Question
Find the equation for the least squares regression line of the data described below. Joseph is a crime scene investigator. He found a footprint at the site of a recent murder and believes the footprint belongs to the culprit. To help identify possible suspects, he is investigating the relationship between a person's height and the length of his or her footprint. He consulted his agency's database and found cases in which detectives had recorded the length of people's footprints, x, and their heights (in centimeters), y. Footprint length (in centimeters) Height (in centimeters) 24.01 169 24.92 175 28.33 172 28.68 184 29.85 189 Round your answers to the nearest thousandth. y=x+
Answers
GPT-4o mini
To find the least squares regression line for the data provided, we need to perform the following steps:
1. **Organize the Data**: Let’s create two lists from the given footprint lengths (x) and heights (y):
- Footprint Length (x):
- 24.01,
- 24.92,
- 28.33,
- 28.68,
- 29.85
- Heights (y):
- 169,
- 175,
- 172,
- 184,
- 189
2. **Calculate Necessary Summations**: To compute the regression line equation \(y = mx + b\), we need to find the following:
- \(n\): the number of data points
- \(\sum x\): the sum of all x values
- \(\sum y\): the sum of all y values
- \(\sum xy\): the sum of the products of x and y values
- \(\sum x^2\): the sum of the squares of x values
3. **Calculate Each Value**:
- \(n = 5\)
- \(\sum x = 24.01 + 24.92 + 28.33 + 28.68 + 29.85 = 135.79\)
- \(\sum y = 169 + 175 + 172 + 184 + 189 = 889\)
- \(\sum xy = (24.01 \cdot 169) + (24.92 \cdot 175) + (28.33 \cdot 172) + (28.68 \cdot 184) + (29.85 \cdot 189)\)
- \(= 4055.69 + 4361.00 + 4877.96 + 5276.32 + 5646.65 = 24017.62\)
- \(\sum x^2 = (24.01^2) + (24.92^2) + (28.33^2) + (28.68^2) + (29.85^2)\)
- \(= 576.4801 + 620.8064 + 802.6889 + 822.5424 + 891.6225 = 2913.13\)
4. **Calculate the Slope (m)** and Intercept (b) Using Formulas:
- \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\)
- \(b = \frac{\sum y - m(\sum x)}{n}\)
5. **Calculate Slope (m)**:
- \(m = \frac{5(24017.62) - (135.79)(889)}{5(2913.13) - (135.79)^2}\)
- \(= \frac{120088.1 - 120482.91}{14565.65 - 18450.0641}\)
- \(= \frac{-394.81}{-3878.4141}\)
- \(= 0.1018\) (rounded to four decimal places)
6. **Calculate Intercept (b)**:
- \(b = \frac{889 - (0.1018 \cdot 135.79)}{5}\)
- \(= \frac{889 - 13.8107}{5}\)
- \(= \frac{875.1893}{5}\)
- \(= 175.0379\) (rounded to four decimal places)
7. **Formulate the Regression Equation**:
- The regression line equation can thus be represented as:
\[
y = 0.102x + 175.038
\]
So rounding to the nearest thousandth, the final least squares regression line is:
\[
y = 0.102x + 175.038
\]
1. **Organize the Data**: Let’s create two lists from the given footprint lengths (x) and heights (y):
- Footprint Length (x):
- 24.01,
- 24.92,
- 28.33,
- 28.68,
- 29.85
- Heights (y):
- 169,
- 175,
- 172,
- 184,
- 189
2. **Calculate Necessary Summations**: To compute the regression line equation \(y = mx + b\), we need to find the following:
- \(n\): the number of data points
- \(\sum x\): the sum of all x values
- \(\sum y\): the sum of all y values
- \(\sum xy\): the sum of the products of x and y values
- \(\sum x^2\): the sum of the squares of x values
3. **Calculate Each Value**:
- \(n = 5\)
- \(\sum x = 24.01 + 24.92 + 28.33 + 28.68 + 29.85 = 135.79\)
- \(\sum y = 169 + 175 + 172 + 184 + 189 = 889\)
- \(\sum xy = (24.01 \cdot 169) + (24.92 \cdot 175) + (28.33 \cdot 172) + (28.68 \cdot 184) + (29.85 \cdot 189)\)
- \(= 4055.69 + 4361.00 + 4877.96 + 5276.32 + 5646.65 = 24017.62\)
- \(\sum x^2 = (24.01^2) + (24.92^2) + (28.33^2) + (28.68^2) + (29.85^2)\)
- \(= 576.4801 + 620.8064 + 802.6889 + 822.5424 + 891.6225 = 2913.13\)
4. **Calculate the Slope (m)** and Intercept (b) Using Formulas:
- \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\)
- \(b = \frac{\sum y - m(\sum x)}{n}\)
5. **Calculate Slope (m)**:
- \(m = \frac{5(24017.62) - (135.79)(889)}{5(2913.13) - (135.79)^2}\)
- \(= \frac{120088.1 - 120482.91}{14565.65 - 18450.0641}\)
- \(= \frac{-394.81}{-3878.4141}\)
- \(= 0.1018\) (rounded to four decimal places)
6. **Calculate Intercept (b)**:
- \(b = \frac{889 - (0.1018 \cdot 135.79)}{5}\)
- \(= \frac{889 - 13.8107}{5}\)
- \(= \frac{875.1893}{5}\)
- \(= 175.0379\) (rounded to four decimal places)
7. **Formulate the Regression Equation**:
- The regression line equation can thus be represented as:
\[
y = 0.102x + 175.038
\]
So rounding to the nearest thousandth, the final least squares regression line is:
\[
y = 0.102x + 175.038
\]