Question
Find the equation for the least squares regression line of the data described below.
Anthony manages the student health services department at Oxford University. He suspects that the students' poor breakfast habits are contributing to the school's academic decline. Anthony surveyed several students asking how many times they ate breakfast in the past month. He also checked each student's GPA.
The data shows the number of times each student ate breakfast in the past month, x, and his or her GPA, y.
Breakfasts GPA
4 2.2
8 2.1
13 1.8
23 3.4
29 3.8
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data, we need to determine the slope (m) and the y-intercept (b) of the line in the form \( y = mx + b \). We can do this using the following formulas:
1. **Find the averages of x and y**:
\[
\bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n}
\]
2. **Calculate the slope (m)**:
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
3. **Calculate the intercept (b)**:
\[
b = \bar{y} - m\bar{x}
\]
Now, let's compute these values step-by-step based on the provided data:
### Given Data
| Breakfasts (x) | GPA (y) |
|----------------|---------|
| 4 | 2.2 |
| 8 | 2.1 |
| 13 | 1.8 |
| 23 | 3.4 |
| 29 | 3.8 |
### Step 1: Compute Averages
\[
\sum x = 4 + 8 + 13 + 23 + 29 = 77
\]
\[
\sum y = 2.2 + 2.1 + 1.8 + 3.4 + 3.8 = 13.3
\]
\[
n = 5
\]
Now, calculate the averages:
\[
\bar{x} = \frac{77}{5} = 15.4
\]
\[
\bar{y} = \frac{13.3}{5} = 2.66
\]
### Step 2: Calculate m (slope)
Now we need to compute \( \sum (x_i - \bar{x})(y_i - \bar{y}) \) and \( \sum (x_i - \bar{x})^2 \).
Calculating \( (x_i - \bar{x}) \), \( (y_i - \bar{y}) \), \( (x_i - \bar{x})(y_i - \bar{y}) \), and \( (x_i - \bar{x})^2 \):
| \(x_i\) | \(y_i\) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \( (x_i - \bar{x})(y_i - \bar{y}) \) | \( (x_i - \bar{x})^2 \) |
|---------|---------|-------------------|-------------------|--------------------------------------|---------------------------|
| 4 | 2.2 | 4 - 15.4 = -11.4 | 2.2 - 2.66 = -0.46 | (-11.4)(-0.46) = 5.244 | (-11.4)^2 = 129.96 |
| 8 | 2.1 | 8 - 15.4 = -7.4 | 2.1 - 2.66 = -0.56 | (-7.4)(-0.56) = 4.144 | (-7.4)^2 = 54.76 |
| 13 | 1.8 | 13 - 15.4 = -2.4 | 1.8 - 2.66 = -0.86 | (-2.4)(-0.86) = 2.064 | (-2.4)^2 = 5.76 |
| 23 | 3.4 | 23 - 15.4 = 7.6 | 3.4 - 2.66 = 0.74 | (7.6)(0.74) = 5.624 | (7.6)^2 = 57.76 |
| 29 | 3.8 | 29 - 15.4 = 13.6 | 3.8 - 2.66 = 1.14 | (13.6)(1.14) = 15.504 | (13.6)^2 = 184.96 |
Now sum them:
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 5.244 + 4.144 + 2.064 + 5.624 + 15.504 = 32.58
\]
\[
\sum (x_i - \bar{x})^2 = 129.96 + 54.76 + 5.76 + 57.76 + 184.96 = 433.20
\]
Now calculate the slope \( m \):
\[
m = \frac{32.58}{433.20} \approx 0.075
\]
### Step 3: Calculate b (intercept)
Now calculate the y-intercept \( b \):
\[
b = \bar{y} - m\bar{x} = 2.66 - 0.075(15.4) = 2.66 - 1.155 = 1.505
\]
### Final Result
Thus, the equation for the least squares regression line is:
\[
y = 0.075x + 1.505
\]
Rounding to the nearest thousandth:
\[
y = 0.075x + 1.505
\]
1. **Find the averages of x and y**:
\[
\bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n}
\]
2. **Calculate the slope (m)**:
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
3. **Calculate the intercept (b)**:
\[
b = \bar{y} - m\bar{x}
\]
Now, let's compute these values step-by-step based on the provided data:
### Given Data
| Breakfasts (x) | GPA (y) |
|----------------|---------|
| 4 | 2.2 |
| 8 | 2.1 |
| 13 | 1.8 |
| 23 | 3.4 |
| 29 | 3.8 |
### Step 1: Compute Averages
\[
\sum x = 4 + 8 + 13 + 23 + 29 = 77
\]
\[
\sum y = 2.2 + 2.1 + 1.8 + 3.4 + 3.8 = 13.3
\]
\[
n = 5
\]
Now, calculate the averages:
\[
\bar{x} = \frac{77}{5} = 15.4
\]
\[
\bar{y} = \frac{13.3}{5} = 2.66
\]
### Step 2: Calculate m (slope)
Now we need to compute \( \sum (x_i - \bar{x})(y_i - \bar{y}) \) and \( \sum (x_i - \bar{x})^2 \).
Calculating \( (x_i - \bar{x}) \), \( (y_i - \bar{y}) \), \( (x_i - \bar{x})(y_i - \bar{y}) \), and \( (x_i - \bar{x})^2 \):
| \(x_i\) | \(y_i\) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \( (x_i - \bar{x})(y_i - \bar{y}) \) | \( (x_i - \bar{x})^2 \) |
|---------|---------|-------------------|-------------------|--------------------------------------|---------------------------|
| 4 | 2.2 | 4 - 15.4 = -11.4 | 2.2 - 2.66 = -0.46 | (-11.4)(-0.46) = 5.244 | (-11.4)^2 = 129.96 |
| 8 | 2.1 | 8 - 15.4 = -7.4 | 2.1 - 2.66 = -0.56 | (-7.4)(-0.56) = 4.144 | (-7.4)^2 = 54.76 |
| 13 | 1.8 | 13 - 15.4 = -2.4 | 1.8 - 2.66 = -0.86 | (-2.4)(-0.86) = 2.064 | (-2.4)^2 = 5.76 |
| 23 | 3.4 | 23 - 15.4 = 7.6 | 3.4 - 2.66 = 0.74 | (7.6)(0.74) = 5.624 | (7.6)^2 = 57.76 |
| 29 | 3.8 | 29 - 15.4 = 13.6 | 3.8 - 2.66 = 1.14 | (13.6)(1.14) = 15.504 | (13.6)^2 = 184.96 |
Now sum them:
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 5.244 + 4.144 + 2.064 + 5.624 + 15.504 = 32.58
\]
\[
\sum (x_i - \bar{x})^2 = 129.96 + 54.76 + 5.76 + 57.76 + 184.96 = 433.20
\]
Now calculate the slope \( m \):
\[
m = \frac{32.58}{433.20} \approx 0.075
\]
### Step 3: Calculate b (intercept)
Now calculate the y-intercept \( b \):
\[
b = \bar{y} - m\bar{x} = 2.66 - 0.075(15.4) = 2.66 - 1.155 = 1.505
\]
### Final Result
Thus, the equation for the least squares regression line is:
\[
y = 0.075x + 1.505
\]
Rounding to the nearest thousandth:
\[
y = 0.075x + 1.505
\]