Chapter 1.7, Problem 28E

(a)

To determine

To find: A linear model and the correlation coefficient using a graphing utility.

Answer to Problem 28E

Linear model $y=0.3401x+77.943$

  $\text{Correlation coefficient}\left(r\right)=0.9934$

Explanation of Solution

Given:

The data set is:

  $\begin{array}{l}\left(14,81.2\right)\left(28,87.1\right)\left(42,93.7\right)\\ \left(56,98.3\right)\left(70,102.4\right)\left(84,106.2\right)\\ \left(98,110\right)\end{array}$

Formula used:

The regression line is written as:

  $y=ax+b$

Where,

a is the intercept coefficient

b is the slope coefficient.

The formula to compute the correlation coefficient is:

  $r=\frac{n{\displaystyle \sum XY-{\displaystyle \sum X{\displaystyle \sum Y}}}}{\sqrt{n{\displaystyle \sum X^2-{\left({\displaystyle \sum X}\right)}^2}}\times \sqrt{n{\displaystyle \sum Y^2-{\left({\displaystyle \sum Y}\right)}^2}}}$

Calculation:

The provided data in tabular form could be written as:

  $\begin{array}{cc}x& y\\ 14& 81.2\\ 28& 87.1\\ 42& 93.7\\ 56& 98.3\\ 70& 102.4\\ 84& 106.2\\ 98& 110\end{array}$

Follow the provided steps of Ti-83 plus calculator to compute a linear model and the correlation coefficient:

1. Turn on the calculator.
2. Click on STAT > Edit > Enter.
3. Enter the data of x in L1 and data of y in L2.
4. Click om STAT > CALC >LinReg ( ax +b) > ENTER.
5. Hit 2nd and 1 to choose L1 and Hit 2nd and 2 to choose L2.
6. Hit Enter.

The obtained output is:

  

From the above output, linear model and correlation coefficients are $y=0.3401x+77.943$ and $r=0.9934$ respectively.

(b)

To determine

To explain: If the model is representing the data well using the correlation coefficient.

Answer to Problem 28E

Yes.

Explanation of Solution

From the above part, it is known that $r=0.9934$ . Since, the value of the correlation coefficient is closer to the 1. Thus, it could be said that model is representing the data well.

(c)

To determine

To plot: The provided data.

Explanation of Solution

Graph:

Using the Ti-83 calculator the figure is constructed as:

  

Interpretation:

In the above figure, the model is represented by the line and actual data by dots.

Since, the data points are lying on the line it could be concluded that model is well fitted for the actual data.

(d)

To determine

To predict: The average length of colt 112 days after birth.

Answer to Problem 28E

The average length is 116.0342.

Explanation of Solution

Calculation:

The length could be calculated as:

  $\begin{array}{c}y=0.3401x+77.943\\ =0.3401\left(112\right)+77.943\\ =116.0342\end{array}$

The required average length is 116.0342.