Question

In order for a company's employees to work in a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Find the equation of the regression line for the given data. Round the regression line values to the nearest hundredth.Number of years, x 3 4 4 5 3 6 2 7 3Grades on test, y 61 68 75 82 73 90 58 93 72the equation of the regression line for the data is y=6.9096x+46.241. State the meaning of the slope in this context. Do not say "change in y over change in x." State what it means in relationship to the data in this problem.

Answer 1

Answer:

$\sum x= 37, \sum y= 672, \sum xy =2907, \sum x^2 =173, \sum y^2 = 51320$

Where:  

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$  

With these we can find the sums:  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=173-\frac{37^2}{9}=20.889$  

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=2907-\frac{37*672}{9}=144.333$  

And the slope would be:  

$m=\frac{144.333}{20.889}=6.90953$  

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

$\bar x= \frac{\sum x_i}{n}=\frac{37}{9}=4.111$  

$\bar y= \frac{\sum y_i}{n}=\frac{672}{9}=74.667$  

And we can find the intercept using this:  

$b=\bar y -m \bar x=74.667-(6.9096*4.111)=46.241$  

So the line would be given by:  

$y=6.9096 x +46.241$  

And for this case the value of the slope m = 6.9096 means that for every increase of 1 unit in the number of years we have an increase of approximately 6.9096 in the grades of the test.  

Step-by-step explanation:

Data given:

x: 3, 4, 4, 5, 3, 6, 2, 7, 3

y: 61, 68, 75, 82, 73, 90, 58, 93, 72

$m=\frac{S_{xy}}{S_{xx}}$  

$\sum x= 37, \sum y= 672, \sum xy =2907, \sum x^2 =173, \sum y^2 = 51320$

Where:  

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$  

With these we can find the sums:  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=173-\frac{37^2}{9}=20.889$  

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=2907-\frac{37*672}{9}=144.333$  

And the slope would be:  

$m=\frac{144.333}{20.889}=6.90953$  

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

$\bar x= \frac{\sum x_i}{n}=\frac{37}{9}=4.111$  

$\bar y= \frac{\sum y_i}{n}=\frac{672}{9}=74.667$  

And we can find the intercept using this:  

$b=\bar y -m \bar x=74.667-(6.9096*4.111)=46.241$  

So the line would be given by:  

$y=6.9096 x +46.241$  

And for this case the value of the slope m = 6.9096 means that for every increase of 1 unit in the number of years we have an increase of approximately 6.9096 in the grades of the test.