Video Transcript
What is the most likely value of
the product-moment correlation coefficient for the data shown in the diagram? Is it (A) negative 0.58, (B) zero,
(C) negative 0.94, (D) 0.78, or (E) 0.37?
In estimating Pearson’s correlation
coefficient from a scatter plot, there are two things we look at. The first is the direction of the
linear pattern, which in our case is top left to bottom right. And the second thing is the spread
of the data points around a possible line of best fit, that is, how close our data
points are to a potential line of best fit. Generally speaking, we know that if
a linear pattern of data is from bottom left to top right, then we have positive or
direct correlation. Conversely, if our data follow a
linear pattern from top left to bottom right, we say our data is negatively or
inversely correlated. And if our data is directly
correlated, that’s positively, then our coefficient is between zero and one, whereas
if our data is inversely correlated, the coefficient is between negative one and
zero.
In our case, our linear pattern is
from top left to bottom right, so ours is the second case. This means our correlation
coefficient must be between negative one and zero. And this means we can eliminate
both (D) and (E) since these are both positive. And now if we look at the spread of
the data, we know that the wider the spread away from a potential line of best fit,
the weaker the correlation and that the closer the data points are to a potential
line of best fit, the stronger the correlation. We know that Pearson’s correlation
coefficient takes values from negative one to positive one and that the closer the
coefficient is to positive or negative one, the stronger the correlation. And we know that the closer the
correlation coefficient gets to zero, the weaker the correlation.
In the given plot, most of the data
points are very close to a possible line of best fit. And remembering that our
coefficient is negative, this means our coefficient must be close to negative
one. We can eliminate (B) since we know
that a correlation coefficient of zero means there’s no correlation at all, and we
have very strong correlation. And so we’re left with option (A)
and option (C). Option (A) with the value negative
0.58 would indicate a moderate correlation. That’s because it’s just over
halfway between zero and negative one. And since our correlation is very
strong, we can eliminate option (A). Option (C) is the closest to
negative one with a value negative 0.94. So the most likely product-moment
correlation coefficient for the data shown is (C) is equal to negative 0.94.
