Chapter 12.2, Problem 37E

(a)

To determine

To make a reasonable accurate scatterplot of the data using the time as the explanatory variable and describe what you see.

Explanation of Solution

Now, the scatterplot of the data using the time as the explanatory variable is as follows:

  

Thus, by looking at the scatterplot we can see that,

Direction: Negative, because the scatterplot slopes downwards.

Form: Curved, because the points do not lie on a straight line.

Strength: Strong, because all points lie very close together in the same pattern.

We also note an outlier in the upper left corner of the scatterplot.

(b)

To determine

To explain why it would be reasonable to use an exponential model to describe the relationship between the count of bacteria and time.

Explanation of Solution

Now, the data of the count of bacteria and the time is given in the question. Thus, we can say that it would be reasonable to use an exponential model to describe the relationship between the count of bacteria and time because the corresponding scatterplot has a roughly linear pattern with no notable strong outliers as we can see in the scatterplot in the part (a).

(c)

To determine

To give the equation of the least squares regression line.

Answer to Problem 37E

  $\ln \widehat{y}=5.97316-0.218425t$ .

Explanation of Solution

Now, it is given that t is the time and y is the count of bacteria. Thus the transformation is the natural logarithm of the count as $\ln y$ .

Thus, as we know that the general equation of the regression line is as follows:

  $\ln \widehat{y}=a+bt$

And the slope and the constant given in the computer output is as:

  $\begin{array}{l}a=5.97316\\ b=-0.218425\end{array}$

Thus, the regression line is as follows:

  $\begin{array}{l}\ln \widehat{y}=a+bt\\ \Rightarrow \ln \widehat{y}=5.97316-0.218425t\end{array}$

(d)

To determine

To use your model to predict the number of surviving bacteria after $17$ minutes and explain do you expect this prediction to be too high, too low or about right.

Answer to Problem 37E

  $\widehat{y}=9.58247$ and it is about right.

Explanation of Solution

Now, the regression line is as follows:

  $\begin{array}{l}\ln \widehat{y}=a+bt\\ \Rightarrow \ln \widehat{y}=5.97316-0.218425t\end{array}$

Thus, we will evaluate the equation as:

  $\begin{array}{l}\ln \widehat{y}=a+bt\\ \Rightarrow \ln \widehat{y}=5.97316-0.218425t\\ \Rightarrow \ln \widehat{y}=5.97316-0.218425(17)\\ \Rightarrow \ln \widehat{y}=2.259935\\ \Rightarrow \widehat{y}=e^{2.259935}\\ =9.58247\end{array}$

Since the pattern of the residual in the residual plot is centered at zero we expect the predict to be about right.