A question from a test in statistics?

I got the following question in my finals, but the teacher claims I didn't answer this right, I chose a. as the right answer, what is the right answer and why?

Here is the question:

In a research about the connection between two quantitative variables there were collected observations (more than 2), where each observation has value for x and y and the standard deviation of x and y ,$S_x$ $S_y$, greater than 0.

which one of the following is true:

1. if Pearson product-moment correlation coefficient is $1$ then Spearman's rank correlation coefficient isn't necessarily 1.
2. if $S_x = S_y$ then $\mathrm{Cov}(x,y)$ can't be greater than $S_x$.
3. if $\mathrm{Cov}(x,y) = 0$ then the percentage of variance non explained must be 0.
4. if the regression line's slope of expected $y$ value as function of $x$ is 1, then the Pearson product-moment correlation coefficient is $1$.
• Did the teacher tell you which one was meant to be correct? – markseeto Aug 18 '13 at 9:09
• @mark999 I think all of them are wrong... – 0x90 Aug 18 '13 at 15:17
• In response to your flag, I don't think this question belongs on math.SE. In case you are not happy with current suggestions, you can always offer a bounty to attract users' attention. – chl Aug 18 '13 at 20:58
• I edited the existing answer to emphasize how it fully answers the question. – whuber Aug 19 '13 at 14:20

None are correct.

The first is wrong because Spearman's rank correlation is defined the Pearson correlation using ranks instead of actual values: therefore when the Spearman correlation is not $1$, the Pearson correlation cannot be $1$, either.

In the second answer the covariance and $S_x$ are measured in different units and so are not even comparable.

In the third answer the percentage of variance explained must be 0.

In the fourth answer, the regression slope is not related to the value of Pearson's correlation coefficient. The PCC can be any value between $0$ and $1$ (not including $0$) and when the slope is $1$ the slope can be any positive value when the PCC is $1$

• It holds for variance not standard deviation – SAAN Aug 18 '13 at 6:51
• Yes, then it seems that all are false – O_Devinyak Aug 18 '13 at 7:11
• If $Y = aX + b$, then $a=r\frac{\sigma_y}{\sigma_x}$, where $r$ is the Pearson correlation, $\sigma_{X,Y}$ are the respective standard deviations. – Doctor Dan Jun 24 '14 at 20:19