91Ó°ÊÓ

There are several extensions of linear regression that apply to exponential growth and power law models. Problems \(22-25\) will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fixed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fixed number during each time period. This means that exponential growth increases by a fixed multiple or percentage of the previous amount. College algebra can be used to show that if a variable grows exponentially, then its logarithm grows linearly. The exponential growth model is \(y=\alpha \beta^{x}\), where \(\alpha\) and \(\beta\) are fixed constants to be estimated from data. How do we know when we are dealing with exponential growth, and how can we estimate \(\alpha\) and \(\beta\) ? Please read on. Populations of living things such as bacteria, locust, fish, panda bears, and so on tend to grow (or decline) exponentially. However, these populations can be restricted by outside limitations such as food, space, pollution, disease, hunting, and so on. Suppose we have data pairs \((x, y)\) for which there is reason to believe the scatter plot is not linear, but rather exponential, as described above. This means the increase in \(y\) values begins rather slowly but then seems to explode. Note: For exponential growth models, we assume all \(y>0\). Consider the following data, where \(x=\) time in hours and \(y=\) number of bacteria in a laboratory culture at the end of \(x\) hours. $$ \begin{array}{l|rrrrr} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 3 & 12 & 22 & 51 & 145 \\ \hline \end{array} $$ (a) Look at the Excel graph of the scatter diagram of the \((x, y)\) data pairs. Do you think a straight line will be a good fit to these data? Do the \(y\) values seem almost to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base 10 (however, natural logarithms of base \(e\) would work just as well). $$ \begin{array}{l|lllll} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y^{\prime}=\log y & 0.477 & 1.079 & 1.342 & 1.748 & 2.161 \\ \hline \end{array} $$ Look at the Excel graph of the scatter diagram of the \(\left(x, y^{\prime}\right)\) data pairs and compare this diagram with the diagram in part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to verify the linear regression equation for the \((x, y)\) data pairs, \(\hat{y}=-50.3+32.3 x\), with correlation coefficient \(r=0.882 .\) (d) Use a calculator with regression keys to verify the linear regression equation for the \(\left(x, y^{\prime}\right)\) data pairs, \(y^{\prime}=0.150+0.404 x\), with correlation coefficient \(r=0.994\). The correlation coefficient \(r=0.882\) for the \((x, y)\) pairs is not bad. But the correlation coefficient \(r=0.994\) for the \(\left(x, y^{\prime}\right)\) pairs is a lot better! (e) The exponential growth model is \(y=\alpha \beta^{x}\). Let us use the results of part (d) to estimate \(\alpha\) and \(\beta\) for this strain of laboratory bacteria. The equation \(y^{\prime}=\) \(a+b x\) is the same as \(\log y=a+b x .\) If we raise both sides of this equation to the power 10 and use some college algebra, we get \(y=10^{a}\left(10^{b}\right)^{x}\). Thus \(\alpha \approx 10^{a}\) and \(\beta \approx 10^{b}\). Use these results to approximate \(\alpha\) and \(\beta\) and write the exponential growth equation for our strain of bacteria. Note: The TI-84Plus calculator fully supports the exponential growth model. Place the original \(x\) data in list \(\mathrm{L} 1\) and the corresponding \(y\) data in list \(\mathrm{L} 2 .\) Then press STAT, followed by CALC, and scroll down to option 0: ExpReg. The output gives values for \(\alpha, \beta\), and the correlation coefficient \(r\).

Step by step solution

01

Analyze Scatter Plot for Exponential Growth

Look at the given data values for \( (x, y) \) and create a scatter plot. As \( x \) increases, check if the \( y \) values increase slowly at first and then increase rapidly. This behavior suggests exponential growth, rather than linear growth, indicating that a straight line may not fit well.

02

Apply Logarithmic Transformation

Transform the \( y \) values using the common logarithm (base 10) to linearize the exponential growth. Use the transformation \( y' = \log_{10}(y) \) to create new data \( \{(x, y')\} \). Check the resulting \( y' \) values to see if they form a pattern more suitable for linear regression when plotted against \( x \).

03

Evaluate Linear Fit on Transformed Data

Create a scatter plot for the transformed data \( \{(x, y')\} \) and compare it with the original \( \{(x, y)\} \). For a better fit, the \( \{(x, y')\} \) plot should align more closely with a straight line, indicating that logarithmic transformation has linearized the data.

04

Verify Linear Regression for Original Data

Use a calculator to perform linear regression on the original \( (x, y) \) pairs and verify the given equation \( \hat{y} = -50.3 + 32.3x \). Confirm the correlation coefficient \( r = 0.882 \), indicating a moderate fit.

05

Verify Linear Regression for Transformed Data

Use a calculator to perform linear regression on the transformed \( (x, y') \) pairs to obtain the equation \( y' = 0.150 + 0.404x \). Confirm the correlation coefficient \( r = 0.994 \), indicating a strong fit compared to the original data.

06

Estimate Parameters \( \alpha \) and \( \beta \)

Based on the regression equation \( y' = a + bx \), where \( a = 0.150 \) and \( b = 0.404 \), estimate \( \alpha \) and \( \beta \) using exponentiation: \( \alpha \approx 10^a = 10^{0.150} \) and \( \beta \approx 10^b = 10^{0.404} \). Calculate these values to form the exponential growth model \( y = \alpha \beta^x \).

07

Formulate Exponential Growth Equation

Substitute the calculated values of \( \alpha \) and \( \beta \) back into the exponential growth formula \( y = \alpha \beta^x \) to write the specific model for the bacteria growth. Calculate \( \alpha \approx 1.41 \) and \( \beta \approx 2.54 \), forming the equation \( y = 1.41 \times 2.54^x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression

Linear regression is a statistical method that models the relationship between two variables by fitting a linear equation to observed data. In simple terms, it tries to draw the best straight line (often called the "line of best fit") through a set of points. This line minimizes the distance between itself and all data points, providing a visual representation of the relationship between the variables.

When dealing with exponential growth, like observing bacteria cultures over time, a scatter plot might display a curve rather than a straight line. In such cases, applying linear regression directly would not provide the best fit, as the assumptions of linear relationships are not met.

• The line of best fit is represented as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• Linear regression can be applied after transforming nonlinear data, like exponential data, into a linear form using a technique like logarithmic transformation.
• A strong fit in linear regression is often assessed using the correlation coefficient \( r \), which measures how close the data is to being perfectly linear.

Logarithmic Transformation

Logarithmic transformation is a mathematical strategy used to linearize data that follows exponential or power-law growth patterns. It involves taking the logarithm of each data point to simplify the model into a linear form, making it more suitable for linear regression.

In cases where the relationship between variables grows exponentially, as with bacterial population examples, this method can be crucial. By transforming the data, you change the exponential relationship into a linear one, allowing you to employ linear regression techniques effectively.

• The transformation formula is \( y' = \log_{10} (y) \), where \( y' \) represents the transformed data.
• This transformation aligns the data along a straight line, facilitating easier analysis with regression models.
• It simplifies estimation of model parameters by converting multiplication into addition, a more straightforward arithmetic operation.

Correlation Coefficient

The correlation coefficient, often denoted as \( r \), is a statistical measure that describes the degree and direction of a linear relationship between two variables. It ranges from -1 to 1.

• A correlation of 1 indicates a perfect positive linear relationship, where both variables move in the same direction.
• A correlation of -1 signifies a perfect negative linear relationship, where they move in opposite directions.
• A correlation around 0 suggests no linear relationship between the variables.

For the bacteria growth data, the correlation coefficient provides insight into how well the data points fit the suggested linear model. After applying logarithmic transformation, the correlation coefficient \( r = 0.994 \) signifies a nearly perfect linear fit, demonstrating that the transformation effectively linearizes the exponential growth, making it suitable for linear regression analysis.

The correlation coefficient is pivotal in assessing the strength of the relationship after data transformation, ensuring the chosen model accurately reflects the underlying growth pattern.