Question

Find and Use an Exponential Model to Find Inputs and Outputs Use Regression to Find and Use an Exponential Model to answer the questions The numbers of polio cases in the world are shown in the table for various years.

Year	Number of Polio Cases (thousands)
1988	350
1992	138
1996	33
2000	3
2005	3.2
2007	1.3

Let f(t)f(t) be the number of polio cases in the world t years since 1980.

Use a graphing calculator to draw a scattergram of the data. Is it better to model the data by using a linear or exponential model? Select an answerExponentialLinear Find an equation of f. Hint

f(t)=f(t)= (Round to 2 decimal places.)

The number of polio cases is Select an answerincreasingdecreasing by Select an answer26%74%3701.26%370126% per year.

Predict the number of polio cases in 2018.

Hint (Round to the nearest whole number.)

Predict in which year there will be 1000 cases of polio.

Hint (Round to the nearest year.) Find the approximate half-life of the number of polio cases. Hint

years (Round to one decimal place.)

Answer 1

Solution:

The scatter plot of the data is given below:

From the scatter plot we can see an exponential model will fit the data.

Now we fit an exponential regression model and obtain the following results:

Model Summary
R	R Square	Adjusted R Square	Std. Error of the Estimate
.970	.940	.925	.627
The independent variable is Year.

ANOVA
	Sum of Squares	df	Mean Square	F	Sig.
Regression	24.801	1	24.801	63.048	.001
Residual	1.573	4	.393
Total	26.375	5
The independent variable is Year.

Coefficients
	Unstandardized Coefficients		Standardized Coefficients	t	Sig.
	B	Std. Error	Beta
Year	-.301	.038	-.970	-7.940	.001
(Constant)	1.896E+262	1.435E+264		.013	.990
The dependent variable is ln(Number_of_Poliio_Case).

The exponential regression model is given by:

where,

We put t = 2018 in the exponential fit to predict the number of polio cases in 2018:

So, the estimated number of polio cases in 2018 is = 0.03016843 thousand = 30.16843.

Now we want to find the year when the number of polio cases will be 1000.

We put to estimate t and obtain the following:

Hence year 2006.369 there will be 1000 polio cases.

We know 350/2 = 175.

We put to estimate half-life t and obtain the following:

Answer: The half life = 1989.21 year