Chapter 7.3, Problem 70E

a.

To determine

Tocalculate:the least squares regression parabola by evaluating the values of a, b, and c.

Answer to Problem 70E

1y1y3

  $y=-\frac{9}{800}{x}^2+\frac{437}{200}x-\frac{313}{10}$

Explanation of Solution

Given information:

The system of equation,

  $\begin{array}{l}4c+520b+69600a=228\\ 520c+69600b+9568000a=28160\\ 69600c+9568000b+1346880000a=3575200\end{array}$

Calculation:

Writ the matrix equation for the given system.

  $\left[\begin{array}{ccc}4& 520& 69600\\ 520& 69600& 9568000\\ 69600& 9568000& 1346880000\end{array}\right]\left[\begin{array}{c}c\\ b\\ a\end{array}\right]=\left[\begin{array}{c}228\\ 28160\\ 3575200\end{array}\right]$

The solution of this equation can be found as shown below.

Now, solve for a, b, and c by finding the inverse of the $3\times 3$ matrix.

  $\begin{array}{c}\left[\begin{array}{c}c\\ b\\ a\end{array}\right]={\left[\begin{array}{ccc}4& 520& 69600\\ 520& 69600& 9568000\\ 69600& 9568000& 1346880000\end{array}\right]}^{-1}\left[\begin{array}{c}228\\ 28160\\ 3575200\end{array}\right]\\ =\left(\begin{array}{ccc}\frac{8579}{20}& -\frac{2691}{400}& \frac{41}{1600}\\ -\frac{2691}{400}& \frac{849}{8000}& -\frac{13}{32000}\\ \frac{41}{1600}& -\frac{13}{32000}& \frac{1}{640000}\end{array}\right)\left(\begin{array}{c}228\\ 28160\\ 3575200\end{array}\right)\\ =\left(\begin{array}{c}-\frac{313}{10}\\ \frac{437}{200}\\ -\frac{9}{800}\end{array}\right)\end{array}$

Therefore, $c=-\frac{313}{10},b=\frac{437}{200},a=-\frac{9}{800}$ .

So, the least squares regression parabola is $y=-\frac{9}{800}{x}^2+\frac{437}{200}x-\frac{313}{10}$ .

b.

To determine

To graph: The model found in part (a) by using graphing utility and then plot the given data in the same viewing window.

Explanation of Solution

Given information:

The data set as shown below.

  $\begin{array}{ccccc}\text{Number,}x& 100& 120& 140& 160\\ \text{Percent,}\text{y}& 75& 68& 55& 30\end{array}$

Graph:

The graph of the model and the data set can be drawn as shown below.

  

It can be observed that each data point lies on the graph of the model. This shows that the least squares regression parabola fits the data set.

c.

To determine

To estimate: the percentage of females that had offspring when there were 170 females.

Answer to Problem 70E

  $15.025\%$

Explanation of Solution

Given information:

The number of females is 170.

Calculation:

Substitute $x=170$ in the model found in part (a) to estimate the percentage of females that had offspring.

  $\begin{array}{c}y=-\frac{9}{800}{x}^2+\frac{437}{200}x-\frac{313}{10}\\ =-\frac{9}{800}{\left(170\right)}^2+\frac{437}{200}\left(170\right)-\frac{313}{10}\\ =\frac{601}{40}\\ =15.025\end{array}$

d.

To determine

To estimate: the number of females when there were 40% of the females had offspring.

Answer to Problem 70E

153

Explanation of Solution

Given information:

The percent of females having offspring, 40%.

Calculation:

Substitute $y=40$ in the model found in part (a) and solve for $x$ to estimate the number of females.

  $\begin{array}{c}40=-\frac{9}{800}{x}^2+\frac{437}{200}x-\frac{313}{10}\\ 40\cdot 800=-\frac{9}{800}{x}^2\cdot 800+\frac{437}{200}x\cdot 800-\frac{313}{10}\cdot 800\\ 32000=-9x^2+1748x-25040\\ -9x^2+1748x-25040=32000\\ -9x^2+1748x-25040-32000=32000-32000\\ x=\frac{-1748\pm \sqrt{{1748}^2-4\left(-9\right)\left(-57040\right)}}{2\left(-9\right)}\\ =\frac{-1748\pm \sqrt{1002064}}{-2\cdot 9}\\ x=\frac{-1748+\sqrt{1002064}}{-18},x=\frac{-1748-\sqrt{1002064}}{-18}\\ x=\frac{1748-\sqrt{1002064}}{18},x=\frac{1748+\sqrt{1002064}}{18}\\ x=41.49825,x=152.72397\\ x\approx 41,x\approx 153\end{array}$

Out of these two values, only $x=153$ is relevant to the data set. So, the number of females was 153.