Question

Describe the model that produces a least squares fit of these points by a function of the form y = a.

a) Linear
b) Exponential
c) Quadratic
d) Logarithmic

Answer 1

The student is asked to describe how to perform a least squares fit using linear, exponential, quadratic, and logarithmic models. This involves entering data into a calculator, generating a scatter plot, and using the regression function to find the best-fitting equation. Additionally, the correlation coefficient and presence of outliers are used to evaluate the fit.

Step-by-step explanation:

The question is asking how to produce a least squares fit for a set of data points using different types of functions, such as linear, exponential, quadratic, and logarithmic. Here's how this can be done for each function type:

1. Linear: Use a calculator or software to input the data and generate a scatter plot. Then apply the regression function to calculate the least-squares linear regression line, which will be of the form y = a + bx, where 'a' is the y-intercept and 'b' is the slope.
2. Exponential: Similarly, enter the data into a calculator and use the regression function to find an exponential fit, which has the form y = a * e^(bx), where 'a' is the initial amount and 'b' is the growth rate.
3. Quadratic: For a quadratic fit, the calculator's regression function will give an equation of the form y = ax^2 + bx + c, where 'a', 'b', and 'c' are constants that define the parabola's shape and position.
4. Logarithmic: When expecting a logarithmic relationship, use the regression function to find a logarithmic equation, typically y = a + b * ln(x), where 'a' is the adjustment term and 'b' indicates the rate of increase.

To determine if these fits are appropriate, one can look at the scatter plot and the correlation coefficient provided by the regression function. The correlation coefficient helps to understand how well the data is represented by the model. The absence or presence of outliers can also influence the choice of model.