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Chapter 13: Problem 3

Explain the difference between exact and nonexact relationships between two variables. Give one example of each.

Short Answer

Expert verified
An exact relationship constitutes a predictable and precise change in another variable when one variable changes, exemplified by the formula for the perimeter of a rectangle. A non-exact relationship, such as the correlation between human height and weight, doesn't result in an exact and predictable change due to various influencing factors.

Step by step solution

01

Defining Exact Relationship

An exact relationship in mathematics is one where a change in one variable produces a precise and predictable change in another variable. These relationships can typically be defined using mathematical equations.
02

Example of Exact Relationship

A suitable example of an exact relationship is the formula for the perimeter of a rectangle \(P = 2l + 2w\), where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. If either length or width changes, a predictable change occurs in the perimeter.
03

Defining Non-exact Relationship

A non-exact relationship, unlike exact, does not produce a predictable and precise alteration in another variable when one variable changes. Non-exact relationships can often be represented graphically.
04

Example of Non-exact Relationship

An example is the relationship between height and weight in humans. Although generally taller people weigh more than shorter ones, this is not an absolute rule. Diet, exercise, genetics and many other factors can impact a person's weight in unpredictable ways.

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Most popular questions from this chapter

Explain the least squares method and least squares regression line. Why are they called by these names?

The following table gives the average weekly retail price of a gallon of regular gasoline in the eastern United States over a 9-week period from December 1, 2014, through January 26, 2015. Consider these 9 weeks as a random sample. $$ \begin{array}{l|rrrrrr} \hline \text { Date } & 12 / 1 / 14 & 12 / 8 / 14 & 12 / 15 / 14 & 12 / 22 / 14 & 12 / 29 / 14 & 1 / 5 / 15 \\ \hline \text { Price (\$) } & 2.861 & 2.776 & 2.667 & 2.535 & 2.445 & 2.378 \\\ \hline \text { Date } & 1 / 12 / 15 & 1 / 19 / 15 & 1 / 26 / 15 & & & \\ \hline \text { Price (\$) } & 2.293 & 2.204 & 2.174 & & & \\ \hline \end{array} $$ a. Assign a value of 0 to \(12 / 1 / 14,1\) to \(12 / 8 / 14,2\) to \(12 / 15 / 14\), and so on. Call this new variable Time. Make a new table with the variables Time and Price. b. With time as an independent variable and price as the dependent variable, compute \(S S_{x x}, S S_{y y}\), and \(S S_{x y}\) c. Construct a scatter diagram for these data. Does the scatter diagram exhibit a negative linear relationship between time and price? d. Find the least squares regression line \(\hat{y}=a+b x\). e. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{d}\). f. Compute the correlation coefficient \(r .\) g. Predict the average price of a gallon of regular gasoline in the eastern United States for Time \(=26 .\) Comment on this prediction.

For a sample data set on two variables, the value of the linear correlation coefficient is (close to) zero. Does this mean that these variables are not related? Explain.

Explain each of the following concepts. You may use graphs to illustrate each concept. a. Perfect positive linear correlation b. Perfect negative linear correlation c. Strong positive linear correlation d. Strong negative linear correlation e. Weak positive linear correlation f. Weak negative linear correlation g. No linear correlation

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams. $$ \begin{array}{lcc} \hline \text { Team } & \begin{array}{c} \text { Total Payroll } \\ \text { (millions of dollars) } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Games Won } \end{array} \\ \hline \text { Arizona Diamondbacks } & 92 & 49 \\ \text { Atlanta Braves } & 98 & 41 \\ \text { Chicago Cubs } & 119 & 60 \\ \text { Cincinnati Reds } & 117 & 40 \\ \text { Colorado Rockies } & 102 & 42 \\ \text { Los Angeles Dodgers } & 273 & 57 \\ \text { Miami Marlins } & 68 & 44 \\ \text { Milwaukee Brewers } & 105 & 42 \\ \text { New York Mets } & 101 & 56 \\ \text { Philadelphia Phillies } & 136 & 39 \\ \text { Pittsburgh Pirates } & 88 & 61 \\ \text { San Diego Padres } & 101 & 46 \\ \text { San Francisco Giants } & 173 & 52 \\ \text { St. Louis Cardinals } & 121 & 62 \\ \text { Washington Nationals } & 165 & 51 \\ \hline \end{array} $$ a. Find the least squares regression line with total payroll as the independent variable and percentage of games won as the dependent variable. b. Is the equation of the regression line obtained in part a the population regression line? Why or why not? Do the values of the \(y\) -intercept and the slope of the regression line give \(A\) and \(B\) or \(a\) and \(b ?\) c. Give a brief interpretation of the values of the \(y\) -intercept and the slope obtained in part a. d. Predict the percentage of games won by a team with a total payroll of \(\$ 150\) million.
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