Question

Find the equation for the least squares regression line of the data described below.
Mr. Joseph wants to demonstrate the importance of proofreading to his English class. He had students read the same passage and mark all the spelling and grammar errors they could find.
He recorded how many minutes each student had spent on the exercise, x, and how many errors that student had missed, y.
Minutes Errors
1 16
2 19
4 22
6 18
8 13
Round your answers to the nearest thousandth.
y = ___ x +_____

corect answers only

Answer 1

To find the least squares regression line of the given data, we first need to organize the data:

| Minutes (x) | Errors (y) |
|-------------|------------|
| 1 | 16 |
| 2 | 19 |
| 4 | 22 |
| 6 | 18 |
| 8 | 13 |

Next, we calculate the necessary sums to find the slope \( m \) and y-intercept \( b \) of the regression line. The formulas we will use are:

1. \( m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \)
2. \( b = \frac{\sum y - m(\sum x)}{N} \)

Where:
- \( N \) is the number of data points.
- \( \sum x \) is the sum of the x values.
- \( \sum y \) is the sum of the y values.
- \( \sum xy \) is the sum of the product of each x and y value.
- \( \sum x^2 \) is the sum of the squares of each x value.

Now we can calculate the necessary sums:

- \( \sum x = 1 + 2 + 4 + 6 + 8 = 21 \)
- \( \sum y = 16 + 19 + 22 + 18 + 13 = 88 \)
- \( \sum xy = (1 \cdot 16) + (2 \cdot 19) + (4 \cdot 22) + (6 \cdot 18) + (8 \cdot 13) = 16 + 38 + 88 + 108 + 104 = 354 \)
- \( \sum x^2 = 1^2 + 2^2 + 4^2 + 6^2 + 8^2 = 1 + 4 + 16 + 36 + 64 = 121 \)
- \( N = 5 \)

Now, plugging in these values into the formulas:

1. Calculate \( m \):

\[
m = \frac{5(354) - (21)(88)}{5(121) - (21)^2}
\]

Calculating the above:

- \( 5(354) = 1770 \)
- \( (21)(88) = 1848 \)
- \( 5(121) = 605 \)
- \( (21)^2 = 441 \)

So,

\[
m = \frac{1770 - 1848}{605 - 441} = \frac{-78}{164} \approx -0.476
\]

2. Now calculate \( b \):

\[
b = \frac{88 - (-0.476)(21)}{5}
\]

Calculating this:

\[
b = \frac{88 + 10.596}{5} = \frac{98.596}{5} \approx 19.719
\]

Finally, the least squares regression line can be expressed as:

\[
y = mx + b
\]

So rounding \( m \) and \( b \) to the nearest thousandth gives us:

\[
y = -0.476x + 19.719
\]

Thus, the equation for the least squares regression line is:

**y = -0.476x + 19.719**.