Question

Find the equation for the least squares regression line of the data described below.
Cora is a math teacher at a large school. She wonders if her test problems are too wordy. Cora is curious whether the wordiness is affecting student performance.
For the last several tests, Cora computes the average number of words in each question, x, as well as the average percentage scores on the tests, y.
Average question length Average student score
21.32 61.24
22.19 70.84
25.21 59.15
44.54 60.06
48.54 51.41
49.76 52.15
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the equation for the least squares regression line \( y = mx + b \), we need to calculate the slope \( m \) and the y-intercept \( b \) using the given data points.

The data points are:
- (21.32, 61.24)
- (22.19, 70.84)
- (25.21, 59.15)
- (44.54, 60.06)
- (48.54, 51.41)
- (49.76, 52.15)

### Step 1: Calculate necessary sums
Let \( n \) be the number of data points, \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \).

1. **Calculate \( n \):**
\[
n = 6
\]

2. **Calculate \( \sum x \):**
\[
\sum x = 21.32 + 22.19 + 25.21 + 44.54 + 48.54 + 49.76 = 211.56
\]

3. **Calculate \( \sum y \):**
\[
\sum y = 61.24 + 70.84 + 59.15 + 60.06 + 51.41 + 52.15 = 354.85
\]

4. **Calculate \( \sum xy \):**
\[
\sum xy = (21.32 \cdot 61.24) + (22.19 \cdot 70.84) + (25.21 \cdot 59.15) + (44.54 \cdot 60.06) + (48.54 \cdot 51.41) + (49.76 \cdot 52.15)
\]
\[
= 1303.4848 + 1573.6396 + 1487.8215 + 2672.6724 + 2495.4874 + 2591.984 = 11984.0897
\]

5. **Calculate \( \sum x^2 \):**
\[
\sum x^2 = (21.32^2) + (22.19^2) + (25.21^2) + (44.54^2) + (48.54^2) + (49.76^2)
\]
\[
= 453.3584 + 492.0561 + 636.0641 + 1986.6116 + 2358.0516 + 2476.0576 = 8456.1195
\]

### Step 2: Calculate slope \( m \) and y-intercept \( b \)
Using the formulas:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{(\sum y) - m(\sum x)}{n}
\]

#### Calculate slope \( m \):
\[
m = \frac{6(11984.0897) - (211.56)(354.85)}{6(8456.1195) - (211.56)^2}
\]
Calculating the components:
\[
m = \frac{71904.5382 - 75041.0656}{50736.717 - 44765.6336}
\]
\[
m = \frac{-3136.5274}{5961.0834} \approx -0.526
\]

#### Calculate y-intercept \( b \):
\[
b = \frac{354.85 - (-0.526)(211.56)}{6}
\]
Calculating:
\[
b \approx \frac{354.85 + 111.35736}{6} \approx \frac{466.20736}{6} \approx 77.701
\]

### Final Equation
Thus, the least squares regression line is:
\[
y = -0.526x + 77.701
\]

### Rounding to the nearest thousandth:
\[
y = -0.526x + 77.701
\]
So,
- Slope \( m \) = -0.526
- Intercept \( b \) = 77.701

The final answer with rounding is:
\[
y = -0.526x + 77.701
\]