Question

In a sample linear regression study between two variables x (the independent variable) and y (the dependent variable), a random sample is collected and the coefficient of correlation r=−0.76 is calculated. Which of the following conclusions may be made? Pick one answer and explain.

1. x and y are almost perfectly linearly correlated, and y increases as x is increased.
2. x and y are almost perfectly linearly correlated, and y decreases as x is increased.
3. x and y are moderately linearly correlated, and y increases as x is increased.
4. x and y are moderately linearly correlated, and y decreases as x is increased.
5. x and y are very poorly correlated.

Answer 1

The correct conclusion is that x and y are moderately linearly correlated with a correlation coefficient of -0.76, indicating a strong inverse relationship where y tends to decrease as x increases.

Step-by-step explanation:

The question asks to conclude about the nature of the relationship between an independent variable x and a dependent variable y given a correlation coefficient r of -0.76. This coefficient indicates a negative correlation, meaning as x increases, y is likely to decrease and vice versa. Furthermore, the value of -0.76 suggests a relatively strong correlation, but not nearly perfect.

Thus, the correct answer is: x and y are moderately linearly correlated, and y decreases as x is increased. This conclusion is based on the negative sign of the correlation coefficient indicating an inverse relationship, and the magnitude of r (0.76) which signifies a moderate to strong linear relationship. The term 'moderately' is a subjective interpretation but is typically appropriate for correlations that are strong without being very close to -1 or +1.