Chapter 12.4, Problem 12.13E

(a)

To determine

To find: the independent variable and the dependent variable.

Answer to Problem 12.13E

The variable API is the response variable (y) or the dependent variable and the variable ELL is the predictor variable (x) or independent variable.

Explanation of Solution

Given:

School	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
API	$745$	$\begin{array}{l}808\hfill \end{array}$	$\begin{array}{l}798\hfill \end{array}$	$\begin{array}{l}791\hfill \end{array}$	$\begin{array}{l}854\hfill \end{array}$	$\begin{array}{l}688\hfill \end{array}$	$\begin{array}{l}801\hfill \end{array}$	$\begin{array}{l}751\hfill \end{array}$
EL	$\begin{array}{l}71\hfill \end{array}$	$18$	$24$	$50$	$17$	$\begin{array}{l}71\hfill \end{array}$	$11$	$57$

Calculation:

From the given data,

The variable API is the response variable (y) or the dependent variable and the variable ELL is the predictor variable (x) or independent variable. Because, the Academic performance index (API) depends upon English language learner (ELL). Based on English language learner (ELL), we can measure Academic performance index (API).

Conclusion: The variable API is the response variable (y) or the dependent variable and the variable ELL is the predictor variable (x) or independent variable.

(b)

To determine

To plot:the given data using the scatterplot and explain whether the relationship between x and y is reasonable.

Answer to Problem 12.13E

The assumptions of a linear relationship between x and y are reasonable.

Explanation of Solution

Given:

School	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
API	$745$	$\begin{array}{l}808\hfill \end{array}$	$\begin{array}{l}798\hfill \end{array}$	$\begin{array}{l}791\hfill \end{array}$	$\begin{array}{l}854\hfill \end{array}$	$\begin{array}{l}688\hfill \end{array}$	$\begin{array}{l}801\hfill \end{array}$	$\begin{array}{l}751\hfill \end{array}$
EL	$\begin{array}{l}71\hfill \end{array}$	$18$	$24$	$50$	$17$	$\begin{array}{l}71\hfill \end{array}$	$11$	$57$

Calculation:

  

The assumptions of a linear relationship between x and y are reasonable because there is no strong curvature present in the scatterplot.

Conclusion: Therefore, the assumptions of a linear relationship between x and y are reasonable.

(c)

To determine

To calculate: the least-squares regression line.

Answer to Problem 12.13E

The least squares regression line is $\widehat{y}=847.476656-1.70453x$ .

Explanation of Solution

Given:

School	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
API	$745$	$\begin{array}{l}808\hfill \end{array}$	$\begin{array}{l}798\hfill \end{array}$	$\begin{array}{l}791\hfill \end{array}$	$\begin{array}{l}854\hfill \end{array}$	$\begin{array}{l}688\hfill \end{array}$	$\begin{array}{l}801\hfill \end{array}$	$\begin{array}{l}751\hfill \end{array}$
EL	$\begin{array}{l}71\hfill \end{array}$	$18$	$24$	$50$	$17$	$\begin{array}{l}71\hfill \end{array}$	$11$	$57$

Calculation:

Calculation for the data in first two columns of the table is:

API(y)	ELL(x)	$x^2$	$y^2$	$xy$
$745$	$\begin{array}{l}71\hfill \end{array}$	$\begin{array}{l}5041\hfill \end{array}$	$\begin{array}{l}555025\hfill \end{array}$	$\begin{array}{l}52895\hfill \end{array}$
$\begin{array}{l}808\hfill \end{array}$	$18$	$\begin{array}{l}324\hfill \end{array}$	$\begin{array}{l}652864\hfill \end{array}$	$\begin{array}{l}14544\hfill \end{array}$
$\begin{array}{l}798\hfill \end{array}$	$24$	$\begin{array}{l}576\hfill \end{array}$	$\begin{array}{l}636804\hfill \end{array}$	$\begin{array}{l}19152\hfill \end{array}$
$\begin{array}{l}791\hfill \end{array}$	$50$	$\begin{array}{l}2500\hfill \end{array}$	$\begin{array}{l}625681\hfill \end{array}$	$\begin{array}{l}39550\hfill \end{array}$
$\begin{array}{l}854\hfill \end{array}$	$17$	$\begin{array}{l}289\hfill \end{array}$	$\begin{array}{l}729316\hfill \end{array}$	$\begin{array}{l}14518\hfill \end{array}$
$\begin{array}{l}688\hfill \end{array}$	$\begin{array}{l}71\hfill \end{array}$	$\begin{array}{l}5041\hfill \end{array}$	$\begin{array}{l}473344\hfill \end{array}$	$\begin{array}{l}48848\hfill \end{array}$
$\begin{array}{l}801\hfill \end{array}$	$11$	$\begin{array}{l}121\hfill \end{array}$	$\begin{array}{l}641601\hfill \end{array}$	$\begin{array}{l}8811\hfill \end{array}$
$\begin{array}{l}751\hfill \end{array}$	$57$	$\begin{array}{l}3249\hfill \end{array}$	$\begin{array}{l}564001\hfill \end{array}$	$\begin{array}{l}42807\hfill \end{array}$
${\displaystyle \sum y=6,236}$	${\displaystyle \sum x=319}$	${\displaystyle \sum x^2}=17,141$	${\displaystyle \sum y^2=4,878,636}$	${\displaystyle \sum xy}=241,125$

The sum of squares are,

  $\begin{array}{l}{S}_{xx}={\displaystyle \sum x_i^2}-\frac{({\displaystyle \sum x_i{)}^2}}{n}\\ =17141-\frac{{(319)}^2}{8}\\ =4420.88\end{array}$

  $\begin{array}{l}{S}_{xy}={\displaystyle \sum x_i}{y}_i-\frac{{\displaystyle \sum x_i}{\displaystyle \sum y_i}}{n}\\ =241125-\frac{319\times 6236}{8}\\ =-7535.5\end{array}$

  $\begin{array}{l}{S}_{yy}={\displaystyle \sum y_i^2}-\frac{({\displaystyle \sum y_i{)}^2}}{n}\\ =4878636-\frac{{(6236)}^2}{8}\\ =17674\end{array}$

The slope is,

  $\begin{array}{l}b=\frac{S_{xy}}{S_{xx}}\\ =\frac{-6060}{1993.5}\\ =-1.70453\end{array}$

The mean is the sum of all values divided by the number of values.

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{319}{8}\\ =39.875\end{array}$

  $\begin{array}{l}\overline{y}=\frac{{\displaystyle \sum y_i}}{n}\\ =\frac{6236}{8}\\ =779.5\end{array}$

The estimate $a$ of the intercept is,

  $\begin{array}{l}a=\overline{y}-b\overline{x}\\ =779.50-(-1.70453)\times 39.88\\ =847.476656\end{array}$

The least squares regression line is:

  $\begin{array}{l}\widehat{y}=a+bx\\ =847.476656-1.70453x\end{array}$

Conclusion: Thus, the least squares regression line is $\widehat{y}=847.476656-1.70453x$ .

(d)

To determine

To plot:the line on the scatterplot in part b and explain whether the line fits the data points.

Answer to Problem 12.13E

The line fits through the data points.

Explanation of Solution

Given:

School	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
API	$745$	$\begin{array}{l}808\hfill \end{array}$	$\begin{array}{l}798\hfill \end{array}$	$\begin{array}{l}791\hfill \end{array}$	$\begin{array}{l}854\hfill \end{array}$	$\begin{array}{l}688\hfill \end{array}$	$\begin{array}{l}801\hfill \end{array}$	$\begin{array}{l}751\hfill \end{array}$
EL	$\begin{array}{l}71\hfill \end{array}$	$18$	$24$	$50$	$17$	$\begin{array}{l}71\hfill \end{array}$	$11$	$57$

Calculation:

We plot the line on the scatter plot in the part (b):

  

The line fits through the data points. Since, there is equal deviation from the fitted line.

Conclusion: The line fits through the data points.