The following table shows the annual number of PhD graduates in a country in various fields.   Natural Sciences Engineering Social Sciences Education 1990 70 10 60 30 1995 130 40 100 50 2000 330 130 280 120 2005 490 370 460 210 2010 590 550 830 520 2012 690 590 1,000 900 (a) With x = the number of social science doctorates and y = the number of education doctorates, use technology to obtain the regression equation. (Round coefficients to three significant digits.) y(x) =    What does the slope tell you about the relationship between the number of social science doctorates and the number of education doctorates? The slope tells us the increase in the number of education doctorates for each additional social science doctorate.   The slope tells us the decrease in the number of social science doctorates for each additional education doctorate.    The slope tells us the increase in the number of social science doctorates for each additional education doctorate.   The slope tells us the decrease in the number of education doctorates for each additional social science doctorate.   (c) Use technology to obtain the coefficient of correlation r. (Round your answer to three decimal places.) r =  Does the value of r suggest a strong correlation between x and y? Since r is close to 1, the correlation between x and y is a weak one.   Since r is close to 1, the correlation between x and y is a strong one.      Since r is close to 0, the correlation between x and y is a weak one.   Since r is close to 0, the correlation between x and y is a strong one.   (d) Does the graph suggest a roughly linear relationship between x and y? Why or why not?   No, the graph suggests a concave-up curve rather than a straight line. Yes, the graph suggests a straight line.      Yes, the graph suggests a concave-up curve rather than a straight line.   Yes, the graph suggests a concave-down curve rather than a straight line.   No, the graph suggests a concave-down curve rather than a straight line.

Given : Data represents the annual number of PhD graduates in a country in various fields. 

In linear regression , the relationship between two variables is of the for Y = a + bX , where Y is dependent variable and X is intendent variable . This is called line of regression of dependent variable Y on intendent variable X . 

• R provides useful function lm() . (Linear model) for regression analysis .  
• Following functions are commonly used to extract constants from output of lm() .  

Here,  x = the number of social science doctorates and y = the number of education doctorates 

Use following commands to obtain regression equation : 

> x=c(60,
+     100,
+     280,
+     460,
+     830,
+     1000
+ )
> y=c(30,
+     50,
+     120,
+     210,
+     520,
+     900
+ )
> d=data.frame(x,y)
> d
     x   y
1   60  30
2  100  50
3  280 120
4  460 210
5  830 520
6 1000 900
> r1=lm(y~x)
> r1

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept)            x  
   -80.6297       0.8475  

> coef(r1)                                                #coef() - It gives coefficients of regression equation . 
(Intercept)           x 
-80.6297061   0.8475378  

Linear regression equation = -80.6297 + 0.8476 X

Interpretation : 

The slope tells us the increase in the number of education doctorates for each additional social science doctorate.