Question

Killing bacteria Expose marine bacteria to X-rays for time periods from $1$to $15$minutes. Here is a scatterplot showing the number of surviving bacteria (in hundreds) on a culture plate after each exposure time:

a. Below is a scatterplot of the natural logarithm of the number of surviving bacteria versus time. Based on this graph, explain why it would be reasonable to use an exponential model to describe the relationship between the count of bacteria and the time.

b). Here is the output from a linear regression analysis of the transformed data. Give the equation of the least-squares regression line. Be sure to defne any variables you use.

c. Use your model to predict the number of surviving bacteria after $17$minutes.

Short answer

a). The scatter plot does not have much curvature.

b). The equation of the least-squaresegression line is $\ln \hat{y}=5.97316-0.218425x$.

c). The expected number of bacteria after $17$ minutes is $9.58247$ hundred bacteria or $958.247$ bacteria.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

The scatter plot does not have much curvature, a linear model between the two variables of the scatter plot would be appropriate. As a result, using a linear relationship between $\ln \left(count\right)$and time is reasonable.

$\ln \left(\text{count}\right)=a+b\left(\text{time}\right)$

Taking the exponential

$\text{count}=e^{\ln \left(\text{count}\right)}$$=e^{a+b\left(\text{time}\right)}$$=e^a{e}^{b\left(\text{time}\right)}$

Part (b) Step 1: Given Information

Given data:

Part (b) Step 2: Explanation

General equation of a least square regression line:

$\hat{y}=b_0+b_{1}x$

In the row "constant" and the column "Coef" of the computer's output, the calculated constant $b_0$is given.

$b_0=5.97316$

In the row "Time" and the column "Coef" of the computer output, the slope $b_1$is found.

$b_1=-0.218425$

Part (b) Step 3: Explanation

Substituting the value of $b_0$and $b_1$:

$\hat{y}=b_0+b_{1}x$

$\hat{y}=5.97316-0.218425x$

Where $x$represents the current time and $y$is the ln (count)

$\ln \hat{y}=5.97316-0.218425x$

Part (c) Step 1: Given Information

Given data:

Part (c) Step 2: Explanation

Substituting the value of $x$:

$\ln \hat{y}=5.97316-0.218425x$

$\ln \hat{y}=5.97316-0.218425\left(17\right)$

$\ln \hat{y}=2.259935$

Taking the exponential

$\hat{y}=e^{\ln \hat{y}}$

$=e^{2.259935}$

$=9.58247$