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Chapter 5: Problem 3

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

Short Answer

Expert verified
A scatterplot with \(r=1\) would have a straight ascending line and a scatterplot with \(r=-1\) would have a straight descending line, indicating perfect positive and negative correlations for X and Y respectively.

Step by step solution

01

Draw Scatterplot for \(r=1\)

Lay out the graph with the X and Y axes. The respective values for X and Y would be put such that a linear pattern emerges where each increase in X corresponds to an equal increase in Y. This means all the points would lie on a straight ascending line.
02

Draw Scatterplot for \(r=-1\)

Lay out a new graph again with X and Y axes, but this time the values for X and Y would be set so that an increase in X corresponds with a decrease in Y. This means all the points should lie on a straight descending line.

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Most popular questions from this chapter

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New Yorker [December 17,2001\(]: 86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at \(15.4\) percent, and SAT I was last at \(13.3\) percent." a. If the data from this study were used to fit a leastsquares line with \(y=\) first-year college \(\mathrm{GPA}\) and \(x=\) high school GPA, what would the value of \(r^{2}\) have been? b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would have been very accurate? Explain why or why not.

A study was carried out to investigate the relationship between the hardness of molded plastic (y, in Brinell units) and the amount of time elapsed since termination of the molding process (x, in hours). Summary quantities include n 5 15, SSResid 5 1235.470, and SSTo 5 25,321.368. Calculate and interpret the coefficient of determination.

Percentages of public school students in fourth grade in 1996 and in eighth grade in 2000 who were at or above the proficient level in mathematics were given in the article 鈥淢ixed Progress in Math鈥� (USA Today, August 3, 2001) for eight western states: $$ \begin{array}{lcc} \text { State } & (1996) & \text { (2000) } \\ \hline \text { Arizona } & 15 & 21 \\ \text { California } & 11 & 18 \\ \text { Hawaii } & 16 & 16 \\ \text { Montana } & 22 & 37 \\ \text { New Mexico } & 13 & 13 \\ \text { Oregon } & 21 & 32 \\ \text { Utah } & 23 & 26 \\ \text { Wyoming } & 19 & 25 \\ \hline \end{array} $$ a. Construct a scatterplot, and comment on any interesting features. b. Find the equation of the least-squares line that summarizes the relationship between \(x=1996\) fourth-grade math proficiency percentage and \(y=2000\) eighth-grade math proficiency percentage. c. Nevada, a western state not included in the data set, had a 1996 fourth- grade math proficiency of \(14 \%\). What would you predict for Nevada's 2000 eighth-grade math proficiency percentage? How does your prediction compare to the actual eighth-grade value of 20 for Nevada?

Consider the four \((x, y)\) pairs \((0,0),(1,1),(1,-1)\), and \((2,0)\). a. What is the value of the sample correlation coefficient \(r\) ? b. If a fifth observation is made at the value \(x=6\), find a value of \(y\) for which \(r>.5\). c. If a fifth observation is made at the value \(x=6\), find a value of \(y\) for which \(r<.5\).

The accompanying data resulted from an experiment in which weld diameter \(x\) and shear strength \(y\) (in pounds) were determined for five different spot welds on steel. A scatterplot shows a pronounced linear pattern. With \(\Sigma(x-\bar{x})=1000\) and \(\Sigma(x-\bar{x})(y-\bar{y})=8577\), the least-squares line is \(\hat{y}=-936.22+8.577 x\). $$ \begin{array}{llllrr} x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0 \\ y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2 \end{array} $$ a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new \(y=0.4536(\) old \(y\) ). What is the equation of the least-squares line when \(y\) is expressed in kilograms? b. More generally, suppose that each \(y\) value in a data set consisting of \(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the units of measurement for \(y\) ). What effect does this have on the slope \(b\) (i.e., how does the new value of \(b\) compare to the value before conversion), on the intercept \(a\), and on the equation of the least-squares line? Verify your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace \(y\) with \(c y\), and see what happens - and remember, this conversion will affect \(\bar{y} .\) )
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