Question

two quantities of data have a strong positive correlation. can the line of best fit for the data set have a correlation coefficient of 0.25? explain.

choose the correct answer below.

a. yes. by definition, data that have a strong positive correlation have a strong linear relationship. a correlation coefficient of 0.25 indicates a fairly strong correlation.

b. no. by definition, data that have a strong positive correlation have a strong linear relationship. the correlation coefficient would be closer to 0 than 0.25 is.

c. no. by definition, data that have a strong positive correlation have a strong linear relationship. the correlation coefficient would be closer to -1 than 0.25 is.

d. no. by definition, data that have a strong positive correlation have a strong linear relationship. the correlation coefficient would be closer to 1 than 0.25 is.

e. yes. the sign of the correlation coefficient indicates the type of correlation data have. because 0.25 is positive, it is possible for data with a strong positive correlation to have it as the correlation coefficient.

Explanation:

To determine the answer, we recall the definition of correlation coefficients: A strong positive correlation means the correlation coefficient \( r \) is close to \( 1 \) (since \( r \) ranges from \(-1\) to \( 1 \), with values near \( 1 \) indicating strong positive linear relationships, near \(-1\) strong negative, and near \( 0 \) weak or no linear relationship). A correlation coefficient of \( 0.25 \) is a weak positive correlation (closer to \( 0 \) than \( 1 \)). So, data with a strong positive correlation must have \( r \) closer to \( 1 \), not \( 0.25 \).

• Option A is wrong because \( 0.25 \) is weak, not strong.
• Option B is wrong as strong positive \( r \) is close to \( 1 \), not \( 0 \).
• Option C is wrong because strong positive \( r \) is close to \( 1 \), not \(-1\) (which is strong negative).
• Option D is correct: Strong positive correlation implies \( r \) is closer to \( 1 \), so \( 0.25 \) (weak) is not suitable.
• Option E is wrong because while \( 0.25 \) is positive, its magnitude (weak) contradicts "strong" positive correlation.

Answer:

D. No. By definition, data that have a strong positive correlation have a strong linear relationship. The correlation coefficient would be closer to 1 than 0.25 is.