Background info given: Survey of 23 students, comparing GPA with starting salary model: salary = 21,100 +2,750(GPA) Range of GPA's are 2.23 to 3.85 Range of starting salaries is 22,300 to 37,400 Average GPA = 3.04 and Average Starting Salary = 29,460 Standard Deviation of GPA = 1.5 Standard Deviation of Salary = 5000
Find the correlation coefficient, r.
I cannot figure out how to do this unless I have each actual data point, which I do not have, simply the ranges, means, and standard deviations.
For $y=\beta_0 + \beta_1X+\epsilon$, the OLS estimator of $\beta_1$ is given by $\frac{S_{xy}}{S^2_x}$ and the sample correlation coefficient is $r=\frac{S_{xy}}{S_xS_y}$, thus $r=\frac{S_x}{S_y}\frac{S_{xy}}{S_xS_y}=\frac{S_x}{S_y}\hat{\beta}_1$, so $r = \frac{1.5}{5000}\times2,750=0.825$.