Learning Materials
Features
Discover
Chapter 9: Problem 4
If two variables have a negative linear correlation, is the slope of the least-squares line positive or negative?
Over 30 million students worldwide already upgrade their learning with 91影视!
These are the key concepts you need to understand to accurately answer the question.
All the tools & learning materials you need for study success - in one app.
Get started for free
How much should a healthy Shetland pony weigh? Let \(x\) be the age of the pony (in months), and let \(y\) be the average weight of the pony (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used in most veterinary colleges). $$ \begin{array}{r|rrrrr} \hline x & 3 & 6 & 12 & 18 & 24 \\ \hline y & 60 & 95 & 140 & 170 & 185 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=63, \quad \Sigma x^{2}=1089, \quad \Sigma y=650\) \(\Sigma y^{2}=95,350\), and \(\Sigma x y=9930 .\) Compute \(r .\) As \(x\) increases from 3 to 24 months, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.
When we take measurements of the same general type, a power law of the form \(y=\alpha x^{\beta}\) often gives an excellent fit to the data. A lot of research has been conducted as to why power laws work so well in business, economics, biology, ecology, medicine, engineering, social science, and so on. Let us just say that if you do not have a good straight-line fit to data pairs \((x, y)\), and the scatter plot does not rise dramatically (as in exponential growth), then a power law is often a good choice. College algebra can be used to show that power law models become linear when we apply logarithmic transformations to both variables. To see how this is done, please read on. Note: For power law models, we assume all \(x>0\) and all \(y>0\). Suppose we have data pairs \((x, y)\) and we want to find constants \(\alpha\) and \(\beta\) such that \(y=\alpha x^{\beta}\) is a good fit to the data. First, make the logarithmic transformations \(x^{\prime}=\log x\) and \(y^{\prime}=\log y .\) Next, use the \(\left(x^{\prime}, y^{\prime}\right)\) data pairs and a calculator with linear regression keys to obtain the least-squares equation \(y^{\prime}=a+b x^{\prime} .\) Note that the equation \(y^{\prime}=a+b x^{\prime}\) is the same as \(\log y=a+b(\log x)\). If we raise both sides of this equation to the power 10 and use some college algebra, we get \(y=10^{a}(x)^{b}\). In other words, for the power law model, we have \(\alpha \approx 10^{a}\) and \(\beta \approx b\). In the electronic design of a cell phone circuit, the buildup of electric current \((\mathrm{Amps})\) is an important function of time (microseconds). Let \(x=\) time in microseconds and let \(y=\) Amps built up in the circuit at time \(x .\) $$ \begin{array}{l|lllll} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y & 1.81 & 2.90 & 3.20 & 3.68 & 4.11 \\ \hline \end{array} $$ (a) Make the logarithmic transformations \(x^{\prime}=\log x\) and \(y^{\prime}=\log y .\) Then make a scatter plot of the \(\left(x^{\prime}, y^{\prime}\right)\) values. Does a linear equation seem to be a good fit to this plot? (b) Use the \(\left(x^{\prime}, y^{\prime}\right)\) data points and a calculator with regression keys to find the least-squares equation \(y^{\prime}=a+b x^{\prime} .\) What is the sample correlation coefficient? (c) Use the results of part (b) to find estimates for \(\alpha\) and \(\beta\) in the power law \(y=\alpha x^{\beta}\). Write the power law giving the relationship between time and Amp buildup. Note: The TI-84Plus/TI-83Plus/TI- \(n\) spire calculators fully support the power law model. Place the original \(x\) data in list L1 and the corresponding \(y\) data in list L2. Then press STAT, followed by CALC, and scroll down to option A: PwrReg. The output gives values for \(\alpha, \beta\), and the sample correlation coefficient \(r\).
What is the optimal amount of 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.). $$ \begin{array}{c|ccccccc} \hline x & 14.1 & 24.3 & 30.2 & 38.3 & 51.3 & 20.5 & 22.7 \\ \hline y & 2.58 & 2.08 & 1.58 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=201.4, \quad \Sigma y=12.6, \quad \Sigma x^{2}=6734.46, \quad \Sigma y^{2}=25.607\), \(\Sigma x y=311.292\), 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 its meaning.
In baseball, is there a linear correlation between batting average and home run percentage? Let \(x\) represent the batting average of a professional baseball player, and let \(y\) represent the player's home run percentage (number of home runs per 100 times at bat). A random sample of \(n=7\) professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan Publishing Company). $$ \begin{array}{l|lllllll} \hline x & 0.243 & 0.259 & 0.286 & 0.263 & 0.268 & 0.339 & 0.299 \\ \hline y & 1.4 & 3.6 & 5.5 & 3.8 & 3.5 & 7.3 & 5.0 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or high? positive or negative? (c) Use a calculator to verify that \(\Sigma x=1.957, \Sigma x^{2} \approx 0.553, \Sigma y=30.1\), \(\Sigma y^{2}=150.15\), and \(\Sigma x y \approx 8.753 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.
Over the past 30 years in the United States, there has been a strong negative correlation between the number of infant deaths at birth and the number of people over age 65 . (a) Is the fact that people are living longer causing a decrease in infant mortalities at birth? (b) What lurking variables might be causing the increase in one or both of the variables? Explain.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine