48. TRANSFORMATIONS Consider the following data set: x 5 6 7 7 8 8 8 8 y 4.2 5 5.2 5.9 6 6.2 6.1 6.9 x 9 9 10 10 11 11 12 12 y 7.2 8 8.3 7.4 8.4 7.8 8.5 9.5 a. Draw a scatter diagram with the x-axis starting at 0 and ending at 30 and with the y-axis starting at 0 and ending at 20. b. Compute the linear correlation coefficient. c. Now multiply both x and y by 2. d. Draw a scatter diagram of the new data with the x-axis starting at 0 and ending at 30 and with the y-axis starting at 0 and ending at 20. Compare the scatter diagrams. e. Compute the linear correlation coefficient. f. Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables. Explain why this is the case.
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To draw a scatter diagram, we plot the given data points on a graph. The x-axis represents the independent variable (x) and the y-axis represents the dependent variable (y). The x-axis should start at 0 and end at 30, and the y-axis should start at 0 and end at Show more…
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Consider the following data set: $$ \begin{array}{lllllllll} \hline x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \\ \hline \end{array} $$ (a) Draw a scatter diagram with the $x$ -axis starting at 0 and ending at 30 and with the $y$ -axis starting at 0 and ending at 20 . (b) Compute the linear correlation coefficient. (c) Now multiply both $x$ and $y$ by 2 . (d) Draw a scatter diagram of the new data with the $x$ -axis starting at 0 and ending at 30 and with the $y$ -axis starting at 0 and ending at 20. Compare the scatter diagrams. (e) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables.
Describing the Relation between Two Variables
Scatter Diagrams and Correlation
Consider the following set of data: $$\begin{array}{lllllllll} x & 2.2 & 3.7 & 3.9 & 4.1 & 2.6 & 4.1 & 2.9 & 4.7 \\ \hline y & 3.9 & 4.0 & 1.4 & 2.8 & 1.5 & 3.3 & 3.6 & 4.9 \end{array}$$ (a) Draw a scatter diagram of the data and compute the linear correlation coefficient. (b) Draw a scatter diagram of the data and compute the linear correlation coefficient with the additional data point $(10.4,9.3) .$ Comment on the effect the additional data point has on the linear correlation coefficient. Explain why correlations should always be reported with scatter diagrams.
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