webers law, a concept taught in most introduc...

webers 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 webers law is that \the weber fraction is constant, regardless of the stimulus intensity.\ it turns out that webers law is violated in many situations. for instance, for some auditory stimuli, the weber fraction depends systematically on the stimulus intensity.\nthe 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. (the products, written in the column labelled \xy\, may aid in calculations.)\n| stimulus intensity, x (in decibels) | weber fraction, y (in decibels) | xy |\n| ---- | ---- | ---- |\n| 35 | - 0.48 | - 16.8 |\n| 40 | - 0.68 | - 27.2 |\n| 45 | - 1.10 | - 49.5 |\n| 50 | - 1.01 | - 50.5 |\n| 55 | - 1.8 | - 99 |\n| 60 | - 2.51 | - 150.6 |\n| 65 | - 2.91 | - 189.15 |\n| 70 | - 2.92 | - 204.4 |\n| 75 | - 4.08 | - 306 |\n| 80 | - 4.17 | - 333.6 |\nwhat is the slope of the least - squares regression line for these data? carry your intermediate computations to at least four decimal places and round your answer to at least four decimal places. (if necessary, consult a list of formulas.)

Answer

# Explanation: ## Step1: Recall slope formula The slope $b_1$ of the least - squares regression line for a set of data points $(x_i,y_i)$ with $n$ data points is given by $b_1=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$. First, we need to calculate some sums from the data. Let's assume we have $n = 10$ data points. Let $\sum_{i=1}^{10}x_i=35 + 40+45+50+55+60+65+70+75+80=575$, $\sum_{i = 1}^{10}y_i=- 0.48-0.58 - 1.10-1.01-1.8 - 2.51-2.91-2.92-4.08-4.17=-21.56$, and $\sum_{i = 1}^{10}x_iy_i=-16.8-27.2-52.2-50.5 - 99-150-188.15-204.4-306-333=-1427.25$. Also, $\sum_{i=1}^{10}x_i^{2}=35^{2}+40^{2}+45^{2}+50^{2}+55^{2}+60^{2}+65^{2}+70^{2}+75^{2}+80^{2}=35^{2}+40^{2}+45^{2}+50^{2}+55^{2}+60^{2}+65^{2}+70^{2}+75^{2}+80^{2}=35425$. ## Step2: Substitute values into formula $n = 10$. $n\sum_{i = 1}^{n}x_iy_i=10\times(-1427.25)=-14272.5$, $\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=575\times(-21.56)=-12407$, $n\sum_{i = 1}^{n}x_i^{2}=10\times35425 = 354250$, $(\sum_{i = 1}^{n}x_i)^{2}=575^{2}=330625$. $b_1=\frac{-14272.5-(-12407)}{354250 - 330625}=\frac{-14272.5 + 12407}{23625}=\frac{-1865.5}{23625}\approx - 0.0790$. # Answer: $-0.0790$