/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Plant Resistance Talekar and \(\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 24

Plant Resistance Talekar and \(\operatorname{Lin}^{20}\) collected the data shown in the table that relates the pod diameter (seed size) of soybeans to the percentage of pods damaged by the lima bean pod borer. \begin{tabular}{|c|cccccc|} \hline\(x\) & 3.1 & 3.4 & 3.9 & 4.1 & 4.2 & 4.5 \\ \hline\(y\) & 12 & 28 & 38 & 37 & 44 & 48 \\ \hline \end{tabular} Here \(x\) is the pod diameter in millimeters, and \(y\) is the percentage of damaged pods. a. Use linear regression to find the best-fitting line that relates the pod diameter to the percentage of pods damaged. b. Find the correlation coefficient. c. Interpret what the slope of the line means.

Short Answer

Expert verified
The best-fitting line is approximately \( y = 14.42x - 21.37 \) with a correlation coefficient of 0.97, indicating a strong positive correlation. The slope means an increase in pod diameter by 1 mm increases damage by 14.42%.

Step by step solution

01

Organize the Data

First, list the given data points clearly. We have the x-values representing the pod diameter (3.1, 3.4, 3.9, 4.1, 4.2, 4.5) and the y-values representing the percentage of pods damaged (12, 28, 38, 37, 44, 48).
02

Calculate the Means

Calculate the mean of the x-values and the y-values:\[\bar{x} = \frac{3.1 + 3.4 + 3.9 + 4.1 + 4.2 + 4.5}{6} = 3.87\]\[\bar{y} = \frac{12 + 28 + 38 + 37 + 44 + 48}{6} = 34.5\]
03

Calculate the Slope and Intercept

The formula for the slope \( m \) in linear regression is:\[m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}\]Calculating numerators and denominators, we find approximately:\[m \approx 14.42\]The y-intercept \( b \) is calculated using:\[b = \bar{y} - m \cdot \bar{x} \approx 34.5 - 14.42 \cdot 3.87 = -21.37\]So, the equation of the best-fitting line is approximately:\[y = 14.42x - 21.37\]
04

Calculate the Correlation Coefficient

The correlation coefficient \( r \) is calculated as follows:\[r = \frac{n \sum{(x_i y_i)} - \sum{x_i} \sum{y_i}}{\sqrt{\left[n\sum{x_i^2} - (\sum{x_i})^2\right] \left[n\sum{y_i^2} - (\sum{y_i})^2\right]}}\]Plugging in the values, we calculate an approximate \( r \) value of 0.97. This value indicates a strong positive correlation.
05

Interpret the Slope

The slope of 14.42 in the line \(y = 14.42x - 21.37\) suggests that for each millimeter increase in pod diameter, the percentage of pods damaged increases by approximately 14.42%. This indicates a positive relationship between pod diameter and damage percentage.

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 91影视!

Key Concepts

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

Correlation Coefficient
When exploring the relationship between two variables, such as the pod diameter of soybeans and the percentage of pods damaged, the correlation coefficient is an essential statistic. Its value, denoted as \( r \), measures the strength and direction of a linear relationship:
  • Values range between -1 and 1.
  • \( r = 1 \) indicates a perfect positive correlation, where variables increase together.
  • \( r = -1 \) reflects a perfect negative correlation, where one variable increases as the other decreases.
  • \( r = 0 \) suggests no correlation at all.
In our case, the calculated correlation coefficient is approximately 0.97, indicating a very strong positive correlation. This suggests that as the pod diameter increases, the percentage of pods damaged also tends to increase.In educational settings, understanding the correlation coefficient is vital for students analyzing data in mathematics. It allows them to determine the reliability of predictions made from the data, assisting in hypotheses formation and testing.
Slope Interpretation
In the context of linear regression, the slope of the line provides valuable insight into the relationship between the independent and dependent variables. The slope here is calculated to be approximately 14.42.
  • It signifies the rate at which \( y \) changes with respect to \( x \).
  • For every unit increase in pod diameter (\( x \)), the percentage of pods damaged (\( y \)) increases by roughly 14.42%.
This interpretation of the slope is crucial in educational data analysis, as it gives a clear picture of how one factor might impact another. Students can see how small changes in one variable can lead to significant differences in the outcome.
Data Analysis in Mathematics
Data analysis in mathematics focuses on interpreting and drawing conclusions from data. Linear regression is a common method used to understand relationships between variables, as exemplified by our soybean study.
  • Collecting Data: Accurately collecting and organizing data is the first crucial step that influences the entire analysis.
  • Analyzing Relationships: Tools like the correlation coefficient and regression slope provide students with tangible ways to evaluate associations between variables.
  • Formulating Conclusions: Data analysis culminates in the ability to make informed conclusions and predictions that are supported by statistical evidence.
Educational exercises involving data analysis are vital. They help students improve critical thinking skills by learning to make data-driven decisions. By practicing these techniques, students can improve their ability to interpret complex mathematical relationships in varied real-world contexts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Ecological Entomology Elliott \(^{37}\) studied the temperature effects on the alder fly. In 1967 he collected the data shown in the following table relating the temperature in degrees Celsius to the number of pupae successfully completing pupation. $$ \begin{array}{|c|cccccc|} \hline t & 8 & 10 & 12 & 16 & 20 & 22 \\ \hline y & 18 & 27 & 43 & 44 & 37 & 5 \\ \hline \end{array} $$ a. Use quadratic regression to find \(y\) as a function of \(t\). b. Determine the temperature at which this model predicts the maximum number of successful pupations. c. Determine the two temperatures at which this model predicts there will be no successful pupation.

Cellular Telephones The following table gives the number of people with cellular telephone service for recent years and can be found in Glassman. \({ }^{59}\) Number is in millions. $$ \begin{array}{|l|ccccc|} \hline \text { Year } & 1984 & 1985 & 1986 & 1987 & 1988 \\ \hline \text { Number } & 0.2 & 0.5 & 0.8 & 1.4 & 2.0 \\ \hline \text { Year } & 1989 & 1990 & 1991 & 1992 & 1993 \\ \hline \text { Number } & 3.8 & 5.7 & 8 & 11 & 13.8 \\ \hline \end{array} $$a. On the basis of this data, find the best-fitting exponential function using exponential regression. Let \(x=0\) correspond to 1984 . Graph. Use this model to estimate the numbers in 1997 . b. Using the model in part (a), estimate when the number of people with cellular telephone service will reach 50 million

uarez-Villa and Karlsson \(^{47}\) studied the relationship between the sales in Sweden's electronic industry and production costs (per unit value of product sales). Their data are presented in the following table. $$\begin{array}{|c|ccccc|}\hline x & 7 & 13 & 14 & 15 & 17 \\\\\hline y & 0.91 & 0.72 & 0.91 & 0.81 & 0.72 \\\\\hline x & 20 & 25 & 27 & 35 & 45 \\\\\hline y & 0.90 & 0.77 & 0.65 & 0.73 & 0.70 \\\\\hline x & 45 & 63 & 65 & 82 & \\\\\hline y & 0.78 & 0.82 & 0.92 & 0.96 & \\\\\hline\end{array}$$ Here \(x\) is product sales in millions of krona, and \(y\) is production costs per unit value of product sales. a. Use cubic regression to find the best-fitting cubic to the data and the correlation coefficient. Graph. b. Find the minimum production costs.

Cost Curve Johnston \(^{12}\) made a statistical estimation of the cost-output relationship for 40 firms. The data for the fifth firm is given in the following table. \begin{tabular}{|c|ccccc|} \hline\(x\) & 180 & 210 & 215 & 230 & 260 \\ \hline\(y\) & 130 & 180 & 205 & 190 & 215 \\ \hline\(x\) & 290 & 340 & 400 & 405 & 430 \\ \hline\(y\) & 220 & 250 & 300 & 285 & 305 \\ \hline\(x\) & 430 & 450 & 470 & 490 & 510 \\ \hline\(y\) & 325 & 330 & 330 & 340 & 375 \\ \hline \end{tabular} Here \(x\) is the output in millions of units, and \(y\) is the total cost in thousands of pounds sterling. a. Determine the best-fitting line using least squares and the correlation coefficient. Graph the line. b. What does this model predict the total cost will be when the output is 300 million units? c. What does this model predict the output will be if the total cost is 200,000 of pounds sterling?

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 \(\$ 30.00 ?\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.