Homework2

Due Feb 19, 2020 by 11:59pm
Points 100
Submitting a file upload

Homework 2

For this assignment, you can either use a single Rmd source to solve all 3 problems (embedding LaTeX as usual) and then output an HTML, PDF, DOCX report , or alternatively, use RMD+HTML for the computational Problem 1 and use a separate LaTeX/PDF for problems 2 & 3.

Problem 1 (Linear Modeling):

Use the SOCR Heart-attack dataset to fit a linear model predicting Y=Hospitalization Charges using X={Length of Stay, Age, Survival (Dead)} as predictors (covariates, or features).

Report your linear model, display some appropriate plots, and interpret the findings in the context of the case-study Links to an external site..

[Hint: Use this R code to load the data into RStudio and see the Linear Modeling Section.]

heart_attack<-read.csv("https://umich.instructure.com/files/1644953/download?download_frd=1", stringsAsFactors = F)
str(heart_attack)

Problem 2 (Inflationary Models):

Assume a grad student stipend is $20,000/yr. The student wants to estimate what this stipend may be after inflation adjustment 50 years later.
Assume the average annual inflation rate is estimated to be 3%. Ignoring other factors like taxes, political factors, environmental conditions, family size, etc., the student wants to project what the grad student stipends may be half a century later.

Problem 3 (Exponential Growth Models):

Suppose LaTeX: Y(t) $Y(t)$ is the amount of a resource at time t, with LaTeX: Y_o=Y(t=0) $Y_o=Y(t=0)$ . The rate at which the resource is consumed is $LaTeX: r = -\frac{dY}{dt}$ $r = -\frac{dY}{dt}$ . Let $LaTeX: r = r_o e^{bt}$ $r = r_o e^{bt}$ be the exponential rate of increase (actually decay) of the resource use over time.

Show that the resource amount remaining at time $t$ is $LaTeX: Y(t) = y_o -\frac{r_o}{b}(e^{bt} -1)$ $Y(t) = y_o -\frac{r_o}{b}(e^{bt} -1)$ .
If the supply of the resource is used at constant rate $r_o$ , it would last for a time $T_c$ . Show the effect of a Global increase with an exponentially growing rate of consumption with a rate $b$ . Specifically, show that the projected resource will last a much shorter time at this accelerated consumption: $LaTeX: T_b = \frac{1}{b} \ln(1 + bT_c)$ $T_b = \frac{1}{b} \ln(1 + bT_c)$ . Give some specific numerical examples of this shorter time until the resource is exhausted.