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Chapter 2: Problem 9

Population Ecology Wagner \(^{31}\) collected the data found in the following table relating the percent mortality \(y\) of eggs of the sweet-potato whitefly and the temperature \(T\) in degrees Celsius. $$\begin{array}{|c|cccccccc|}\hline T & 20 & 22 & 24 & 26 & 28 & 30 & 32 & 34 \\\\\hline y & 7 & 6 & 3.5 & 5 & 7 & 4 & 6.8 & 18 \\\\\hline\end{array}$$ Use quartic regression to find the best-fitting fourth-order polynomial to the data and the correlation coefficient. Graph.

Short Answer

Expert verified
Use statistical software for quartic regression to find a fourth-degree polynomial and correlation coefficient \( R \), then graph the data and curve.

Step by step solution

01

Understand the Problem

We need to perform a quartic regression analysis on the given data. This means we want to find a polynomial of degree 4 that fits the relationship between temperature \( T \) and egg mortality \( y \). Additionally, we will calculate the correlation coefficient to understand how well the model fits the data.
02

Data Organization

We start by organizing our data into two vectors: \( T = [20, 22, 24, 26, 28, 30, 32, 34] \) and \( y = [7, 6, 3.5, 5, 7, 4, 6.8, 18] \). These represent the independent and dependent variables respectively.
03

Perform Quartic Regression

Using statistical software or a graphing calculator, input the vectors \( T \) and \( y \). Select `Quartic Regression` (or polynomial regression with degree 4) from the analysis options. The software will output a quartic polynomial of the form \( y = aT^4 + bT^3 + cT^2 + dT + e \), where \( a, b, c, d, \) and \( e \) are coefficients that minimize the difference between the data points and the model.
04

Calculate the Correlation Coefficient

Most statistical tools will provide a correlation coefficient, \( R \), along with the polynomial. \( R \) indicates the strength and direction of the relationship between \( T \) and \( y \). An \( R \) value close to 1 or -1 suggests a strong relationship, while a value near 0 suggests a weak relationship.
05

Plot the Data and Regression Curve

Plot the temperature values \( T \) on the x-axis and mortality rates \( y \) on the y-axis. Then, overlay the quartic polynomial curve on this plot. Whether you use software or manually sketch the graph, this step helps visualize how closely the polynomial fits the actual data.

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Key Concepts

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

Population Ecology
Population ecology is the study of how species populations interact with their environment. In this case, the focus is on the percent mortality of eggs of the sweet-potato whitefly and its relationship with temperature. This topic is crucial because it helps scientists understand factors affecting population sizes and survival rates. For example, higher temperatures might increase mortality rates, affecting the whitefly's population. By analyzing this data using mathematical methods like quartic regression, scientists can predict trends and make informed decisions about pest control and ecological balance.
Correlation Coefficient
The correlation coefficient, often represented as \( R \), is a numerical measure that describes the strength and direction of a linear relationship between two variables. In the context of regression analysis, \( R \) helps determine how well the polynomial model fits the real-world data.

- A correlation coefficient close to 1 indicates a strong positive relationship.
- A correlation coefficient close to -1 indicates a strong negative relationship.
- A coefficient near 0 suggests a weak or no linear relationship.

For this exercise, evaluating \( R \) provides insight into how temperature changes influence egg mortality, allowing researchers to see if their model successfully captures the underlying patterns.
Polynomial Regression
Polynomial regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables using a polynomial equation. Specifically, a quartic polynomial regression involves a polynomial of degree four. The general form of a quartic polynomial is given by:
\[ y = aT^4 + bT^3 + cT^2 + dT + e \]
This type of regression is capable of capturing more complex relationships, ideal for data like ours where a straight line does not fit the data well. Using quartic regression in population ecology studies allows for detailed modeling of how variables such as temperature affect species mortality, giving a clearer understanding of biological phenomena.
Data Organization
Organizing data is a critical first step in any analysis because it lays a strong foundation for accurate modeling. In this exercise, data organization involves organizing temperature and mortality rate data into vectors.

The vector for temperature \( T = [20, 22, 24, 26, 28, 30, 32, 34] \) and the vector for mortality rates \( y = [7, 6, 3.5, 5, 7, 4, 6.8, 18] \) allow for clear visualization and understanding.

Proper data organization facilitates input into statistical tools that then perform operations like polynomial regression, enabling an easy and insightful analysis process. It ensures that researchers can approach complex phenomena methodically, thus extracting meaningful correlations from the chaos of raw data.

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Most popular questions from this chapter

Population Ecology Lactin and colleagues \(^{35}\) collected the following data relating the feeding rate \(y\) of the first-instar Colorado potato beetle and the temperature \(T\) in degrees Celsius. $$ \begin{array}{|c|ccccccc|} \hline T & 14 & 17 & 20 & 23 & 29 & 32 & 38 \\\ \hline y & 0.15 & 0.35 & 1.05 & 1.15 & 1.55 & 1.55 & 1.45 \\ \hline \end{array} $$ a. Use quadratic regression to find \(y\) as a function of \(T\). b. Find the temperature for which the feeding rate is maximum.

Productivity Bernstein \(^{14}\) also studied the correlation between investment as a percent of GNP and productivity growth of six countries: France (F), Germany (G), Italy (I), Japan (J), the United Kingdom (UK), and the United States (US). Productivity is given as output per employeehour in manufacturing. The data they collected for the years \(1960-1977\) is given in the following table. \begin{tabular}{|c|cccccc|} \hline Country & US & UK & I & F & G & J \\ \hline\(x\) & 17 & 18 & 22 & 23 & 24 & 34 \\ \hline\(y\) & 2.8 & 3.0 & 5.6 & 5.5 & 5.8 & 8.3 \\ \hline \end{tabular} Here \(x\) is investment as percent of GNP, and \(y\) is the productivity growth (\%). a. Determine the best-fitting line using least squares and the correlation coefficient. b. What does this model predict the productivity growth will be when investment is \(20 \%\) of GNP? c. What does this model predict investment as a percent of GNP will be if productivity growth is \(7 \% ?\)

Wylie and Ezekiel \(^{43}\) obtained actual data from the United States Steel Corporation relating the percent of capacity at which the plants operated to cost per ton of steel. The following table gives their data. $$\begin{array}{|c|ccccc|}\hline x & 20 & 30 & 40 & 50 & 60 \\\\\hline y & 33.20 & 27.50 & 24.40 & 22.50 & 21.45 \\ \hline x & 70 & 80 & 90 & 100 & \\\\\hline y & 20.80 & 20.35 & 20.10 & 19.95 & \\\\\hline\end{array}$$ Here \(x\) is the percent of capacity at which the plants operated, and \(y\) is cost per ton. a. Use power regression to find the best-fitting power function to the data and the correlation coefficient. Graph. b. What does this model predict the cost per ton will be when plants are operated at \(75 \% ?\) c. What does this model predict the operating percent of capacity will be if the cost per ton is \(\$ 30.00 ?\)

Population of Northeast The table gives the population in millions of the northeastern part of the United States for some selected early years. \(^{70}\) a. On the basis of the data given for the years \(1790-1890\), find the best- fitting exponential function using exponential regression. Determine the correlation coefficient. Graph. Using this model, estimate the population in \(1990 .\) b. Now find the best-fitting logistic curve. Graph. Using this model, estimate the population in \(1990 .\) Note that the actual population of the Northeast in 1990 was 50.8 million. $$ \begin{array}{|l|ccc|} \hline \text { Year } & 1790 & 1810 & 1830 \\ \hline \text { Population } & 2.0 & 3.5 & 5.5 \\ \hline \text { Year } & 1850 & 1870 & 1890 \\ \hline \text { Population } & 8.6 & 12.3 & 17.4 \\ \hline \end{array} $$

Nordin \(^{49}\) obtained the following data relating output and total final cost for an electric utility in Iowa. (Instead of using the original data set for 541 eight-hour shifts, we have just given the array means.) $$\begin{array}{|l|llllllll}\hline x & 25 & 28 & 33 & 38 & 41 & 45 & 50 & 53 \\\\\hline y & 24 & 24 & 26 & 28 & 27 & 31 & 34 & 38 \\\\\hline x & 57 & 63 & 66 & 71 & 74 & 79 & 83 & 89 \\\\\hline y & 39 & 42 & 43 & 51 & 53 & 52 & 54 & 52 \\\\\hline\end{array} $$ Here \(x\) is the output as a percent of capacity, and \(y\) is final total cost in dollars (multiplied by a constant not given to avoid showing the exact level of the costs.) Use cubic regression to find the best-fitting cubic to the data and the correlation coefficient. Graph. Also find the correlation coefficients associated with both linear and quadratic regression, and compare them to that found for cubic regression.
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