \((6, 1.8), (11.1, 3.2), (13.7, 3.7), (14.9, 4.3), (17.6, 5.2)\). These represent the dry weights of the whole crop and rice respectively.

02

Calculate the Slope of the Best-Fit Line

To find the slope \(m\) of the line, use the formula for the slope given two points: \[m = \frac{\sum (W_i - \bar{W})(B_i - \bar{B})}{\sum (W_i - \bar{W})^2}\]. Firstly, calculate the means \(\bar{W} = 12.66\) and \(\bar{B} = 3.64\). Then, perform the calculations for the sums, yielding \(m \approx 0.265\).

03

Determine the Y-intercept

Use the average values calculated and the slope to determine the y-intercept \(b\) of the line with the formula \(b = \bar{B} - m \bar{W}\). Substituting the values, we find \(b \approx 0.303\).

04

Formulate the Linear Model

Using the slope \(m\) and the y-intercept \(b\), the linear relationship is expressed as: \(B = 0.265W + 0.303\).

05

Analyze Rice Weight vs. Expected Values

For each \(W\), calculate the expected \(B\) based on the linear model. Compare actual \(B\) to expected \(B\):- For \(W = 11.1\), expected \(B = 3.24\); actual \(B = 3.2\), slightly lower than expected, making it the sample with lower rice weight than expected.

06

Calculate Additional Rice from Whole Crop

From the slope of the model \(m = 0.265\), we determine that each additional ton per hectare of the whole crop is expected to increase the rice weight by approximately 0.265 tons per hectare.

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Key Concepts

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

Data Analysis

Data analysis is the careful examination, transformation, and organization of collected data with the objective of discovering useful information. For this exercise, it is essential to understand the link between different weights measured in tons per hectare for agricultural fields. By closely looking at these numbers, you can start to identify patterns or correlations between the weights of the whole crop and the rice.

• The primary data to analyze are the dry weights of both whole crops and rice.
• The relationship between these two variables is key to understanding the problem.

Starting with data extraction, take out the information given as coordinate pairs like o (6, 1.8) for easy handling.

Next comes the calculation of means, variances, and other statistical metrics. These metrics help in gaining insights into the dispersion and central tendency of the data. After data preparation, forming a mathematical model becomes easier, allowing you to see potential trends and dependencies.

Linear Regression

Linear regression is a statistical method used to model the relationship between two quantitative variables. In this context, it helps in predicting the rice weight \( B \) given the whole crop weight \( W \). This approach is particularly valuable because:

• It provides a simple yet powerful model for understanding how changes in one variable influence another.
• The simplicity of linear regression makes it easily interpretable.

To construct the linear model, you start with calculating the slope \( m \) of the line. This is achieved using the formula:\[ m = \frac{\sum (W_i - \bar{W})(B_i - \bar{B})}{\sum (W_i - \bar{W})^2} \]Once the slope is known, use it to calculate the y-intercept \( b \) using \( b = \bar{B} - m \bar{W} \).

The linear model ultimately is:\[ B = 0.265W + 0.303 \]Such a model allows prediction of rice yield based on known whole crop weights, while also providing insight into how well the model fits the actual data observed.

Agricultural Yield Modeling

Agricultural yield modeling involves predicting or understanding the outcome of planting and harvesting in farming. In this problem, we are focusing on how the weight of the entire crop, which includes stems and roots, relates to the yield of rice.

• This helps farmers make informed decisions on how to optimize growth conditions to maximize yield.
• Poorly informed decisions could mean under or overestimating yield, which could lead to economic losses.

For practical purposes, the regression model aids in predicting the increase in rice weight with each additional ton of whole crop weight. Specifically, our model indicates that an additional ton per hectare of the whole crop is expected to increase the rice weight by approximately \(0.265 \) tons per hectare.

Such predictive capabilities are invaluable, especially in contexts where planning and resource allocation are crucial for successful harvests. Also, understanding deviations (such as with the sample at \( W = 11.1 \)) can lead to insights on other influencing factors like soil quality or water availability, paving the way for more refined models.