To calculate the slope of the least-squares regression line, we use the formula:

$m = \frac{n \sum(xy) - \sum(x) \sum(y)}{n \sum(x^2) - (\sum(x))^2}$

Step-by-step breakdown:

1. Extract the given data:

   • $x$ = Stimulus Intensity (in decibels)
   • $y$ = Weber Fraction (in decibels)
   • $xy$ and $x^2$ are given in the table or can be calculated.

2. Sum up the necessary columns:

   • Calculate $\sum x$, $\sum y$, $\sum(xy)$, and $\sum(x^2)$ using the table values.

3. Substitute into the formula for $m$:

I'll calculate this now for you using the data in the table. Data from the table import pandas as pd

Stimulus Intensity (x), Weber Fraction (y), and xy product

data = { "x": [35, 40, 45, 50, 55, 60, 65, 70, 75, 80], "y": [-0.63, -0.26, -1.56, -1.15, -1.96, -2.57, -3.29, -3.26, -4.07, -3.97], "xy": [-22.05, -10.4, -70.2, -57.5, -107.8, -154.2, -213.85, -228.2, -305.25, -317.6] }

df = pd.DataFrame(data)

Add x^2 column

df["x^2"] = df["x"] ** 2

Calculate sums needed for the regression formula

sum_x = df["x"].sum() sum_y = df["y"].sum() sum_xy = df["xy"].sum() sum_x2 = df["x^2"].sum() n = len(df)

Calculate slope (m) of the regression line

slope = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) sum_x, sum_y, sum_xy, sum_x2, slope### Calculation Results:

• $\sum x = 575$
• $\sum y = -22.72$
• $\sum (xy) = -1487.05$
• $\sum (x^2) = 35125$
• Slope of the least-squares regression line:
  $m = -0.0876 \, (\text{rounded to four decimal places})$

Would you like a detailed explanation of how these values were calculated or the next steps in regression analysis?

Related Questions:

1. How do we calculate the intercept of the regression line using this slope?
2. What does the slope indicate in the context of Weber's law violation?
3. How can we use this regression equation to predict Weber fractions for other stimulus intensities?
4. What statistical tests can we perform to check the fit of this regression line?
5. How would the regression line differ if we considered a different listener's data?

Tip:

Always double-check intermediate values when performing regression calculations manually to avoid propagation of errors.