Question

McDonald's Corporation has reported the following values for total revenues and net income during the 1998 to 2005 period. All data are in billions of dollars: Source: McDonald's Corporation, 2005 Annual Report. 1998 1999 2000 2001 Net Income 1.55 1.95 1.98 1.64Total Revenues 12.42 13.26 14.24 14.87 2002 2003 2004 2005 Net Income 0.89 1.47 2.28 2.60Total Revenues 15.41 17.14 19.07 20.46 Determine the least-squares regression equation line for estimating net income and interpret its slope.

Answer 1

Answer:

$y=0.0998 x +0.212$

For this case the slope means that for every increase of 1 unit in the Revenues we will have an increase of approximately 0.0998 in the net income.

We assume that Net Income (Y) and the Revenues represent (X)

See explanation below.

Step-by-step explanation:

For this case w ehave the following data:

Year                1998    1999    2000      2001     2002    2003   2004     2005

Net Income     1.55      1.95     1.98        1.64       0.89      1.47      2.28      2.6

Revenues       12.42    13.26   14.24     14.87     15.41     17.14      19.07     20.46

We assume that Net Income (Y) and the Revenues represent (X)

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

$m=\frac{S_{xy}}{S_{xx}}$

Where:

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

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

So we can find the sums like this:

$\sum_{i=1}^n x_i =126.87$

$\sum_{i=1}^n y_i =14.36$

$\sum_{i=1}^n x^2_i =2067.503$

$\sum_{i=1}^n y^2_i =27.7264$

$\sum_{i=1}^n x_i y_i =233.2763$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=2067.503-\frac{126.87^2}{8}=55.503$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=233.2763-\frac{126.87*14.36}{8}=5.544$

And the slope would be:

$m=\frac{5.544}{55.503}=0.0998$

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

$\bar x= \frac{\sum x_i}{n}=\frac{126.87}{8}=15.859$

$\bar y= \frac{\sum y_i}{n}=\frac{14.36}{8}=1.795$

And we can find the intercept using this:

$b=\bar y -m \bar x=1.795-(0.0998*15.859)=0.212$

So the line would be given by:

$y=0.0998 x +0.212$

For this case the slope means that for every increase of 1 unit in the Revenues we will have an increase of approximately 0.0998 in the net income.