gainejavima

2021-11-17

Find the GPA of a student if he gives average hours to study by using regression alysis. A random sample of ten students is selected from a class of 60 students, find regression line and also find the GPA of a student if he gives average hours to study by using this regression analysis. Their GPA and the hours to study are given below:

$$\begin{array}{|ccccccccccc|}\hline GPA& 2.5& 3.1& 3.6& 3.2& 2.9& 3.7& 2.9& 2.4& 3.3& 3.5\\ Hours\text{}to\text{}study& 3& 5& 7& 5& 3& 8& 4& 2& 5& 7\\ \hline\end{array}$$

Also interpret the regression coefficient.

Also interpret the regression coefficient.

Elizabeth Witte

Beginner2021-11-18Added 24 answers

Step 1

In simple linear regression, there will be exactly one dependent variable and independent variable.

Step 2

The regression analysis can be done in any of the statistical software or by hand. Here GPA is the dependent variable (X) and amount of time study is the independent variable (Y).

So the regression line can be calculated as follows.

Sum of

Sum of

Sum of squares

Sum of products =7.51

=0.21519

=2.05559

So the regression equation is given below.

Step 3

Here the average hours of study is 4.9. Substitute this in the regression equation.

=3.11

So the GPA is 3.11 when the the study time is average.

Her the regression coefficient is 0.21519. This indicates that for every one unit change in time, the GPA also change by 0.21519 in the same direction.

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Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by:

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$f(x,y)=\{\begin{array}{ll}({x}^{2}+{y}^{2})\mathrm{cos}\frac{1}{\sqrt{{x}^{2}+{y}^{2}}},& \text{for}(x,y)\ne (0,0)\\ 0,& \text{for}(x,y)=(0,0)\end{array}$

then check whether its differentiable and also whether its partial derivatives ie ${f}_{x},{f}_{y}$ are continuous at $(0,0)$. I dont know how to check the differentiability of a multivariable function as I am just beginning to learn it. For continuity of partial derivative I just checked for ${f}_{x}$ as function is symmetric in $y$ and $x$. So ${f}_{x}$ turns out to be

${f}_{x}(x,y)=2x\mathrm{cos}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)+\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\mathrm{sin}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)$

which is definitely not $0$ as $(x,y)\to (0,0)$. Same can be stated for ${f}_{y}$. But how to proceed with the first part?