Question: Question 4 (2 points) Saved
The following data show the gain in reading speed (in words per minute) and the
number of weeks in a speed-reading program for six students.
No. of Weeks Gain In Reading Speed
30 50
5 39
30 24
9 49
5 23
49 9
What is the slope of the regression line, b? Leave your answers in 2 decimal places.
1. **State the problem:** We are given data for number of weeks ($x$) and gain in reading speed ($y$) for six students. We need to find the slope $b$ of the regression line $y = a + bx$.
2. **Formula for slope $b$ of regression line:**
$$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$
where $n$ is the number of data points.
3. **List the data points:**
$$(x,y) = (30,50), (5,39), (30,24), (9,49), (5,23), (49,9)$$
4. **Calculate sums:**
$$\sum x = 30 + 5 + 30 + 9 + 5 + 49 = 128$$
$$\sum y = 50 + 39 + 24 + 49 + 23 + 9 = 194$$
$$\sum xy = (30)(50) + (5)(39) + (30)(24) + (9)(49) + (5)(23) + (49)(9) = 1500 + 195 + 720 + 441 + 115 + 441 = 3412$$
$$\sum x^2 = 30^2 + 5^2 + 30^2 + 9^2 + 5^2 + 49^2 = 900 + 25 + 900 + 81 + 25 + 2401 = 4332$$
5. **Plug values into formula:**
$$b = \frac{6(3412) - (128)(194)}{6(4332) - (128)^2} = \frac{20472 - 24832}{25992 - 16384} = \frac{-4360}{9608}$$
6. **Simplify fraction:**
$$b = \frac{\cancel{-4360}}{\cancel{9608}} = -0.45$$
7. **Interpretation:** The slope $b$ is approximately $-0.45$, meaning for each additional week, the gain in reading speed decreases by about 0.45 words per minute on average.
**Final answer:**
$$b = -0.45$$