All Flashcards
How do you determine the best type of function (linear, exponential, or quadratic) to model a given dataset?
- Examine the rate of change: constant (linear), increasing/decreasing (exponential), changing direction (quadratic). 2. Plot the data to visualize the pattern.
How do you interpret a residual plot to assess the fit of a model?
- Examine the scatter of residuals. 2. Random scatter indicates a good fit. 3. A pattern indicates a poor fit.
How do you calculate and interpret residuals?
- Calculate: (Residual = Actual - Predicted). 2. Interpret: Positive residual = underestimation; Negative residual = overestimation.
Given a set of data and a proposed linear model, how do you calculate the residuals?
- For each data point, use the linear model to predict the y-value. 2. Subtract the predicted y-value from the actual y-value to find the residual.
Given a set of data and a proposed exponential model, how do you calculate the residuals?
- For each data point, use the exponential model to predict the y-value. 2. Subtract the predicted y-value from the actual y-value to find the residual.
Given a set of data and a proposed quadratic model, how do you calculate the residuals?
- For each data point, use the quadratic model to predict the y-value. 2. Subtract the predicted y-value from the actual y-value to find the residual.
How do you choose between overestimating and underestimating in a real-world scenario?
Consider the consequences of each. Choose the prediction that minimizes the potential negative impact.
How do you build a model to fit a given dataset?
- Plot the data. 2. Determine the type of function (linear, exponential, or quadratic) that best represents the data. 3. Find the equation of the function.
How do you validate a model?
- Calculate the residuals. 2. Plot the residuals. 3. Check for random scatter.
How do you determine if an exponential model is a good fit for a dataset?
- Calculate the residuals. 2. Plot the residuals. 3. Check for random scatter.
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.
Explain when a linear model is most appropriate.
When the data exhibits a constant rate of change, forming a straight-line pattern.
Explain when an exponential model is most appropriate.
When the data exhibits growth or decay patterns, with a changing rate of change.
Explain when a quadratic model is most appropriate.
When the data exhibits a parabolic (U-shaped) pattern, with a changing rate of change.
What does a random scatter of residuals indicate?
A good model fit, where the model's errors are randomly distributed around zero.
What does a pattern in the residuals indicate?
A poor model fit, suggesting the model is not accurately capturing the underlying trend in the data.
Why is the context of the problem important when considering overestimation vs. underestimation?
The consequences of over or under predicting can vary greatly depending on the real-world scenario. For example, overestimating hospital resources is preferable to underestimating.
Explain the significance of residuals in determining the appropriateness of a model.
Residuals indicate how well the model fits the data. If residuals are randomly scattered, the model is a good fit; if they form a pattern, the model is not a good fit.
What does the sign of a residual tell you?
A positive residual indicates the model underestimated the actual value; a negative residual indicates the model overestimated the actual value.
How can you visually assess if a linear model is appropriate for a given dataset?
Plot the data points on a scatter plot. If the points appear to form a straight line, a linear model may be appropriate.
How can you visually assess if an exponential model is appropriate for a given dataset?
Plot the data points on a scatter plot. If the points appear to follow a curve that increases or decreases rapidly, an exponential model may be appropriate.