I have a question about residual plots and regression equations.
The data is:
X Y
1 6.1
2 5.5
3 9.8
4 10.6
5 14.2
6 21.5
7 29.9
8 37.2
9 50.6
10 64
11 77.6
What function fits this data the best, between Exponential, Linear, Quadratic, and Power.
Also how do you make residual plots on a TI-83 Graphing Calculator?
Thank you so much for your help in advance. This is crucial to my grade.
First, graph it out. what does the graph look like. not straight so toss out Linear. Constantly rising (no humps), so (probably) toss out quadratic. Could be a power function, but offhand id say it was an exponential function (in that the function seems to be growing by e^(ax))
So, next take the natural log of your y values, and graph that. I get something resembling a straight line. Estimate an OLS regression on the logged values. Plot the predicted from the regression, and actual values; the difference between the two are the residuals (which could easily be plotted).
To determine the best-fitting function for the given data, we can first start by graphing the data points. Looking at the graph, we can see that the data does not form a straight line, so we can rule out a linear function as the best fit.
Next, let's check if the data follows a quadratic function. Quadratic functions have a distinct concave shape with a single hump. However, looking at the data, we do not see a clear hump in the graph, so we can also rule out a quadratic function as the best fit.
Now, let's consider a power function. Power functions have the general form y = ax^b, where a and b are constants. Power functions can represent different types of growth or decay, depending on the values of a and b. Although the data seems to be growing, it does not follow a strict power function pattern, so we cannot definitively say that a power function is the best fit.
Finally, let's consider an exponential function. Exponential functions have the general form y = ab^x, where a and b are constants. Exponential growth can be recognized by a rapid upward curve, which seems to be present in the data graph. Based on this observation, an exponential function is a strong candidate for the best fit.
To confirm if the data follows an exponential function, we can take the natural logarithm (ln) of the y-values and graph the result. If the resulting graph forms a straight line, it would indicate that the original data follows an exponential function.
To create a residual plot on a TI-83 Graphing Calculator, follow these steps:
1. Enter the data into a list on the calculator. For the x-values, enter them into L1, and for the y-values, enter them into L2.
2. Go to the "STAT" menu by pressing the "STAT" button.
3. Select "Edit" to access the lists.
4. Highlight L1 and L2, and then press the "GRAPH" button to plot the data points on the graph.
5. To find the natural logarithm of the y-values, go to the "MATH" menu by pressing the "2nd" button and then "LN" for natural logarithm.
6. Enter "LN(L2)" and press "ENTER". This will create a new list with the natural logarithm of the y-values.
7. Go back to the "STAT" menu and select "Plot1".
8. Highlight "Plot1", and under "Type," select "Scatter".
9. For "Xlist," enter L1, and for "Ylist," enter the list obtained from taking the natural logarithm, which could be, for example, L3.
10. Press "GRAPH" to plot the logged data points on the same graph as the original data points.
11. To generate the exponential regression equation, go to the "STAT" menu, select "CALC," and then choose "LnReg" for a logarithmic regression.
12. Press "ENTER" and specify the x-values as L1 and the y-values as L3 or the logged y-values list.
13. The calculator will display the exponential regression equation in the form y = a*b^x, along with the values of a and b.
To create the residual plot, subtract the predicted y-values from the actual y-values obtained from the regression equation. Plot the resulting differences, which are the residuals, against the x-values.
It is important to note that the decision on the best-fitting function may also depend on the context and underlying theory of the data.