Chapter 3.5, Problem 54E

Solution

To plot: The type of regression model suitable for the given data and explain as needed.

The exponential regression model is the best fir for the given data and is given as shown below

  $y=(11.7781){(0.9278)}^x$

Given:

The given data is shown below

X	Y
1	11
8	6
15	4.8
16	4
17	2.5

Concept used:

The different types of regression models are shown below

Logarithmic regression model $y=a+b\ln x$

Linear regression model $y=a+bx$

Exponential regression model $y=a{b}^x$

Where,

  $\begin{array}{l}a=\text{Intercept coefficient}\\ b=\text{Slope coefficient}\\ x=\text{Independent variable}\\ y=\text{Dependent variable}\end{array}$

Plot:

The data x, y is entered in L1 and L2 and $\ln x,\text{ lny}$ values in L3 and L4 in the calculator as shown below

L1 (x)	L2 (y)	L3 (lnx)	L4 (lny)
1	11	0.0000	2.3979
8	6	2.0794	1.7918
15	4.8	2.7081	1.5686
16	4	2.7726	1.3863
17	2.5	2.8332	0.9163

Using the data of $\ln x$ and y a scatterplot is inserted in the calculator and get the following

  

Using the data of $\ln x$ and $\ln y$ a scatterplot is inserted in the calculator and get the following

  

Using the data of x and $\ln y$ a scatterplot is inserted in the calculator and get the following

  

Interpretation:

With reference to scatterplots, the plot of x and $\ln y$ seems to be approximately best fit among all plots, so an exponential model would be the best fit.

Calculation:

Press STAT button in the calculator and go to CALC menu to select Exp Reg option. Select the appropriate data from the lists and the following regression model output and its plot are observed.

    

Conclusion:

The given data has a best fit exponential regression model with equation $y=(11.7781){(0.9278)}^x$ and coefficient of determination of 0.875 which indicate that it is a good model.