All Flashcards
Define a linear function.
A function of the form (f(x) = b + mx) with a constant rate of change.
Define an exponential function.
A function of the form (f(x) = ab^x) with a changing rate of change dependent on the base 'b'.
Define a quadratic function.
A function of the form (f(x) = ax^2 + bx + c) with a changing rate of change dependent on the coefficient 'a'.
What are residuals in model validation?
The differences between the actual data values and the values predicted by the model. (Residual = Actual - Predicted)
What does a residual represent?
The error or difference between an observed value and the value predicted by a model.
What does 'overestimate' mean in modeling?
When a model's prediction is higher than the actual value.
What does 'underestimate' mean in modeling?
When a model's prediction is lower than the actual value.
Define 'error' in the context of model validation.
The difference between the predicted value and the actual value.
What is a residual plot?
A graph that displays the residuals on the y-axis and the independent variable on the x-axis.
What is model validation?
The process of checking whether a statistical model accurately represents the data and makes reliable predictions.
What is the general form of a linear function?
(f(x) = b + mx)
What is the general form of an exponential function?
(f(x) = ab^x)
What is the general form of a quadratic function?
(f(x) = ax^2 + bx + c)
How do you calculate a residual?
(Residual = Actual - Predicted)
Given data points, how do you determine the equation of an exponential function?
Use two points ((x_1, y_1)) and ((x_2, y_2)) to solve for (a) and (b) in (f(x) = ab^x).
How to determine the equation of a linear function?
Use the slope-intercept form: (y = mx + b), where (m) is the slope and (b) is the y-intercept.
How to determine the equation of a quadratic function from its vertex form?
Use the vertex form: (y = a(x - h)^2 + k), where ((h, k)) is the vertex of the parabola.
How to calculate predicted population in a exponential model?
Use the exponential model equation: (f(x) = ab^x), where (x) is the time, (a) is the initial population, and (b) is the growth factor.
How to calculate predicted population in a linear model?
Use the linear model equation: (f(x) = b + mx), where (x) is the time, (b) is the initial population, and (m) is the rate of change.
How to calculate predicted population in a quadratic model?
Use the quadratic model equation: (f(x) = ax^2 + bx + c), where (x) is the time, and (a), (b), and (c) are constants.
What are the key differences between linear and exponential functions in the context of data modeling?
Linear: Constant rate of change | Exponential: Changing rate of change, growth/decay patterns.
What are the key differences between quadratic and exponential functions in the context of data modeling?
Quadratic: Parabolic shape, changing direction | Exponential: Growth/decay patterns, rapidly increasing/decreasing.
Compare the residual plots of a good model vs. a bad model.
Good Model: Residuals randomly scattered around zero | Bad Model: Residuals show a pattern (curve or line).
Compare the appropriateness of linear vs. exponential models for population growth.
Linear: Suitable for short-term, constant growth | Exponential: Suitable for long-term, accelerating growth.
Compare the appropriateness of linear vs. quadratic models for modeling projectile motion.
Linear: Not suitable | Quadratic: Suitable for modeling the parabolic path of a projectile.
Compare the effect of overestimation vs. underestimation in financial forecasting.
Overestimation: Might lead to overspending | Underestimation: Might lead to insufficient budgeting.
Compare the effect of overestimation vs. underestimation in resource allocation.
Overestimation: Might lead to waste of resources | Underestimation: Might lead to shortage of resources.
Compare the effect of overestimation vs. underestimation in medical diagnosis.
Overestimation: Might lead to unnecessary treatment | Underestimation: Might lead to delayed treatment.
Compare the rate of change in linear vs. quadratic functions.
Linear: Constant rate of change | Quadratic: Rate of change varies linearly.
Compare the rate of change in exponential vs. quadratic functions.
Exponential: Rate of change varies exponentially | Quadratic: Rate of change varies linearly.