4. What is R2 , what additional information does this give us? 5. a. Use SPSS to find the regression equation. b. What score on the exam could we predict someone would receive if they scored a 38 on the anxiety questionnaire?

Q4 )

The provided data are shown in the table below

Also, the following calculations are needed to compute the correlation coefficient:

The correlation coefficient $r$ is computed using the following expression:

$r = \frac{ SS_{XY}}{\sqrt{ SS_{XX} SS_{YY}}}$

where

$SS_{XY} = \sum_{i=1}^n {X_i Y_i}-\frac{1}{n}\left(\sum_{i=1}^n X_i\right) \left(\sum_{i=1}^n Y_i\right)$

$SS_{XX} = \sum_{i=1}^n {X_i^2}-\frac{1}{n}\left(\sum_{i=1}^n X_i\right)^2$

$SS_{YY} = \sum_{i=1}^n {Y_i^2}-\frac{1}{n}\left(\sum_{i=1}^n Y_i\right)^2$

In this case, based on the data provided, we get that

$SS_{XY} = 25671 - \frac{1}{ 10}( 394 \times 655) = -136$

$SS_{XX} = 15946 - \frac{1}{ 10} (394)^2 = 422.4$

$SS_{YY} = 45205 - \frac{1}{ 10} (655)^2 = 2302.5$

Therefore, based on this information, the sample correlation coefficient is computed as follows

$r = \frac{ SS_{XY}}{\sqrt{ SS_{XX} SS_{YY}}} = \frac{ -136}{\sqrt{ 422.4 \times 2302.5}} = -0.138$

Cofficient of Determination , $R^2$ = (-0.138${)}^2$ = 0.019

The coefficient of determination is a measurement used to explain how much variability of one factor can be caused by its relationship to another related factor. 

which means 1.9% of variability in the Data is explained by this model which is very less as the value of correlation coefficient is 0.138 which lies in lowest zone

Q5)

The independent variable is marks in the Anxiety paper represented by $x$, and the dependent variable is marks in Final exams represented by $y$. In order to compute the regression coefficients, the following table needs to be used:

Based on the above table, the following is calculated:

$\bar X = \frac{1}{n} \sum_{i=1}^{n} X_i = \frac{ 394}{ 10} = 39.4$

$\bar Y = \frac{1}{n} \sum_{i=1}^{n} Y_i = \frac{ 655}{ 10} = 65.5$

$SS_{XX} = \sum_{i=1}^{n} X_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right)^2 = 15946 - 394^2/10 = 422.4$

$SS_{YY} = \sum_{i=1}^{n} Y_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} Y_i\right)^2 = 45205 - 655^2/10 = 2302.5$$SS_{XY} = \sum_{i=1}^{n} X_i Y_i - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right) \left(\sum_{i=1}^{n} Y_i\right) = 25671 - 394 \times 655/10 = -136$

Therefore, based on the above calculations, the regression coefficients (the slope $m$, and the y-intercept $n$) are obtained as follows:

$m = \frac{SS_{XY}}{SS_{XX}} = \frac{ -136}{ 422.4} = -0.322$

 $n = \bar Y - \bar X \cdot m = 65.5 - 39.4 \times (-0.322) = 78.1856$

Therefore, we find that the regression equation is:

$\mathit{y}\mathbf{=}\mathbf{78}\mathbf{.1856}\mathbf{-}\mathbf{0}\mathbf{.322}\mathit{x}$

Given :

Anxiety Score =38

Then , Final Marks , y = 78.1856 - 0.322*(38) = 65.9496