91Ó°ÊÓ

Horticultural Entomology The brown citrus aphid and the melon aphid both spread the citrus tristeza virus to fruit and thus have become important pests. Yokomi and coworkers \({ }^{19}\) collected the data found in the following table. \begin{tabular}{|c|cccc|} \hline\(x\) & 1 & 5 & 10 & 20 \\ \hline\(y\) & 25 & 22 & 50 & 85 \\ \hline\(z\) & 10 & 5 & 18 & 45 \\ \hline \end{tabular} Here \(x\) is the number of aphids per plant, and \(y\) and \(z\) are the percentage of times the virus is transmitted to the fruit for the brown and melon aphid, respectively. a. Use linear regression for each aphid to find the bestfitting line that relates the number of aphids per plant to the percentage of times the virus is transmitted to the fruit. b. Find the correlation coefficients. c. Interpret what the slope of the line means in each case. d. Which aphid is more destructive? Why?

Step by step solution

01

Organize the Data

We have two sets of data: \( (x, y) \) for the brown citrus aphid and \( (x, z) \) for the melon aphid, where \( x \) is the number of aphids per plant. Rewrite the data points as pairs: \( (1, 25), (5, 22), (10, 50), (20, 85) \) for \( (x, y) \) and \( (1, 10), (5, 5), (10, 18), (20, 45) \) for \( (x, z) \).

02

Calculate the Line of Best Fit for Brown Citrus Aphid

Apply the linear regression formula to the data set \( (x, y) \). The formula for the slope \( m \) is \( m = \frac{N(\sum{xy}) - (\sum{x})(\sum{y})}{N(\sum{x^2}) - (\sum{x})^2} \) and for the y-intercept \( b \) is \( b = \frac{(\sum{y}) - m(\sum{x})}{N} \). Calculate these values to determine the line \( y = mx + b \).

03

Calculate the Line of Best Fit for Melon Aphid

Apply the same linear regression process to the \((x, z)\) dataset to find the equation of the line \( z = mx + b \). Follow the same steps as in Step 2.

04

Find the Correlation Coefficients

Calculate the Pearson correlation coefficient \( r \) for both datasets. Use the formula \( r = \frac{N(\sum{xy}) - (\sum{x})(\sum{y})}{\sqrt{[N\sum{x^2} - (\sum{x})^2][N\sum{y^2} - (\sum{y})^2]}} \) for the brown citrus aphid, and a similar formula for the melon aphid using \( z \) values.

05

Interpret the Slopes

The slope in each linear equation represents the change in the percentage of virus transmission per additional aphid. Calculate the slope from the equations found in Steps 2 and 3 and interpret them: a higher slope implies a greater increase in virus transmission for each additional aphid.

06

Determine the More Destructive Aphid

Compare both the slopes and correlation coefficients obtained in Steps 3 and 4. A higher slope and/or a stronger correlation suggest higher destructiveness. Thus, interpret which aphid is more destructive.

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.

Correlation Coefficient

The correlation coefficient is a statistical measure that helps us understand the strength and direction of the relationship between two variables. In our context, it explains how strongly the number of aphids per plant (\( x \)) is related to the virus transmission percentage (\( y \) or \( z \)). A correlation coefficient, denoted as \( r \), ranges from -1 to 1.

• If \( r = 1 \), there is a perfect positive linear relationship.
• If \( r = -1 \), there is a perfect negative linear relationship.
• If \( r = 0 \), there's no linear relationship at all.

In research like this, higher absolute values indicate stronger relationships between the two variables. In our exercise, calculating the correlation coefficient for both aphids provides insights into how consistently the number of aphids correlates with the increased percentage of virus transmission.

Horticultural Entomology

Horticultural entomology is a branch of science focusing on the study of insects that affect garden crops or fruit plants. In this context, we're looking at two particular pests: the brown citrus aphid and the melon aphid. Both are known to spread the citrus tristeza virus, causing considerable damage to fruits like oranges and grapefruits.

These insects are threats because they directly influence the health and yield of the plants they infest. In managing horticultural pests, scientists like Yokomi and coworkers use data to monitor and predict the severity of pest-related issues in crops. Understanding how these pests spread the virus can lead to better prevention and control strategies. By analyzing patterns, researchers aim to mitigate the destructive impact of these aphids on fruit production, ensuring healthier crops and plant sustainability.

Slope Interpretation

In linear regression, the slope of the line represents the relationship between the two variables plotted on the graph. Specifically, it describes how much the dependent variable (\( y \) or \( z \)) changes for each unit increase in the independent variable (\( x \)). In this exercise, the slope indicates the change in virus transmission percentage for each additional aphid per plant.

• A positive slope indicates that as the number of aphids increases, the percentage of virus transmission also increases.
• A steeper slope means that the virus transmission rate increases more significantly with each added aphid.

By comparing the slopes for the two datasets, we gain insights into which aphid has a more significant impact per unit increase. The interpretation helps in determining the more potent transmitter of the virus, which poses a greater threat to plant health.