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What does a negative sign tell you about the relationship between two variables, $X$ and $Y$?

Select all that apply

$1$) a negative correlation means $X$ and $Y$ change in the same direction

$2$) a negative correlation means as $X$ increases, $Y$ decreases

$3$) a negative correlation means $X$ and $Y$ change in opposite directions

$4$) a negative correlation means that $X$ and $Y$ are weakly related to each other

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  • $\begingroup$ I had only thought of (3) being one of the answers but am unsure of any others. $\endgroup$ – B.E. Nov 30 '17 at 1:46
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Both (2) and (3) are meant to apply.

However, the question is not a very good one, since clearly $X$ and $Y$ are meant to be random variates which have a negative correlation. It is easy to construct cases where $X$ and $Y$ have a negative correlation yet for some particular values $X$ increases while $Y$ also increases.

For example, let random variate $X$ be distributed uniformly on $(-10,10)$ and let $Y$ be a deterministic function of $X$ with $Y = 9X-X^2$. Then the correlation coefficient of $X$ and $Y$ is $-17\sqrt{21/7669} \approx -.89$. Yet if you change $X$ from $0$ to $1$, $Y$ changes from $0$ to $8$. Of course, through most of the domain, a positive change in $X$ results in a negative change in $Y$, but that does not hold everywhere.

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  • $\begingroup$ Very good point, the question was given to me to answer and it was more confusing with what was being asked. Thank you for your help! $\endgroup$ – B.E. Nov 30 '17 at 2:09
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Guide:

If you can answer the following question, it should help you answer your question.

  • If $(3)$ is correct, can $(1)$ be an answer?

  • What is the relationship between $(2)$ and $(3)$. What happens to $Y$ if $X$ decrease?

  • What can you say if the correlation is $-1$? What can you say if the correlation is $-0.01$?

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