Concept:
SS is a method to find the error between observed data and an expression or function representing the data.
Example:
Suppose you perform an experiment and get data points (x1, y1), (x2, y2), ... (xn, yn). You assume a linear expression \( y = mx + c \).
Concept:
Linear regression fits a linear model to a set of data points to minimize the sum of squared errors.
Example:
Given data points (1,2), (2,3), and (3,4), the linear regression line \( y = mx + c \) is calculated to minimize SS.
Concept:
Quadratic expressions form parabolas, and their minimum or maximum can be found using vertex form.
Example:
For \( f(x) = ax^2 + bx + c \), the vertex (minimum or maximum) occurs at \( x = -\frac{b}{2a} \).
Concept:
The goal is to find the values of parameters (like m and c) that minimize SS.
Example:
Given an expression \( y = mx + c \), compute SS for various values of m and c. Use calculus (derivatives) to find the minimum SS.
1. Sum of Squares (SS):
\[ SS = \sum_{i=1}^n (y_i - (mx_i + c))^2 \]
2. Linear Regression Line:
\[ y = mx + c \]
3. Vertex of a Parabola:
\[ x = -\frac{b}{2a} \]
4. Quadratic Formula:
\[ ax^2 + bx + c = 0 \]
1. What is the purpose of the Sum of Squares (SS) method?
a) To find the average of data points.
b) To calculate the mean of data points.
c) To find the error between observed data and a function.
d) To determine the median of data points.
Answer: c) To find the error between observed data and a function.
2. In linear regression, the line \( y = mx + c \) is chosen to:
a) Maximize the sum of squares.
b) Minimize the sum of squares.
c) Equalize the sum of squares.
d) None of the above.
Answer: b) Minimize the sum of squares.
3. The vertex of the parabola \( y = ax^2 + bx + c \) occurs at:
a) \( x = \frac{b}{2a} \)
b) \( x = -\frac{b}{a} \)
c) \( x = -\frac{b}{2a} \)
d) \( x = \frac{b}{a} \)
Answer: c) \( x = -\frac{b}{2a} \)
4. In the formula \( y = mx + c \), what does 'm' represent?
a) The y-intercept.
b) The slope of the line.
c) The x-intercept.
d) The error term.
Answer: b) The slope of the line.
5. When is the sum of squares (SS) considered minimized?
a) When SS equals zero.
b) When SS is at its highest value.
c) When SS is at its lowest value.
d) When SS equals the mean of the data.
Answer: c) When SS is at its lowest value.
6. What does the quadratic expression \( ax^2 + bx + c \) represent in relation to a parabola?
a) The area under the parabola.
b) The equation of the parabola.
c) The height of the parabola.
d) The focus of the parabola.
Answer: b) The equation of the parabola.
7. Which method is used to find the minimum value of a quadratic expression?
a) Differentiation.
b) Integration.
c) Addition.
d) Subtraction.
Answer: a) Differentiation.
8. In the context of SS, what does the term \( (y_i - \hat{y_i})^2 \) represent?
a) The total error.
b) The squared error.
c) The average error.
d) The sum of the errors.
Answer: b) The squared error.
9. For the equation \( y = mx + c \), which term represents the y-intercept?
a) y
b) x
c) m
d) c
Answer: d) c
10. What is the main goal of using the SS method in experiments?
a) To find the median of data points.
b) To identify the best-fit expression for the data.
c) To calculate the range of data points.
d) To determine the mode of data points.
Answer: b) To identify the best-fit expression for the data.
1. Sum of Squares (SS):
\[ SS = \sum_{i=1}^n (y_i - \hat{y_i})^2 \]
2. Linear Regression Line:
\[ y = mx + c \]
3. Quadratic Formula:
\[ ax^2 + bx + c = 0 \]
4. Vertex of a Parabola:
\[ x = -\frac{b}{2a} \]