/*! 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 32 Northern flying squirrels eat li... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 32

Northern flying squirrels eat lichen and fungi, which makes for a relatively low quality diet. The authors of the paper "Nutritional Value and Diet Preference of Arboreal Lichens and Hypogeous Fungi for Small Mammals in the Rocky Mountain" (Canadian Journal of Zoology [2008]: 851-862) measured nitrogen intake and nitrogen retention in six flying squirrels that were fed the fungus Rhizopogon. Data read from a graph that appeared in the paper are given in the table below. (The negative value for nitrogen retention for the first squirrel represents a net loss in nitrogen.) \begin{tabular}{cc} Nitrogen Intake. \(x\) (grams) & Nitrogen Retention. \(y\) (grams) \\ \hline \(0.03\) & \(-0.04\) \\ \(0.10\) & \(0.00\) \\ \(0.07\) & \(0.01\) \\ \(0.06\) & \(0.01\) \\ \(0.07\) & \(0.04\) \\ \(0.25\) & \(0.11\) \\ \hline \end{tabular} a. Construct a scatterplot of these data. b. Find the equation of the least-squares regression line. Based on this line, what would you predict nitrogen retention to be for a flying squirrel whose nitrogen intake is \(0.06\) grams? What is the residual associated with the observation \((0.06,0.01) ?\) c. Look again at the scatterplot from Part (a). Which observation is potentially influential? Explain the reason for your choice. d. When the potentially influential observation is deleted from the data set, the equation of the leastsquares regression line fit to the remaining five observations is \(\hat{y}=-0.037+0.627 x\). Use this equation to predict nitrogen retention for a flying squirrel whose nitrogen intake is \(0.06\). Is this prediction much different than the prediction made in Part (b)?

Short Answer

Expert verified
The actual numerical values for variables a, b, residuals and nitrogen predictions depend on the data analysis tool used for computing. After following these steps, you will have constructed a scatterplot, completed regression analysis, identified potentially influential data, and made predictions based on the given dataset without including the potentially influential observation.

Step by step solution

01

Construct a scatterplot

Using a graphing tool, each x,y point in the table represents a point on the scatterplot. The x-axis will have nitrogen intake and the y-axis will have nitrogen retention. Plot all six points on this scatterplot.
02

Calculate least-squares regression line

The least-squares regression line is calculated using statistical software or calculator, which minimizes the sum of the squared residuals. The general form of the equation is \(y = a + bx\), where `b` is the slope and `a` the y-intercept. This line will allow us to predict nitrogen retention (`y`) based on nitrogen intake (`x`).
03

Predict nitrogen retention

Using the y=a+bx formula acquired from step 2, substitute x with 0.06. The resulting `y` value will be the predicted nitrogen retention for a squirrel with a nitrogen intake of 0.06 g.
04

Calculate the residual

The residual associated with observation (0.06, 0.01) is the observed y value minus the predicted y value from the regression line.
05

Identify influential observations

Reviewing the scatterplot from step 1, identify the observation that seems to have an outside influence as compared to others. Typically, observations are considered potentially influential if they are far away from the other points in some dimension.
06

Predict nitrogen retention without potential influential observation

Use the new regression line \(\hat{y}=-0.037+0.627 x\) to predict nitrogen retention for a squirrel with nitrogen intake of 0.06. Compare this prediction to that made in step 3.

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.

Scatterplot
A scatterplot is a visual representation that helps us understand the relationship between two quantitative variables. In this exercise, we are plotting nitrogen intake on the x-axis and nitrogen retention on the y-axis for six flying squirrels. To create a scatterplot:
  • Mark each pair of values ( 0.03, -0.04), (0.10, 0.00), (0.07, 0.01), (0.06, 0.01), (0.07, 0.04), and (0.25, 0.11) as points on a graph.
  • Choose a consistent scale for both axes to represent the entire range of data.
A well-drawn scatterplot makes it easier to see patterns or trends! For instance, you might observe that as nitrogen intake increases, so does nitrogen retention, suggesting a positive correlation. To improve your scatterplot, ensure that it is clearly labeled with descriptive axes titles. This visual will be a vital tool in subsequent steps, particularly when identifying influential points.
Least-Squares Method
The least-squares method helps us find the best-fitting straight line through a set of data points on a scatterplot. This line minimizes the sum of the squares of the vertical distances (residuals) from the points to the line. The idea is to capture the overall trend of the data, allowing us to predict one variable based on another.To apply the least-squares method:
  • The equation for the line is usually in the form \(y = a + bx\), where \(b\) is the slope and \(a\) is the y-intercept.
  • We use statistical tools or calculators to compute these values, ensuring the best fit by minimizing squared differences.
For the squirrels, this method is used to predict nitrogen retention based on intake. Once calculated, the regression line provides insights into how changes in intake might generally affect retention. Compared to the actual values, our line might not pass through every point, but aims to be as close as possible to all.
Residuals
Residuals allow us to assess the accuracy of our regression line by highlighting discrepancies between observed and predicted values. A residual is the difference between the actual value of the dependent variable and the value predicted by our regression line.To calculate a residual:
  • Subtract the predicted value \(\hat{y}\) (from the regression line equation) from the actual observed value \(y\).
  • For example, with an observed nitrogen retention of 0.01 and a predicted value from the regression line, the residual quantifies the error.
Smaller residuals indicate predictions close to actual values, suggesting the line fits well. Conversely, large residuals could imply outliers or potential influential observations. Evaluating residuals is crucial for understanding the quality of our regression model and identifying unusual data points that might skew the results.

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

Anabolic steroid abuse has been increasing despite increased press reports of adverse medical and psychiatric consequences. In a recent study, medical researchers studied the potential for addiction to testosterone in hamsters (Neuroscience \([2004]: 971-981)\). Hamsters were allowed to self-administer testosterone over a period of days, resulting in the death of some of the animals. The data below show the proportion of hamsters surviving versus the peak self-administration of testosterone \((\mu \mathrm{g}) .\) Fit a logistic regression equation and use the equation to predict the probability of survival for a hamster with a peak intake of \(40 \mu \mathrm{g}\). \begin{tabular}{cccc} \multicolumn{4}{c} { Survival } \\ Peak Intake (micrograms) & Proportion \((p)\) & \(\frac{p}{1-p}\) & \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) \\ \hline 10 & \(0.980\) & \(49.0000\) & \(3.8918\) \\ 30 & \(0.900\) & \(9.0000\) & \(2.1972\) \\ 50 & \(0.880\) & \(7.3333\) & \(1.9924\) \\ 70 & \(0.500\) & \(1.0000\) & \(0.0000\) \\ 90 & \(0.170\) & \(0.2048\) & \(-1.5856\) \\ \hline \end{tabular}

Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\).

As part of a study of the effects of timber management strategies (Ecological Applications [2003]: IIIOII123) investigators used satellite imagery to study abundance of the lichen Lobaria oregano at different elevations. Abundance of a species was classified as "common" if there were more than 10 individuals in a plot of land. In the table below, approximate proportions of plots in which Lobaria oregano were common are given. Proportions of Plots Where Lobaria oregano Are Common \begin{tabular}{lrrrrrrr} \hline Elevation (m) & 400 & 600 & 800 & 1000 & 1200 & 1400 & 1600 \\ Prop. of plots & \(0.99\) & \(0.96\) & \(0.75\) & \(0.29\) & \(0.077\) & \(0.035\) & \(0.01\) \\ with lichen & & & & \end{tabular} with lichen \begin{tabular}{l} with lichen \\ common \\ \hline \end{tabular} a. As elevation increases, does the proportion of plots for which lichen is common become larger or smaller? What aspect(s) of the table support your answer? b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the elevations and fit the line \(y^{\prime}=a+b(\) Elevation). What is the equation of the best-fit line? c. Using the best-fit line from Part (b), estimate the proportion of plots of land on which Lobaria oregano are classified as "common" at an elevation of \(900 \mathrm{~m} .\)

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that no relationship exists between the two variables under study.

Penicillin was administered orally to five different horses, and the concentration of penicillin in the blood was determined after five different lengths of time. The following data on \(x=\) elapsed time (in hours) and \(y=\) penicillin concentration (in \(\mathrm{mg} / \mathrm{ml}\) ) appeared in the paper "Absorption and Distribution Patterns of Oral Phenoxymethyl Penicillin in the Horse" (Cornell Veterinarian [1983]: \(314-323\) ): 8 1 $$ \begin{array}{lrrrrr} \boldsymbol{x} & 1 & 2 & 3 & 6 \\ \boldsymbol{y} & 1.8 & 1.0 & 0.5 & 0.1 & 0.1 \end{array} $$ Construct scatterplots using the following variables. Which transformation, if any, would you recommend? a. \(x\) and \(y\) c. \(x\) and \(\sqrt{y}\) b. \(\sqrt{x}\) and \(y\) d. \(\sqrt{x}\) and \(\sqrt{y}\) e. \(x\) and \(\log (y)\) (values of \(\log (y)\) are \(0.26,0,-0.30\), \(-1\), and \(-1\) )
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.