Penicillin Doctors studying how the human body assimilates medication inject some patients with penicillin, and then monitor the concentration of the drug (in units/cc) in the patients' blood for seven hours. The data are shown in the scatterplot. First they tried to fit a linear model. The regression analysis and residuals plot are shown. Dependent variable is: Concentration
No Selector
R squared

=90.8 \% \quad

R squared (adjusted)

=90.6 \%

s=3.472

with

43-2=41

degrees of freedom

\begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 4900.55 & 1 & 4900.55 & 407 \\ \text { Residual } & 494.199 & 41 & 12.0536 & \end{array}

\begin{array}{lllrc}\text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 40.3266 & 1.295 & 31.1 & \leq 0.0001 \\ \text { Time } & -5.95956 & 0.2956 & -20.2 & \leq 0.0001\end{array}

$Penicillin Doctors studying how the human body assimilates medication inject some patients with penicillin, and then monitor the concentration of the drug (in units/cc) in the patients' blood for seven hours. The data are shown in the scatterplot. First they tried to fit a linear model. The regression analysis and residuals plot are shown. Dependent variable is: Concentration No Selector R squared =90.8 \% \quad R squared (adjusted) =90.6 \% s=3.472 with 43-2=41 degrees of freedom \begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 4900.55 & 1 & 4900.55 & 407 \\ \text { Residual } & 494.199 & 41 & 12.0536 & \end{array} \begin{array}{lllrc}\text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 40.3266 & 1.295 & 31.1 & \leq 0.0001 \\ \text { Time } & -5.95956 & 0.2956 & -20.2 & \leq 0.0001\end{array} a. Find the correlation between time and concentration. b. Using this model, estimate what the concentration of penicillin will be after 4 hours. c. Is that estimate likely to be accurate, too low, or too high? Explain. Now the researchers try a new model, using the re-expression log(Concentration). Examine the regression analysis and the residuals plot below. Dependent variable is: LogCnn No Selector R squared =98.0 \% \quad R squared (adjusted) =98.0 \% s=0.0451 with 43-2=41 degrees of freedom \begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 4.11395 & 1 & 4.11395 & 2022 \\ \text { Residual } & 0.083412 & 41 & 0.002034 & \\ & & & &\\ \\ \text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 1.80184 & 0.0168 & 107 & \leq 0.0001 \\ \text { Time } & -0.172672 & 0.0038 & -45.0 & \leq 0.0001\end{array} d. Explain why you think this model is better than the original linear model. e. Using this new model, estimate the concentration of penicillin after 4 hours.$

a. Find the correlation between time and concentration.
b. Using this model, estimate what the concentration of penicillin will be after 4 hours.
c. Is that estimate likely to be accurate, too low, or too high? Explain.
Now the researchers try a new model, using the re-expression log(Concentration). Examine the regression analysis and the residuals plot below. Dependent variable is: LogCnn
No Selector

R

squared

=98.0 \% \quad R

squared (adjusted)

=98.0 \%

s=0.0451

with

43-2=41

degrees of freedom

\begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 4.11395 & 1 & 4.11395 & 2022 \\ \text { Residual } & 0.083412 & 41 & 0.002034 & \\ & & & &\\ \\ \text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 1.80184 & 0.0168 & 107 & \leq 0.0001 \\ \text { Time } & -0.172672 & 0.0038 & -45.0 & \leq 0.0001\end{array}

$Penicillin Doctors studying how the human body assimilates medication inject some patients with penicillin, and then monitor the concentration of the drug (in units/cc) in the patients' blood for seven hours. The data are shown in the scatterplot. First they tried to fit a linear model. The regression analysis and residuals plot are shown. Dependent variable is: Concentration No Selector R squared =90.8 \% \quad R squared (adjusted) =90.6 \% s=3.472 with 43-2=41 degrees of freedom \begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 4900.55 & 1 & 4900.55 & 407 \\ \text { Residual } & 494.199 & 41 & 12.0536 & \end{array} \begin{array}{lllrc}\text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 40.3266 & 1.295 & 31.1 & \leq 0.0001 \\ \text { Time } & -5.95956 & 0.2956 & -20.2 & \leq 0.0001\end{array} a. Find the correlation between time and concentration. b. Using this model, estimate what the concentration of penicillin will be after 4 hours. c. Is that estimate likely to be accurate, too low, or too high? Explain. Now the researchers try a new model, using the re-expression log(Concentration). Examine the regression analysis and the residuals plot below. Dependent variable is: LogCnn No Selector R squared =98.0 \% \quad R squared (adjusted) =98.0 \% s=0.0451 with 43-2=41 degrees of freedom \begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 4.11395 & 1 & 4.11395 & 2022 \\ \text { Residual } & 0.083412 & 41 & 0.002034 & \\ & & & &\\ \\ \text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 1.80184 & 0.0168 & 107 & \leq 0.0001 \\ \text { Time } & -0.172672 & 0.0038 & -45.0 & \leq 0.0001\end{array} d. Explain why you think this model is better than the original linear model. e. Using this new model, estimate the concentration of penicillin after 4 hours.$
d. Explain why you think this model is better than the original linear model.
e. Using this new model, estimate the concentration of penicillin after 4 hours.

A Residual Plot That Has No Pattern Is a Sign

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