Weber's law, a concept taught in most. Introduction to Psychology courses, states that the ratio of the intensity of a stimulus to the "just noticeable" increment in intensity is constant. The ratio is called the "Weber fraction, "so a concise statement of Weber's law is that "the Weber fraction is constant, regardless of the stimulus intensity. "It turns out that Weber's law is violated in many situations. For instance, for some auditory stimuli, the Weber fraction depends systematically on the stimulus intensity. The following bivariate data are the experimental data obtained for one listener in an auditory intensity discrimination task. For each of the ten stimulus intensities $x$ (in decibels), the Weber fraction $y$ (in decibels) is shown. Figure 1 is a scatter plot of the data. Also given is the product of the stimulus intensity and the Weber fraction for each of the ten stimuli. (These products, written in the column labelled " $x y$ ", may aid in calculations.) \begin{tabular}{|c|c|c|} \hline \begin{tabular}{c} Stimulus \\ intensity, $x$ \\ (in decibels) \end{tabular} & \begin{tabular}{c} Weber \\ fraction, $\boldsymbol{y}$ \\ (in decibels) \end{tabular} & $\boldsymbol{x} \boldsymbol{y}$ \\ \hline 35 & -0.51 & -17.85 \\ \hline 40 & -0.35 & -14 \\ \hline 45 & -1.14 & -51.3 \\ \hline 50 & -0.89 & -44.5 \\ \hline 55 & -2.23 & -122.65 \\ \hline 60 & -2.93 & -175.8 \\ \hline 65 & -3.11 & -202.15 \\ \hline 70 & -3 & -210 \\ \hline 75 & -4.05 & -303.75 \\ \hline 80 & -3.88 & -310.4 \\ \hline \end{tabular} Send data to calculator Stimulus intensity (in decibels) Figure 1 What is the sample correlation coefficient for these data? Carry your irsermed ase computations to at least four decimal places and round your answer to at least three decimal places. (If necessary, consult a list of formulas.)
Solution
Step 1 ：Given the data for stimulus intensity (x), Weber fraction (y), and their product (xy), we can calculate the sample correlation coefficient using the formula: \[r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\]
Step 2 ：First, we calculate the sums of x, y, x², y² and xy. The sum of x (sum_x) is 575, the sum of y (sum_y) is -22.09, the sum of xy (sum_xy) is -1452.4, the sum of x² (sum_x2) is 35125, and the sum of y² (sum_y2) is 66.1611.
Step 3 ：Next, we substitute these values into the formula. The number of pairs of data (n) is 10.
Step 4 ：Substituting these values into the formula, we get \[r = \frac{10(-1452.4) - (575)(-22.09)}{\sqrt{[10*35125 - (575)^2][10*66.1611 - (-22.09)^2]}}\]
Step 5 ：After calculating the above expression, we find that the sample correlation coefficient (r) is approximately -0.9629028344354716.
Step 6 ：Rounding this to three decimal places, we get \(\boxed{-0.963}\) as the final answer.
