Question

If the correlation coefficient has a negative value, then the coefficient of determination:_________.a. can take on either a negative or positive value.b. must be positive.c. will also have a negative value.d. will equal zero.

Answer 1

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Answer 2

Answer:

Step-by-step explanation:

The sign of r depends on the sign of the estimated slope coefficient b1:

If b1 is negative, then r takes a negative sign.

If b1 is positive, then r takes a positive sign.

That is, the estimated slope and the correlation coefficient r always share the same sign. Furthermore, because r2 is always a number between 0 and 1, the correlation coefficient r is always a number between -1 and 1.

One advantage of r is that it is unitless, allowing researchers to make sense of correlation coefficients calculated on different data sets with different units. The "unitless-ness" of the measure can be seen from an alternative formula for r, namely:

r=∑ni=1(xi−x¯)(yi−y¯)∑ni=1(xi−x¯)2∑ni=1(yi−y¯)2−−−−−−−−−−−−−−−−−−−−−−−√

If x is the height of an individual measured in inches and y is the weight of the individual measured in pounds, then the units for the numerator is inches × pounds. Similarly, the units for the denominator is inches × pounds. Because they are the same, the units in the numerator and denominator cancel each other out, yielding a "unitless" measure.

Another formula for r that you might see in the regression literature is one that illustrates how the correlation coefficient r is a function of the estimated slope coefficient b1:

r=∑ni=1(xi−x¯)2−−−−−−−−−−−−√∑ni=1(yi−y¯)2−−−−−−−−−−−√×b1