Regresion lineal simple

RESEARCH PROJECT 1

INTRODUCTION
I am going to study the possible relationship between the economic growth in Spain, specifically the GDP per capita, and the growth in the birth rate in Spain between the years 1991 and 2005. For this research, I have collected data from the Spanish National Statistics Institute webpage (www.ine.es).
The birth rate is measured per 1000 inhabitants and isdenoted by “x” and the GDP per capita is measured in Euros € and is denoted as “y”.

| x | y |
1991 | 4,28 | 8.798,10 |
1992 | 4,54 | 9.444,55 |
1993 | 4,54 | 9.740,83 |
1994 | 4,74 | 10.332,14 |
1995 | 4,85 | 11.353,85 |
1996 | 4,99 | 12.002,66 |
1997 | 4,83 | 12.730,62 |
1998 | 5,23 | 13.581,69 |
1999 | 5,64 | 14.524,98 |
2000 | 6,11 | 15.653,20 |
2001 | 6,62 |16.715,47 |
2002 | 7,22 | 17.650,33 |
2003 | 7,36 | 18.629,66 |
2004 | 7,74 | 19.678,42 |
2005 | 8,24 | 20.863,89 |
| 86,93 | 211.700,39 |

On a first impression, we can say that both the GDP and the birth rate increase at the same time but we have to use statistics to see whether there is a correlation between these two growths or not, whether our hypothesis is reliable or not.
Now I amgoing to present summary statistics using Excel that will help us to analyze the data:

  | x | y | X2 | Y2 | XY | (Xi- ̅) | (Xi-x)2 | (Yi-y) | (Yi-y)2 |
1991 | 4,28 | 8.798,10 | 18,32 | 77.406.563,61 | 37.655,87 | -1,52 | 2,30 | -5.315,26 | 28.251.981,78 |
1992 | 4,54 | 9.444,55 | 20,61 | 89.199.524,70 | 42.878,26 | -1,26 | 1,58 | -4.668,81 | 21.797.780,59 |
1993 | 4,54 | 9.740,83 |20,61 | 94.883.769,09 | 44.223,37 | -1,26 | 1,58 | -4.372,53 | 19.119.012,77 |
1994 | 4,74 | 10.332,14 | 22,47 | 106.753.116,98 | 48.974,34 | -1,06 | 1,11 | -3.781,22 | 14.297.619,65 |
1995 | 4,85 | 11.353,85 | 23,52 | 128.909.909,82 | 55.066,17 | -0,95 | 0,89 | -2.759,51 | 7.614.891,76 |
1996 | 4,99 | 12.002,66 | 24,90 | 144.063.847,08 | 59.893,27 | -0,81 | 0,65 | -2.110,70 | 4.455.051,68 |1997 | 4,83 | 12.730,62 | 23,33 | 162.068.685,58 | 61.488,89 | -0,97 | 0,93 | -1.382,74 | 1.911.968,06 |
1998 | 5,23 | 13.581,69 | 27,35 | 184.462.303,26 | 71.032,24 | -0,57 | 0,32 | -531,67 | 282.672,28 |
1999 | 5,64 | 14.524,98 | 31,81 | 210.975.044,00 | 81.920,89 | -0,16 | 0,02 | 411,62 | 169.431,57 |
2000 | 6,11 | 15.653,20 | 37,33 | 245.022.670,24 | 95.641,05 | 0,31 | 0,10 | 1.539,84 |2.371.109,28 |
2001 | 6,62 | 16.715,47 | 43,82 | 279.406.937,32 | 110.656,41 | 0,82 | 0,68 | 2.602,11 | 6.770.979,92 |
2002 | 7,22 | 17.650,33 | 52,13 | 311.534.149,11 | 127.435,38 | 1,42 | 2,03 | 3.536,97 | 12.510.161,50 |
2003 | 7,36 | 18.629,66 | 54,17 | 347.064.231,72 | 137.114,30 | 1,56 | 2,45 | 4.516,30 | 20.396.971,71 |
2004 | 7,74 | 19.678,42 | 59,91 | 387.240.213,70 | 152.310,97 |1,94 | 3,78 | 5.565,06 | 30.969.900,22 |
2005 | 8,24 | 20.863,89 | 67,90 | 435.301.905,93 | 171.918,45 | 2,44 | 5,98 | 6.750,53 | 45.569.664,28 |
  | 86,93 | 211.700,39 | 528,18 | 3.204.292.872,13 | 1.298.209,87 | 0,00 | 24,39 | 0,00 | 216.489.197,06 |

* Mean x: | 5,79533333 |
* Mean y: | 14113,3593 |
* Standard deviation x: | 1,2752666 |
* Standard deviation y: |3799,02792 |
* var(x): | 1,74246952 |
* Var(y): | 15463514,1 |
* Cov(x,y): | 4755,70296 |
* Correlation coefficient: | 0,98161515 |
* Min (x): | 4,28 |
* Max (x): | 8,24 |
* Min(y): | 8.798,10 |
* Max(y): | 20.863,89 |
* Median(x): | 5,23 |
* Median(y): | 13.581,69 |

Since the linear correlation coefficient value is very close to 1, we can say that itis a positive and strong correlation between the two variables, the birth rate and the GDP. This means that when Spanish wealth increases, the birth rate also increases.

LINEAR REGRESSION:
I have used excel again to make the simple linear regression between the GDP and the birth rate.

We can see that the slope is positive, meaning that it is a positive correlation between the two...