Question

This week, we are covering relationships that can be approximated by linear equations. For instance, y = 453x + 3768 represents the increase in cost of tuition and fees at public four-year universities. Now think about how this might apply in your chosen field and answer the following question: What is an example of a relationship (in your chosen field) that can be represented by a linear equation? Show the equation and explain the relationship. Please provide a reference for your example (website, book, etc.)

Answer 1

See explanation below.

Explanation:

We assume that the data is given by :

x: 30, 30, 30, 50, 50, 50, 70,70, 70,90,90,90

y: 38, 43, 29, 32, 26, 33, 19, 27, 23, 14, 19, 21.

Where X represent the cost for scholarships in thousands of dollars and y represent the cost of life for an academic semester (The data comes from the web)

We can find the least-squares line appropriate for this data.

For this case we need to calculate the slope with the following formula:

`m=(S_(xy))/(S_(xx))`

Where:

`S_(xy)=\sum_(i=1)^n x_i y_i -((\sum_(i=1)^n x_i)(\sum_(i=1)^n y_i))/(n)`

`S_(xx)=\sum_(i=1)^n x^2_i -((\sum_(i=1)^n x_i)^2)/(n)`

So we can find the sums like this:

`\sum_(i=1)^n x_i = 30+30+30+50+50+50+70+70+70+90+90+90=720`

`\sum_(i=1)^n y_i =38+43+29+32+26+33+19+27+23+14+19+21=324`

`\sum_(i=1)^n x^2_i =30^2+30^2+30^2+50^2+50^2+50^2+70^2+70^2+70^2+90^2+90^2+90^2=49200`

`\sum_(i=1)^n y^2_i =38^2+43^2+29^2+32^2+26^2+33^2+19^2+27^2+23^2+14^2+19^2+21^2=9540`

`\sum_(i=1)^n x_i y_i =30*38+30*43+30*29+50*32+50*26+50*33+70*19+70*27+70*23+90*14+90*19+90*21=17540`

With these we can find the sums:

`S_(xx)=\sum_(i=1)^n x^2_i -((\sum_(i=1)^n x_i)^2)/(n)=49200-(720^2)/(12)=6000`

`S_(xy)=\sum_(i=1)^n x_i y_i -\frac{(\sum_(i=1)^n x_i)(\sum_(i=1)^n y_i)}=17540-(720*324)/(12){12}=-1900`

And the slope would be:

`m=-(1900)/(6000)=-0.317`

Nowe we can find the means for x and y like this:

`\bar x= (\sum x_i)/(n)=(720)/(12)=60`

`\bar y= (\sum y_i)/(n)=(324)/(12)=27`

And we can find the intercept using this:

`b=\bar y -m \bar x=27-(-0.317*60)=46.02`

So the line would be given by:

`y=-0.317 x +46.02`

We have an inverse linear relationship since the slope is negative between the variables of interest.