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Killing bacteria Expose marine bacteria to X-rays for time periods from 1 to 15 minutes. Here are the number of surviving bacteria (in hundreds) on a culture plate after each exposure time: \(^{21}\) $$ \begin{array}{cccc} \hline \text { Time } t & \text { Count } y & \text { Time } t & \text { Count } y \\ 1 & 355 & 9 & 56 \\ 2 & 211 & 10 & 38 \\ 3 & 197 & 11 & 36 \\ 4 & 166 & 12 & 32 \\ 5 & 142 & 13 & 21 \\ 6 & 106 & 14 & 19 \\ 7 & 104 & 15 & 15 \\ 8 & 60 & & \\ \hline \end{array} $$ (a) Make a reasonably accurate scatterplot of the data by hand, using time as the explanatory variable. Describe what you see. (b) A scatterplot of the natural logarithm of the number of surviving bacteria versus time is shown below. Based on this graph, explain why it would be reasonable to use an exponential model to describe the relationship between count of bacteria and time. (c) Minitab output from a linear regression analysis on the transformed data is shown below. $$ \begin{aligned} &\begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 5.97316 & 0.05978 & 99.92 & 0.000 \\ \text { Time } & -0.218425 & 0.006575 & -33.22 & 0.000 \end{array}\\\ &\begin{array}{lll} S=0.110016 & R-S q=98.8 \frac{8}{6} & R-S q(a d j)=98.7 \% \end{array} \end{aligned} $$ Give the equation of the least-squares regression line. Be sure to define any variables you use. (d) Use your model to predict the number of surviving bacteria after 17 minutes. Show your work.

Step by step solution

01

Constructing a Scatterplot

To construct a scatterplot, we will plot the exposure time, \( t \), on the x-axis and the number of surviving bacteria \( y \) (in hundreds) on the y-axis. When looking at the scatterplot, we might observe a downward trend because the number of surviving bacteria seems to decrease as the exposure time increases.

02

Analyzing Logarithmic Transformation

A scatterplot of \( \ln(y) \) versus \( t \) should show a linear relationship if using an exponential model is appropriate. In this context, if the points roughly form a straight line, an exponential relationship can be assumed between \( y \) and \( t \).

03

Understanding Minitab Output

From Minitab, the linear regression equation for \( \ln(y) \) is displayed as \( \ln(y) = 5.97316 - 0.218425t \). In this equation, the coefficient of \(t\) is negative, indicating a decrease in the number of bacteria over time, while the constant provides the intercept. \( R^2 \) indicates a high degree of correlation (98.8%), suggesting a good fit for the model.

04

Formulating the Exponential Model

To convert the model back to bacteria count, take the exponential of both sides: \( y = e^{5.97316 - 0.218425t} \). Here, \( y \) represents the number of surviving bacteria and \( t \) the exposure time in minutes.

05

Predicting for 17 Minutes

Substitute \( t = 17 \) into the exponential model: \( y = e^{5.97316 - 0.218425 \times 17} \). After calculating, this yields \( y \approx 10.77 \), indicating approximately 108 bacteria (since \( y \) represents hundreds of bacteria).

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Key Concepts

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

Scatterplot

Creating a scatterplot is one of the best ways to visualize the relationship between two variables. In the case of the exercise about killing bacteria, a scatterplot can help us observe how the bacterial count changes over time when exposed to X-rays. To make a scatterplot:

• Plot time (in minutes) on the x-axis.
• Plot the count of surviving bacteria (in hundreds) on the y-axis.

By analyzing this scatterplot, you'll notice that as the exposure time increases, the number of surviving bacteria decreases. The points may form a downward trend, revealing a negative correlation between the two variables. This trend suggests that longer exposure to X-rays reduces the number of bacteria, which is important for determining the effectiveness of the exposure over time.

Natural Logarithm Transformation

A natural logarithm transformation is a mathematical technique used to linearize data that follows an exponential trend. In this exercise, the number of surviving bacteria decreases exponentially with time. However, the relationship is not initially linear, making it difficult to use simple linear regression. To understand if an exponential model will work:

• Take the natural logarithm of the bacterial count, denoted as \( \ln(y) \).
• Create a scatterplot with \( \ln(y) \) versus time \( t \).

If the transformed data forms a roughly straight line when graphed, it suggests a linear relationship is present in the transformed data. This alignment implies that an exponential model is justified, as it shows that the data can be represented as an exponential function \( y = e^{a - bt} \). This simplification makes data analysis more manageable and regression more accurate.

Linear Regression

Linear regression is a statistical method used to model and analyze the relationships between variables. With the bacteria data, after logging the counts, we utilize linear regression on the transformed data to find a linear equation.Minitab output gives us the equation:\[ \ln(y) = 5.97316 - 0.218425t \]This equation helps us understand that:- The constant \(5.97316\) is the intercept, which represents the hypothetical scenario of zero exposure.- The coefficient of \( t \) (\(-0.218425\)) reflects the rate of change, showing how much \( \ln(y) \) decreases per unit time increase.A high \( R^2 \) value of 98.8% signifies an excellent fit, suggesting that our regression line accurately predicts the transformed bacterial count based on exposure time.

Bacterial Count Prediction

Predicting the bacterial count involves converting the regression equation from a natural logarithm scale back to the original bacterial count. Here's how you can predict the count after a certain time, such as 17 minutes:First, use the linear equation:\[ \ln(y) = 5.97316 - 0.218425 \times 17 \]Calculate to find:\[ \ln(y) = 2.37627 \]Then, exponentiate both sides to solve for \( y \):\[ y = e^{2.37627} \approx 10.77 \]Thus, the prediction translates to approximately 108 surviving bacteria (since \( y \) represents hundreds of bacteria). This predictive method allows you to estimate the decrease in surviving bacteria over time effectively, supporting studies on bacterial resistance or eradication through X-ray exposure.