Chapter 5.1, Problem 23E

Special Rounding Instructions When you perform logistic regression, round the $r$ value to three decimal places and the other parameters to two decimal places. Round all answers to two decimal places unless otherwise indicated.

An Inverted Logistic Curve When the parameter $r$ in the logistic formula is negative (resulting in a positive power of $e$), the logistic curve has a different shape. For example, the following formula gives the approximate number of deaths due to tuberculosis as a fraction of all deaths in the United States t years after $1875$:

$T=\frac{0.13}{1+0.07e^{0.05t\text{'}}}$

a. Make a graph of $T$ versus $t$. Include dates up to $2000$.

b. What is the limiting value of $T$? Explain in practical terms what this limiting value means.

c. Estimate when the fraction of deaths due to tuberculosis was decreasing most rapidly. Give your answer accurate to the nearest decade (e.g., $1920$).