What is a non-linear regression model ?

A non-linear regression model is a curve that is used to model bivariate data rather than a straight-line.

True or False? You need to find the equation of any non-linear regression model by hand using algebra.

False. You do not need to find the equation of any non-linear regression model by hand using algebra. You have to use your GDC to find equations.

What is a least squares regression curve ?

For a given type of model, a least squares regression curve is the curve that minimises the sum of the square residuals .

What is the value of the coefficient of determination for a model that is a perfect fit for some data?

The coefficient of determination is equal to 1 for a model that is a perfect fit for some data.

If you have to decide on a model based solely on the coefficients of determination , should you choose the one with the smaller or bigger coefficient?

If you have to decide on a model based solely on the coefficients of determination , you choose the one with the bigger coefficient. A bigger coefficient of determination implies a better fit with the data.

True or False? Logarithmic scales increase exponentially .

True. Logarithmic scales increase exponentially .

What is a log-log graph ?

A log-log graph uses logarithmic scales for both axes .

What is a semi-log graph ?

A semi-log graph only uses a logarithmic scale for the y-axis .

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What is a non-linear regression model?

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What is a non-linear regression model?
A non-linear regression model is a curve that is used to model bivariate data rather than a straight-line.
True or False?
You need to find the equation of any non-linear regression model by hand using algebra.
False.
You do not need to find the equation of any non-linear regression model by hand using algebra.
You have to use your GDC to find equations.
$f (x)$ is a regression model which is used to predict the values of $y$ .
What is meant by a residual?
$f (x)$ is a regression model which is used to predict the values of $y$ .
A residual is the difference between an actual value and the predicted value.
Residual = $y_{i} - f (x_{i})$ .
What is denoted by $S S_{r e s}$ ?
$S S_{r e s}$ denotes the sum of the square residuals of a regression model.
$S S_{r e s} = \sum_{i = 1}^{n} {(y_{i} - f (x_{i}))}^{2}$ .
This is not given in your exam formula booklet.
True or False?
If you have two regression models using the same data then the one with the smaller $S S_{r e s}$ fits the data better.
True.
If you have two regression models using the same data then the one with the smaller $S S_{r e s}$ fits the data better.
What is a least squares regression curve?
For a given type of model, a least squares regression curve is the curve that minimises the sum of the square residuals.
What is the coefficient of determination ( $R^{2}$ )?
The coefficient of determination ( $R^{2}$ ) is a measure of fit for a model.
It is the proportion of the variance of the dependent variable that can be explained by the variance of the independent variable.
What is the value of the coefficient of determination for a model that is a perfect fit for some data?
The coefficient of determination is equal to 1 for a model that is a perfect fit for some data.
True or False?
For a linear regression model, the coefficient of determination is equal to the square of the PMCC, i.e. $R^{2} = r^{2}$ .
True.
For a linear regression model, the coefficient of determination is equal to the square of the PMCC, i.e. $R^{2} = r^{2}$ .
If you have to decide on a model based solely on the coefficients of determination, should you choose the one with the smaller or bigger coefficient?
If you have to decide on a model based solely on the coefficients of determination, you choose the one with the bigger coefficient.
A bigger coefficient of determination implies a better fit with the data.
True or False?
Logarithmic scales increase exponentially.
True.
Logarithmic scales increase exponentially.
What is a log-log graph?
A log-log graph uses logarithmic scales for both axes.
What is a semi-log graph?
A semi-log graph only uses a logarithmic scale for the y-axis.
Which type of graph appears as a straight line when plotted on a log-log graph?
A power graph ( $y = a x^{b}$ ) will appear as a straight line when plotted on a log-log graph.
Which type of graph appears as a straight line when plotted on a semi-log graph?
An exponential graph ( $y = a b^{x}$ ) will appear as a straight line when plotted on a semi-log graph.
The point $(3, 1.74)$ lies on the graph where base 10 is used for the logarithms. How would you find the corresponding pair of data $(x, y)$ ?
To find the corresponding pair of data, you would raise both values as powers of 10, i.e. $(10^{3}, 10^{1.74})$ .
Which variables should you use to linearise the graph of an exponential graph $y = a b^{x}$ ?
To linearise the graph of an exponential graph $y = a b^{x}$ , you should use the variables $x$ and $\log y$ using any base, (although the $\ln$ is frequently used).
Which variables should you use to linearise the graph of a power graph $y = a x^{b}$ ?
To linearise the graph of a power graph $y = a x^{b}$ , you should use the variables $\log x$ and $\log y$ using any base, (although the $\ln$ is frequently used).
If $y = a b^{x}$ , write an expression for $\log y$ in terms of $x$ .
If $y = a b^{x}$ , then $\log y = \log a + x \log b$ .
If $y = a x^{b}$ , then write an expression for $\log y$ in terms of $\log x$ .
If $y = a x^{b}$ , then $\log y = \log a + b \log x$ .
If $\log y = a x + b$ then which type of function is $y$ : exponential or power?
If $\log y = a x + b$ then $y$ is an exponential graph.
If you know the equation of the straight line that is produced when a power graph is plotted on a log-log graph, how can you find the equation of the power function?
For example, if you know $\ln y = 2 + 3 \ln x$ , how can you find the values of $a$ and $b$ where $y = a x^{b}$ ?
If you know the equation of the straight line that is produced when a power graph is plotted on a log-log graph, then to find the equation of the power function:
- linearise the general form of a power function, $\ln y = \ln a + b \ln x$ ,
- equate coefficients with the equation of the straight line,
- solve to find the values of $a$ and $b$ .
For example, if you know $\ln y = 2 + 3 \ln x$ , then $\ln a = 2 \Rightarrow a = e^{2}$ and $b = 3$ .

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