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.

$\begin{array}{ccccc}\text{Time}\mathit{t}& \text{Count}\mathit{y}& & \text{Time}\mathit{t}& \text{Count}\mathit{y}\\ 1& 355& & 9& 56\\ 2& 211& & 10& 38\\ 3& 197& & 11& 36\\ 4& 166& & 12& 32\\ 5& 142& & 13& 21\\ 6& 106& & 14& 19\\ 7& 104& & 15& 15\\ 8& 60& & & \\ & & & & \end{array}$

(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 the count of bacteria and time.

Minitab output from a linear regression analysis on the transformed data is shown below.

$\begin{array}{lcccc}\text{Predictor}& \text{Coef}& \text{SE Coef}& T& \text{P}\\ \text{Constant}& 5.97316& 0.05978& 99.92& 0.000\\ \text{Time}& -0.218425& 0.006575& -33.22& 0.000\end{array}$

$\mathrm{S}=0.110016\mathrm{R}-\mathrm{Sq}=98.8\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=98.7\%$

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. Do you expect this prediction to be too high, too low, or about right? Explain

$\begin{array}{ccccc}\text{Time}\mathit{t}& \text{Count}\mathit{y}& & \text{Time}\mathit{t}& \text{Count}\mathit{y}\\ 1& 355& & 9& 56\\ 2& 211& & 10& 38\\ 3& 197& & 11& 36\\ 4& 166& & 12& 32\\ 5& 142& & 13& 21\\ 6& 106& & 14& 19\\ 7& 104& & 15& 15\\ 8& 60& & & \\ & & & & \end{array}$

The scatter plot is:

As a result, we can see from the scatterplot that,

Because the scatterplot slopes downhill, the direction is negative.

Because the points are not in a straight line, the shape is curved.

Strength: Strong because all points in the same pattern are relatively near together.

We also see an anomaly in the scatterplot's upper left corner.

The values for the bacteria count and the time are now provided in the question. As a result, we can argue that using an exponential model to describe the relationship between the count of bacteria and time would be plausible because the related scatterplot has an approximately linear pattern with no apparent strong outliers, as seen in the scatterplot in part $1$. (a).

Minitab output from a linear regression analysis on the transformed data is shown below.

$\begin{array}{lcccc}\text{Predictor}& \text{Coef}& \text{SE Coef}& T& \text{P}\\ \text{Constant}& 5.97316& 0.05978& 99.92& 0.000\\ \text{Time}& -0.218425& 0.006575& -33.22& 0.000\end{array}$

$\mathrm{S}=0.110016\mathrm{R}-\mathrm{Sq}=98.8\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=98.7\%$

Now $t$denotes the current time and $y$denotes the number of microorganisms. As a result, the transformation is $Iny$, where is the count's natural logarithm.

As we already know, the regression line's general equation is as follows:

$\ln \hat{y}=a+bt$

The slope and constant in the computer output are as follows:

$a=5.97316$

$b=-0.218425$

As a result, the regression line looks like this:

localid="1652849978599" $$\begin{gathered} \ln \hat{y}=a+bt \\ =5.97316-0.218425t \end{gathered}$$

Minitab output from a linear regression analysis on the transformed data is shown below.

$\begin{array}{lcccc}\text{Predictor}& \text{Coef}& \text{SE Coef}& T& \text{P}\\ \text{Constant}& 5.97316& 0.05978& 99.92& 0.000\\ \text{Time}& -0.218425& 0.006575& -33.22& 0.000\end{array}$

$\mathrm{S}=0.110016\mathrm{R}-\mathrm{Sq}=98.8\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=98.7\%$

The regression line is

$$\begin{gathered} \ln \hat{y}=a+bt \\ =5.97316-0.218425t \end{gathered}$$

Calculation of equation is

$$\begin{gathered} \ln \hat{y}=a+bt \\ =5.97316-0.218425t \\ =5.97316-0.218425\left(17\right) \\ =2.259935 \\ =e^{2.259935} \\ =9.58247 \end{gathered}$$