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Penicillin Doctors Studying How the Human Body Assimilates Medication Inject =90.8% =90.8 \% \quad

Question 71

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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% =90.8 \% \quad R squared (adjusted) =90.6% =90.6 \%
s=3.472 s=3.472 with 43−2=41 43-2=41 degrees of freedom

 Source  Sum of Squares  df  Mean Square  F-ratio  Regression 4900.5514900.55407 Residual 494.1994112.0536 \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}

 Variable  Coefficient  s.e. of Coeff  t-ratio  prob  Constant 40.32661.29531.1≤0.0001 Time −5.959560.2956−20.2≤0.0001 \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 R squared =98.0%R =98.0 \% \quad R squared (adjusted) =98.0% =98.0 \%
s=0.0451 s=0.0451 with 43−2=41 43-2=41 degrees of freedom

 Source  Sum of Squares  df  Mean Square  F-ratio  Regression 4.1139514.113952022 Residual 0.083412410.002034 Variable  Coefficient  s.e. of Coeff  t-ratio  prob  Constant 1.801840.0168107≤0.0001 Time −0.1726720.0038−45.0≤0.0001 \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.

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a. -0.953
b. 16.5 units/cc
c. ...

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