Chapter 8.1, Problem 101E

To determine

The Health and Wellness.

Answer to Problem 101E

  $\boxed{y=7.5t+28}$

The estimate reasonable

Explanation of Solution

Given information: From 2000 through 2011, the numbers of new cases of a waterborne disease in a small city increased in a pattern that was approximately linear (see figure). Find the least squares regression line

  $y=at+b$

for the data shown in the figure by solving the following system using matrices. Let $t$ represent the year, with $t=0$ corresponding to $2000$

  $\{\begin{array}{l}12b+66a=831\\ 66b+506a=5643\end{array}$

Use the result to predict the number of new cases of the waterborne disease in 2014. Is the estimate reasonable? Explain.

  

Calculation: The system of equations is,

  $\{\begin{array}{l}12b+66a=831\\ 66b+506a=5643\end{array}$

The augmented matrix of above system of equations,

  $\left[\begin{array}{cccc}12& 66& ⋮& 831\\ 66& 506& ⋮& 5643\end{array}\right]$

Now apply Gauss-Jordan elimination method.

  $\begin{array}{l}1/2R_1\to R_1\\ R_2-11R_1\to R_2\end{array}$ $\left[\begin{array}{cccc}6& 33& ⋮& 415.5\\ 0& 143& ⋮& 1072.5\end{array}\right]$

  $1/143R_2\to R_2$ $\left[\begin{array}{cccc}6& 33& ⋮& 415.5\\ 0& 1& ⋮& 7.5\end{array}\right]$

  $R_1-33R_2\to R_1$ $\left[\begin{array}{cccc}6& 0& ⋮& 168\\ 0& 1& ⋮& 7.5\end{array}\right]$

  $1/6R_1\to R_1$ $\left[\begin{array}{cccc}1& 0& ⋮& 28\\ 0& 1& ⋮& 7.5\end{array}\right]$

  $\begin{array}{l}b=28\\ a=7.5\end{array}$

Substitute the values of variables in $y=at+b$

  $y=7.5t+28$

Hence the least square regression model is,

  $\boxed{y=7.5t+28}$

New cases of the waterborne disease in 2014

Here $t=14$ corresponding to the year $2000-2014$

  $\begin{array}{l}y\left(14\right)=7.5\left(14\right)+28\\ =133\end{array}$

Hence New cases of the waterborne disease in 2014 is $133$

The estimate reasonable