/*! 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 8 Please do the following. (a) D... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 8

Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums \(\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2}\), and \(\sum x y\) and the value of the sample correlation coefficient \(\underline{r}\) (c) Find \(\bar{x}, \bar{y}, a\), and \(b .\) Then find the equation of the least- squares line \(\hat{y}=a+b x\) (d) Graph the least-squares line on your scatter diagram. Be sure to use the point \((\bar{x}, \bar{y})\) as one of the points on the line. (e) Find the value of the coefficient of determination \(r^{2} .\) What percentage of the variation in \(y\) can be explained by the corresponding variation in \(x\) and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar-S herd. How much should a healthy calf weigh? Let \(x\) be the age of the calf (in weeks), and let \(y\) be the weight of the calf (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used by many ranchers). $$ \begin{array}{r|rrrrrr} \hline x & 1 & 3 & 10 & 16 & 26 & 36 \\ \hline y & 42 & 50 & 75 & 100 & 150 & 200 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=92, \Sigma y=617, \Sigma x^{2}=2338, \Sigma y^{2}=\) \(82,389, \Sigma x y=13,642\), and \(r \approx 0.998 .\) (f) The calves you want to buy are 12 weeks old. What does the least- squares line predict for a healthy weight?

Short Answer

Expert verified
A 12-week-old calf is predicted to weigh approximately 84.36 kg.

Step by step solution

01

Draw a Scatter Diagram

Plot the given data points on a graph with the x-axis representing the age in weeks (x) and the y-axis representing the weight in kilograms (y). Mark each point corresponding to the data pairs: (1,42), (3,50), (10,75), (16,100), (26,150), and (36,200).
02

Verify Sums and Correlation Coefficient

Given sums are \(\Sigma x = 92\), \(\Sigma y = 617\), \(\Sigma x^2 = 2338\), \(\Sigma y^2 = 82389\), and \(\Sigma xy = 13642\). The sample correlation coefficient \(r\) is approximately 0.998, which indicates a strong positive linear correlation between age and weight.
03

Calculate Averages and Coefficients

Calculate the averages \(\bar{x}\) and \(\bar{y}\):\[ \bar{x} = \frac{\Sigma x}{n} = \frac{92}{6} \approx 15.33 \]\[ \bar{y} = \frac{\Sigma y}{n} = \frac{617}{6} \approx 102.83 \]Using the formulas for the slope \(b\) and intercept \(a\):\[ b = \frac{n\Sigma xy - \Sigma x \Sigma y}{n\Sigma x^2 - (\Sigma x)^2} \]\[ a = \bar{y} - b\bar{x} \]Calculating these:\[ b = \frac{6 \times 13642 - 92 \times 617}{6 \times 2338 - 92^2} \approx 5.17 \]\[ a = 102.83 - 5.17 \times 15.33 \approx 22.32 \]
04

Write the Equation of the Least-Squares Line

Substitute \(a\) and \(b\) into the equation of the line:\[ \hat{y} = 22.32 + 5.17x \]
05

Graph the Least-Squares Line

Draw the line \(\hat{y} = 22.32 + 5.17x\) on the scatter diagram. Use the point \((\bar{x}, \bar{y}) = (15.33, 102.83)\) to help position the line accurately. Extend the line to cover the range of data points.
06

Calculate the Coefficient of Determination

The coefficient of determination \(r^2\) is given by:\[ r^2 = (0.998)^2 \approx 0.996 \]Thus, approximately 99.6% of the variation in weight can be explained by age using this linear model, leaving about 0.4% unexplained.
07

Predict Weight for a 12-week-old Calf

Using the least-squares equation:\[ \hat{y} = 22.32 + 5.17 \times 12 \]Calculate \(\hat{y}\):\[ \hat{y} \approx 84.36 \]So, a healthy 12-week-old calf is predicted to weigh about 84.36 kilograms.

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.

Scatter Plot
A scatter plot is a powerful tool for visualizing the relationship between two variables. In this context, the x-axis typically represents the independent variable, such as the age of calves in weeks, while the y-axis represents the dependent variable, like their weight in kilograms.

To create a scatter plot, you plot each pair of data as a point on the graph. In this exercise, the data points (1,42), (3,50), (10,75), (16,100), (26,150), and (36,200) represent the ages and weights of calves. Each point on the graph corresponds to a specific pair of age and weight values.

  • A scatter plot helps to visually assess the overall trend.
  • It can indicate whether a linear relationship might exist between the variables.
  • Patterns like clusters or outliers can also be observed.
This visualization provides a foundation for further statistical analysis, such as calculating the least-squares regression line to quantify the relationship.
Correlation Coefficient
The correlation coefficient, denoted as \( r \), is a statistical measure that describes the strength and direction of a linear relationship between two variables. It varies from -1 to 1:
- A value of 1 signifies a perfect positive linear relationship.
- A value of -1 signifies a perfect negative linear relationship.
- A value of 0 suggests no linear correlation at all.
In the given exercise, the correlation coefficient is approximately 0.998. This value indicates a very strong positive linear relationship between the age of the calf and its weight. A higher value close to 1 suggests that as one variable increases, the other also tends to increase in a roughly linear fashion.
Keeping in mind that the correlation coefficient doesn't imply causation, it is still a vital tool for understanding how closely related two variables are. In predictive modeling, a high correlation value implies that one variable is a good predictor of the other.
Coefficient of Determination
The coefficient of determination, \( r^2 \), is another important statistic that helps to quantify how well data fits a statistical model. It is the square of the correlation coefficient and is expressed as a percentage.
The \( r^2 \) value tells you how much of the variation in the dependent variable can be explained by the independent variable using the model. In this exercise, the \( r^2 \) value is approximately 0.996, or 99.6%.

  • This means that 99.6% of the variation in the calf's weight can be explained by its age using the least-squares line.
  • The remaining 0.4% of the variance is due to other factors not included in the model.
A high \( r^2 \) value, like in this example, suggests that the linear regression model provides a good fit to the data, making it a reliable tool for prediction.
Predictive Modeling
Predictive modeling involves using statistical techniques to create a mathematical model that can predict future outcomes based on historical data. In this exercise, the goal is to predict the weight of a calf given its age using a least-squares regression line.
The equation \( \hat{y} = 22.32 + 5.17x \) represents the least-squares line derived from the data:
  • \( \hat{y} \) is the predicted weight.
  • \( x \) is the age of the calf.
  • The slope (5.17) indicates how much the weight is expected to increase for each additional week of age.
  • The intercept (22.32) is the predicted weight of a calf at 0 weeks old, though this is mostly theoretical in this biological context.
Utilizing this equation, you can predict the weight of a 12-week-old calf by substituting x = 12. The calculation results in about 84.36 kg. Predictive modeling, therefore, provides insight into future scenarios based on existing trends, aiding decision-making processes effectively.

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

What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let \(x=\) depth of dive in meters, and let \(y=\) optimal time in hours. A random sample of divers gave the following data (based on information taken from Medical Physiology by A. C. Guyton, M.D.). $$ \begin{array}{l|lllllll} \hline x & 14.1 & 24.3 & 30.2 & 38.3 & 51.3 & 20.5 & 22.7 \\ \hline y & 2.58 & 2.08 & 1.58 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=201.4, \Sigma y=12.6, \Sigma x^{2}=6735.46, \Sigma y^{2}=25.607, \Sigma x y=\) 311.292, and \(r \approx-0.976\). (b) Use a \(1 \%\) level of significance to test the claim that \(\rho<0\). (c) Verify that \(S_{e} \approx 0.1660, a \approx 3.366\), and \(b \approx-0.0544\). (d) Find the predicted optimal time in hours for a dive depth of \(x=18\) meters. (e) Find an \(80 \%\) confidence interval for \(y\) when \(x=18\) meters. (f) Use a \(1 \%\) level of significance to test the claim that \(\beta<0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and interpret its meaning.

In Section \(5.1\), we studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let \(x\) and \(y\) be random variables with means \(\mu_{x}\) and \(\mu_{y}\), variances \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) and population correlation coefficient \(\rho\) (the Greek letter rho). Let \(a\) and \(b\) be any constants and let \(w=a x+b y .\) Then $$ \begin{aligned} &\mu_{w}=a \mu_{x}+b \mu_{y} \\ &\sigma_{w}^{2}=a^{2} \sigma_{x}^{2}+b^{2} \sigma_{y}^{2}+2 a b \sigma_{x} \sigma_{y} \rho \end{aligned} $$ In this formula, \(\rho\) is the population correlation coefficient, theoretically computed using the population of all \((x, y)\) data pairs. The expression \(\sigma_{x} \sigma_{y} \rho\) is called the covariance of \(x\) and \(y\). If \(x\) and \(y\) are independent, then \(\rho=0\) and the formula for \(\sigma_{w}^{2}\) reduces to the appropriate formula for independent variables (see Section 5.1). In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let \(x\) represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let \(y\) represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates (based on Morningstar Mutual Fund Report). $$ \mu_{x} \approx 7.32 \quad \sigma_{x} \approx 6.59 \quad \mu_{y} \approx 13.19 \quad \sigma_{y} \approx 18.56 \quad \rho \approx 0.424 $$ (a) Do you think the variables \(x\) and \(y\) are independent? Explain. (b) Suppose you decide to put \(60 \%\) of your investment in bonds and \(40 \%\) in real estate. This means you will use a weighted average \(w=0.6 x+0.4 y\). Estimate your expected percentage return \(\mu_{w}\) and risk \(\sigma_{w}\). (c) Repeat part (b) if \(w=0.4 x+0.6 y\). (d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by \(\sigma_{w} ?\)

Suppose two variables are positively correlated. Does the response variable increase or decrease as the explanatory variable increases?

Let \(x=\) day of observation and \(y=\) number of locusts per square meter during a locust infestation in a region of North Africa. $$ \begin{array}{r|rrrrr} \hline x & 2 & 3 & 5 & 8 & 10 \\ \hline y & 2 & 3 & 12 & 125 & 630 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the \((x, y)\) data pairs. Do you think a straight line will be a good fit to these data? Do the \(y\) values almost seem to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base \(10 .\) Draw a scatter diagram of the \(\left(x, y^{\prime}\right)\) data pairs and compare this diagram with the diagram of part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to find the linear regression equation for the data pairs \(\left(x, y^{\prime}\right) .\) What is the correlation coefficient? (d) The exponential growth model is \(y=\alpha \beta^{x}\). Estimate \(\alpha\) and \(\beta\) and write the exponential growth equation. Hint: See Problem 22 .

The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let \(x\) be the number of different research programs, and let \(y\) be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs: $$ \begin{array}{c|rrrrrr} \hline x & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline y & 1.8 & 1.7 & 1.5 & 1.4 & 1.0 & 0.7 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=90, \Sigma y=8.1, \Sigma x^{2}=1420\), \(\Sigma y^{2}=11.83, \Sigma x y=113.8\), and \(r \approx-0.973 .\) (f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for \(y=\) mean number of patents per program?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.