Answer :
To find the equation of the least squares regression line for the given data, we'll be determining the line that best fits the points in the dataset. Here, the dosage (in milligrams) is our independent variable [tex]\( x \)[/tex], and the rise in blood sugar level (in milligrams per deciliter) is our dependent variable [tex]\( y \)[/tex].
Here’s the step-by-step process of how you could find the least squares regression line:
1. Data Points:
We have the following pairs of data:
- (161, 23.83)
- (177, 26.81)
- (272, 19.77)
- (483, 23.00)
- (593, 17.70)
- (794, 15.11)
2. Least Squares Regression Line Equation:
The equation of the least squares line is typically given as:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope of the line, and [tex]\( b \)[/tex] is the y-intercept.
3. Calculate Slope (m) and Intercept (b):
- The slope [tex]\( m \)[/tex] is calculated as follows:
[tex]\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\][/tex]
- The y-intercept [tex]\( b \)[/tex] is calculated using:
[tex]\[
b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}
\][/tex]
However, these calculations can become complex and are typically performed using computational methods or statistical software.
4. Final Results:
After performing these calculations, the slope [tex]\( m \)[/tex] and intercept [tex]\( b \)[/tex] were found to be:
- Slope ([tex]\( m \)[/tex]): [tex]\(-0.014\)[/tex]
- Intercept ([tex]\( b \)[/tex]): [tex]\(26.961\)[/tex]
5. Regression Line Equation:
Substituting these values into the equation format gives us:
[tex]\[
y = -0.014x + 26.961
\][/tex]
This is the equation for the least squares regression line that best fits the data, rounded to the nearest thousandth.
Here’s the step-by-step process of how you could find the least squares regression line:
1. Data Points:
We have the following pairs of data:
- (161, 23.83)
- (177, 26.81)
- (272, 19.77)
- (483, 23.00)
- (593, 17.70)
- (794, 15.11)
2. Least Squares Regression Line Equation:
The equation of the least squares line is typically given as:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope of the line, and [tex]\( b \)[/tex] is the y-intercept.
3. Calculate Slope (m) and Intercept (b):
- The slope [tex]\( m \)[/tex] is calculated as follows:
[tex]\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\][/tex]
- The y-intercept [tex]\( b \)[/tex] is calculated using:
[tex]\[
b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}
\][/tex]
However, these calculations can become complex and are typically performed using computational methods or statistical software.
4. Final Results:
After performing these calculations, the slope [tex]\( m \)[/tex] and intercept [tex]\( b \)[/tex] were found to be:
- Slope ([tex]\( m \)[/tex]): [tex]\(-0.014\)[/tex]
- Intercept ([tex]\( b \)[/tex]): [tex]\(26.961\)[/tex]
5. Regression Line Equation:
Substituting these values into the equation format gives us:
[tex]\[
y = -0.014x + 26.961
\][/tex]
This is the equation for the least squares regression line that best fits the data, rounded to the nearest thousandth.
