In Example 1 on page 731, we used two data points and an...

In Example 1 on page 731, we used two data points and an exponential function to model the population of the United States from 1970 through 2009. The data are shown again in the table. Use all five data points. $$\begin{array}{c|c} x, \text { Number of Years after } 1969 & y, \text { U.S. Population (millions) } \\ \hline 1(1970) & 203.3 \\ \hline 11(1980) & 226.5 \\ \hline 21(1990) & 248.7 \\ \hline 31(2000) & 281.4 \\ \hline 40(2009) & 307.0 \\ \hline \end{array}$$ a. Use your graphing utility's exponential regression option to obtain a model of the form $y=a b^{x}$ that fits the data. How well does the correlation coefficient, $r,$ indicate that the model fits the data? b. Rewrite the model in terms of base e. By what percentage is the population of the United States increasing each year?

In Example 1 on page 731, we used two data points and an exponential function to model the population of the United States from 1970 through 2009. The data are shown again in the table. Use all five data points.
$$\begin{array}{c|c}
x, \text { Number of Years after } 1969 & y, \text { U.S. Population (millions) } \\
\hline 1(1970) & 203.3 \\
\hline 11(1980) & 226.5 \\
\hline 21(1990) & 248.7 \\
\hline 31(2000) & 281.4 \\
\hline 40(2009) & 307.0 \\
\hline
\end{array}$$
a. Use your graphing utility's exponential regression option to obtain a model of the form $y=a b^{x}$ that fits the data. How well does the correlation coefficient, $r,$ indicate that the model fits the data?
b. Rewrite the model in terms of base
e. By what percentage is the population of the United States increasing each year?

Key Concepts

Exponential Regression

Exponential regression is a statistical method used to determine the exponential function that best fits a set of data points. It involves finding constants in the model y = a * b^x such that the model minimizes deviation from the actual data, typically by using a least squares approach. This method is essential when analyzing growth or decay processes, like population dynamics, where change is proportional to the current state.

Correlation Coefficient

The correlation coefficient in the context of exponential regression measures the strength and direction of the linear relationship after a transformation (or, in some software, directly in the exponential scale) between the model predictions and the observed data. A value close to 1 or -1 indicates that the model fits the data very well, suggesting that the chosen exponential function accurately captures the trend in the data.

Conversion to Base e

Many exponential models are rewritten in terms of the natural base e because of its unique properties in calculus and continuous growth modeling. Transforming the model from the form y = a * b^x to y = a * e^(kx) involves using the logarithm (k = ln(b)), which simplifies analysis, particularly when dealing with differentiation and integration, and provides a clear interpretation of the continuous growth rate.

Percentage Growth Rate

The percentage growth rate in an exponential model represents the continuous rate at which the quantity increases per unit time. In the context of an exponential function, converting the base to e allows us to interpret the parameter k as the continuous growth rate, meaning that the population increases by approximately 100k percent per year. This rate is crucial for understanding the incremental changes in populations or other similar datasets.

Transcript

00:01 This question asks us to use a calculator to model the population growth in the us.

00:06 I've already entered the data that we're given into this calculator.

00:11 The way i did that is i pressed the stat button and then chose option 1 for edit.

00:16 You can see in list 1 i have the years and in list 2 i have the population in millions.

00:22 Now we're going to use an exponential regression on this, but before we do that, we need to turn a setting on.

00:28 It may already be on in your calculators, but it's always good to make sure to do that.

00:33 I'm going to quit out of this, and then i'm going to press second zero, which will bring us to the catalog.

00:40 Then i'm going to press the x to the negative first button because that's the d button.

00:47 It'll navigate us to the d's, and i'm going to scroll down until i find diagnostic on.

00:56 I want to turn diagnostic on because that's going to, when we run our regression, that will tell us what the r squared and r values are.

01:07 We need that.

01:08 Without diagnostic on, we would just get values for a and b.

01:11 So i press enter on that, press enter again, and now i've turned the diagnostic on.

01:16 So now what i'm going to do is i'm going to run our exponential regression.

01:20 And to do that, i go back to the stat button, and i navigate over to the calc selection, and i scroll down until i can.

01:27 Get to option zero, which is exponential regression.

01:31 When i press enter, it'll bring up this little list.

01:36 For my x variable, i know i want list one because that's where i put my years for my y variable.

01:41 I know i want list two...

Please give Ace some feedback