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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
Two scatterplots are drawn. One where the points form a positively sloped straight line, representing \(r=1\) (perfect positive linear relationship). Another where the points form a negatively sloped straight line, representing \(r=-1\) (perfect negative linear relationship).

Step by step solution

01

Sketch Scatterplot for \(r=1\)

To sketch the scatterplot, we start by drawing a straight horizontal line (the x-axis) and a straight vertical line (the y-axis). Then, we plot several points starting from the lower left-hand corner and moving towards the upper right-hand corner in a straight line. This would result in a positively sloped line indicating that as one variable (represented by points on x-axis) increases, the other variable (represented by points on y-axis) also increases.
02

Sketch Scatterplot for \(r=-1\)

Again, we start by drawing a straight horizontal line (the x-axis) and a straight vertical line (the y-axis). But this time, we plot several points starting from the upper left-hand corner and moving towards the lower right-hand corner in a straight line. This would result in a negatively sloped line indicating that as one variable (represented by points on x-axis) increases, the other variable (represented by points on y-axis) decreases.

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Most popular questions from this chapter

Both \(r^{2}\) and \(s\), are used to assess the fit of a line. a. Is it possible that both \(r^{2}\) and \(s_{e}\) could be large for a bivariate data set? Explain. (A picture might be helpful.) b. Is it possible that a bivariate data set could yield values of \(r^{2}\) and \(s_{e}\) that are both small? Explain. (Again, a picture might be helpful.) c. Explain why it is desirable to have \(r^{2}\) large and \(s_{s}\) small if the relationship between two variables \(x\) and \(\gamma\) is to be described using a straight line.

\(5.19\) - The accompanying data on \(x=\) head circumference \(z\) score (a comparison score with peers of the same age - a positive score suggests a larger size than for peers) at age 6 to 14 months and \(y=\) volume of cerebral grey matter (in \(\mathrm{ml}\) ) at age 2 to 5 years were read from a graph in the article described in the chapter introduction (journal of the American Medical Association [2003]). \begin{tabular}{lc} Cerebral Grey Matter (ml) \(2-5\) yr & Head Circumference \(z\) Scores at \(6-14\) Months \\ \hline 680 & \(-.75\) \\ 690 & \(1.2\) \\ 700 & \(-.3\) \\ 720 & \(.25\) \\ 740 & \(.3\) \\ 740 & \(1.5\) \\ 750 & \(1.1\) \\ 750 & \(2.0\) \\ 760 & \(1.1\) \\ 780 & \(1.1\) \\ 790 & \(2.0\) \\ 810 & \(2.1\) \\ 815 & \(2.8\) \\ 820 & \(2.2\) \\ 825 & \(.9\) \\ 835 & \(2.35\) \\ 840 & \(2.3\) \\ 845 & \(2.2\) \\ \hline \end{tabular} a. Construct a scatterplot for these data. b. What is the value of the correlation coefficient? c. Find the equation of the least-squares line. d. Predict the volume of cerebral grey matter at age 2 to 5 years for a child whose head circumference \(z\) score at age 12 months was \(1.8\). e. Explain why it would not be a good idea to use the least-squares line to predict the volume of grey matter for a child whose head circumference \(z\) score was \(3.0\).

The accompanying data represent \(x=\) amount of catalyst added to accelerate a chemical reaction and \(y\) \(=\) resulting reaction time: \(\begin{array}{cccccc}x & 1 & 2 & 3 & 4 & 5 \\ y & 49 & 46 & 41 & 34 & 25\end{array}\) a. Calculate \(r\). Does the value of \(r\) suggest a strong linear relationship? b. Construct a scatterplot. From the plot, does the word linear provide the most effective description of the relationship between \(x\) and \(y\) ? Explain.

Cost-to-charge ratio (the percentage of the amount billed that represents the actual cost) for inpatient and outpatient services at 11 Oregon hospitals is shown in the following table (Oregon Department of Health Services, 2002 ). A scatterplot of the data is also shown. \begin{tabular}{ccc} & \multicolumn{2}{c} { Cost-to-Charge Ratio } \\ \cline { 2 - 3 } Hospital & Outpatient Care & Inpatient Care \\ \hline 1 & 62 & 80 \\ 2 & 66 & 76 \\ 3 & 63 & 75 \\ 4 & 51 & 62 \\ 5 & 75 & 100 \\ 6 & 65 & 88 \\ 7 & 56 & 64 \\ 8 & 45 & 50 \\ 9 & 48 & 54 \\ 10 & 71 & 83 \\ 11 & 54 & 100 \\ \hline \end{tabular} The least-squares regression line with \(y=\) inpatient costto-charge ratio and \(x=\) outpatient cost-to-charge ratio is \(\hat{y}=-1.1+1.29 x\). a. Is the observation for Hospital 11 an influential observation? Justify your answer. b. Is the observation for Hospital 11 an outlier? Explain. c. Is the observation for Hospital 5 an influential observation? Justify your answer. d. Is the observation for Hospital 5 an outlier? Explain.

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.
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