Question
Find the equation for the least squares regression line of the data described below.
Dakota teaches a standardized test prep course. Students take a diagnostic test at the beginning of the course and then take the real test at the end. Dakota wanted to know how well the diagnostic test predicts a student's score on the real test.
She recorded each student's score on the diagnostic test, x, and on the real test, y. Both tests were graded on a scale from 120 to 180.
Diagnostic test score Real test score
140 127
154 147
155 171
162 177
165 163
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the equation for the least squares regression line, we first need to calculate the necessary statistical values. The general form of the least squares regression line is:
\[
y = mx + b
\]
Where:
- \(m\) is the slope of the line,
- \(b\) is the y-intercept.
### Step 1: Calculate the Means
First, we need to compute the means of the \(x\) and \(y\) values.
**Data Points:**
- \(x\) (Diagnostic test scores): 140, 154, 155, 162, 165
- \(y\) (Real test scores): 127, 147, 171, 177, 163
**Mean of \(x\) (\(\overline{x}\))**:
\[
\overline{x} = \frac{140 + 154 + 155 + 162 + 165}{5} = \frac{776}{5} = 155.2
\]
**Mean of \(y\) (\(\overline{y}\))**:
\[
\overline{y} = \frac{127 + 147 + 171 + 177 + 163}{5} = \frac{785}{5} = 157.0
\]
### Step 2: Calculate the Slope (m)
The slope \(m\) can be calculated using the formula:
\[
m = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}}
\]
#### Calculate Necessary Summations:
1. Compute \( (x_i - \overline{x}) \) and \( (y_i - \overline{y}) \) for each point:
| \(x_i\) | \(y_i\) | \(x_i - \overline{x}\) | \(y_i - \overline{y}\) | \( (x_i - \overline{x})(y_i - \overline{y}) \) | \( (x_i - \overline{x})^2 \) |
|---------|---------|-------------------------|-------------------------|-----------------------------------------------|--------------------------------|
| 140 | 127 | -15.2 | -30 | 456 | 231.04 |
| 154 | 147 | -1.2 | -10 | 12 | 1.44 |
| 155 | 171 | -0.2 | 14 | -2.8 | 0.04 |
| 162 | 177 | 6.8 | 20 | 136 | 46.24 |
| 165 | 163 | 9.8 | 6 | 58.8 | 96.04 |
2. Now sum the values in the last two columns:
\[
\sum{(x_i - \overline{x})(y_i - \overline{y})} = 456 + 12 - 2.8 + 136 + 58.8 = 660
\]
\[
\sum{(x_i - \overline{x})^2} = 231.04 + 1.44 + 0.04 + 46.24 + 96.04 = 374.8
\]
3. Now we can calculate \(m\):
\[
m = \frac{660}{374.8} \approx 1.759
\]
### Step 3: Calculate the Y-Intercept (b)
The y-intercept \(b\) can be calculated using:
\[
b = \overline{y} - m\overline{x}
\]
\[
b = 157.0 - (1.759)(155.2)
\]
Calculating \(1.759 \times 155.2\):
\[
1.759 \times 155.2 \approx 273.4393
\]
Now calculate \(b\):
\[
b = 157.0 - 273.4393 \approx -116.4393
\]
### Conclusion: Regression Line Equation
Substituting the values of \(m\) and \(b\) into the regression line equation:
\[
y \approx 1.759x - 116.439
\]
Rounding to the nearest thousandth:
\[
y = 1.759x - 116.439
\]
So the final regression equation is:
\[
\boxed{y = 1.759x - 116.439}
\]
\[
y = mx + b
\]
Where:
- \(m\) is the slope of the line,
- \(b\) is the y-intercept.
### Step 1: Calculate the Means
First, we need to compute the means of the \(x\) and \(y\) values.
**Data Points:**
- \(x\) (Diagnostic test scores): 140, 154, 155, 162, 165
- \(y\) (Real test scores): 127, 147, 171, 177, 163
**Mean of \(x\) (\(\overline{x}\))**:
\[
\overline{x} = \frac{140 + 154 + 155 + 162 + 165}{5} = \frac{776}{5} = 155.2
\]
**Mean of \(y\) (\(\overline{y}\))**:
\[
\overline{y} = \frac{127 + 147 + 171 + 177 + 163}{5} = \frac{785}{5} = 157.0
\]
### Step 2: Calculate the Slope (m)
The slope \(m\) can be calculated using the formula:
\[
m = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}}
\]
#### Calculate Necessary Summations:
1. Compute \( (x_i - \overline{x}) \) and \( (y_i - \overline{y}) \) for each point:
| \(x_i\) | \(y_i\) | \(x_i - \overline{x}\) | \(y_i - \overline{y}\) | \( (x_i - \overline{x})(y_i - \overline{y}) \) | \( (x_i - \overline{x})^2 \) |
|---------|---------|-------------------------|-------------------------|-----------------------------------------------|--------------------------------|
| 140 | 127 | -15.2 | -30 | 456 | 231.04 |
| 154 | 147 | -1.2 | -10 | 12 | 1.44 |
| 155 | 171 | -0.2 | 14 | -2.8 | 0.04 |
| 162 | 177 | 6.8 | 20 | 136 | 46.24 |
| 165 | 163 | 9.8 | 6 | 58.8 | 96.04 |
2. Now sum the values in the last two columns:
\[
\sum{(x_i - \overline{x})(y_i - \overline{y})} = 456 + 12 - 2.8 + 136 + 58.8 = 660
\]
\[
\sum{(x_i - \overline{x})^2} = 231.04 + 1.44 + 0.04 + 46.24 + 96.04 = 374.8
\]
3. Now we can calculate \(m\):
\[
m = \frac{660}{374.8} \approx 1.759
\]
### Step 3: Calculate the Y-Intercept (b)
The y-intercept \(b\) can be calculated using:
\[
b = \overline{y} - m\overline{x}
\]
\[
b = 157.0 - (1.759)(155.2)
\]
Calculating \(1.759 \times 155.2\):
\[
1.759 \times 155.2 \approx 273.4393
\]
Now calculate \(b\):
\[
b = 157.0 - 273.4393 \approx -116.4393
\]
### Conclusion: Regression Line Equation
Substituting the values of \(m\) and \(b\) into the regression line equation:
\[
y \approx 1.759x - 116.439
\]
Rounding to the nearest thousandth:
\[
y = 1.759x - 116.439
\]
So the final regression equation is:
\[
\boxed{y = 1.759x - 116.439}
\]