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Environmental Entomology The beet armyworm has become one of the most serious pests of the tomato in southern California. Eigenbrode and Trumble \({ }^{18}\) collected the data shown in the table that relates the nine-day survival of larva of this insect to the percentage of fruit damaged. \begin{tabular}{|c|ccccccc|} \hline\(x\) & 39.2 & 29.6 & 95.4 & 113.7 & 179.3 & 42.4 & 60.0 \\ \hline\(y\) & 0.5 & 0.1 & 4.9 & 4.0 & 10.0 & 5.3 & 1.8 \\ \hline \end{tabular} Here \(x\) is the nine-day weight of the larvae in milligrams, and \(y\) is the percentage of fruit damaged. a. Use linear regression to find the best-fitting line that relates the nine- day weight of the larvae to the percentage of fruit damaged. b. Find the correlation coefficient. c. Interpret what the slope of the line means.

Step by step solution

01

Gather and List the Data Points

We have the data points for the nine-day weight of the larvae \(x\) and the percentage of fruit damaged \(y\). The data points are as follows: \((39.2, 0.5), (29.6, 0.1), (95.4, 4.9), (113.7, 4.0), (179.3, 10.0), (42.4, 5.3), (60.0, 1.8)\).

02

Calculate the Mean of X and Y

Calculate the mean of the nine-day weight \(\bar{x}\) and the mean of the percentage of fruit damaged \(\bar{y}\). \[\bar{x} = \frac{39.2 + 29.6 + 95.4 + 113.7 + 179.3 + 42.4 + 60.0}{7},\ \bar{y} = \frac{0.5 + 0.1 + 4.9 + 4.0 + 10.0 + 5.3 + 1.8}{7}\]

03

Compute the Slope (m) of the Best-Fitting Line

Use the formula for the slope \(m\): \[m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\] Compute the values \((x_i - \bar{x})(y_i - \bar{y})\) and \((x_i - \bar{x})^2\) for each data point, sum them, and substitute them into the formula.

04

Calculate the Y-Intercept (b) of the Line

Apply the y-intercept formula: \[b = \bar{y} - m\bar{x}\] Substitute the mean values and the slope obtained from the previous step into this formula to find \(b\).

05

Formulate the Equation of the Line

The equation of the best-fitting line is \(y = mx + b\). Substitute the values of \(m\) and \(b\) to express the line equation.

06

Calculate the Correlation Coefficient (r)

The correlation coefficient \(r\) is given by: \[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \times \sum (y_i - \bar{y})^2}}\]. Compute both numerator and denominator parts of the formula and proceed with the division.

07

Interpret the Slope of the Line

The slope \(m\) represents the change in the percentage of fruit damaged per milligram of larvae weight gained over nine days. A positive slope would show an increase in fruit damage percentage with each milligram increase in larvae weight, whereas a negative slope would indicate a decrease.

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Key Concepts

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

Correlation Coefficient

In linear regression, the correlation coefficient, denoted as \( r \), is a numerical value that indicates the strength and direction of the linear relationship between two variables. It ranges from -1 to 1. Here's what these values mean:

• If \( r = 1 \), there is a perfect positive linear relationship between variables. This means that as one variable increases, the other does as well, following a straight line.
• If \( r = -1 \), there is a perfect negative linear relationship, meaning as one variable increases, the other decreases in a straight linear fashion.
• If \( r = 0 \), there is no linear relationship between the variables. However, they might still have a nonlinear relation.

A strong correlation close to -1 or 1 shows that the data points closely follow a linear trend, which is crucial in predictions. In this exercise, calculating \( r \) involves substituting the deviation products and squaring the deviations into the correlation coefficient formula. This measure is essential for determining how well the larvae weight and fruit damage percentages are related.

Slope Interpretation

The slope in a linear equation, represented by \( m \), describes the rate at which the dependent variable \( y \) (percentage of fruit damaged) changes concerning the independent variable \( x \) (weight of larvae). It is calculated from the line's equation \( y = mx + b \).

Here's how you can interpret its meaning:

• If the slope is positive, an increase in the larvae weight will likely correspond to an increase in the fruit damage percentage.
• If the slope is negative, an increase in larvae weight indicates a decrease in fruit damage.
• A larger slope value suggests a more significant change in the percentage of damage for each milligram increase in larvae weight.

In applied settings like this, understanding the slope's implication helps in predicting outcomes. For example, if the slope value is 0.05, it indicates that every additional milligram in larvae weight corresponds to a 5% increase in fruit damage, helping us understand the pest's impact effectively.

Problem Solving

Solving linear regression problems involves several steps to arrive at a reliable prediction model. Here's a straightforward approach:

• First, gather and arrange your data as given. This involves identifying the independent and dependent variables, such as the larvae weight and percentage of fruit damage.
• Calculate the mean for both \( x \) and \( y \) values. This average helps in determining deviations and is crucial for finding the slope and intercept.
• Compute both the slope \( m \) and y-intercept \( b \) using the formulas and calculate the best-fitting line, which predicts \( y \) based on \( x \).
• Determine the correlation coefficient \( r \), which validates how strongly your data follow the linear regression line.
• Finally, interpret the results. Understanding the slope's meaning in practical terms provides insights into the relationship between variables.

By following these steps methodically, you solve regressions systematically, making it easier to handle real-world data and derive meaningful conclusions from your analysis.