Questions: Suppose that two variables, X and Y, are negatively associated. Does this mean that above-average values of X will always be associated with below-average values of Y? Explain. Choose the correct answer below. A. No, because when two variables, X and Y, are negatively associated, above-average values of X are associated with above-average values of Y. B. No, because there will always be at least one point that does not fit the trend. C. No, because association does not mean that every point fits the trend. The negative association only means that above-average values of X are generally associated with below-average values of Y. D. Yes, because if one or more above-average values of X are associated with above-average values of Y, the variables cannot be negatively associated.

Suppose that two variables, X and Y, are negatively associated. Does this mean that above-average values of X will always be associated with below-average values of Y? Explain.

Choose the correct answer below.
A. No, because when two variables, X and Y, are negatively associated, above-average values of X are associated with above-average values of Y.
B. No, because there will always be at least one point that does not fit the trend.
C. No, because association does not mean that every point fits the trend. The negative association only means that above-average values of X are generally associated with below-average values of Y.
D. Yes, because if one or more above-average values of X are associated with above-average values of Y, the variables cannot be negatively associated.

Transcript text: oter 4: Describing the Two Variables Question 5 of 10 This quiz: 10 point(s) possible This question: 1 point(s) possible Submit quiz Suppose that two variables, X and Y , are negatively associated. Does this mean that above-average values of X will always be associated with below-average values of Y? Explain. Choose the correct answer below. A. No, because when two variables, $X$ and $Y$, are negatively associated, above-average values of $X$ are associated with above-average values of $Y$. B. No, because there will always be at least one point that does not fit the trend. C. No, because association does not mean that every point fits the trend. The negative association only means that above-average values of X are generally associated with below-average values of Y . D. Yes, because if one or more above-average values of $X$ are associated with above-average values of $Y$, the variables cannot be negatively associated.

Solution

Solution Steps

Step 1: Calculate Covariance and Standard Deviations

To determine the relationship between the variables $ X $ and $ Y $, we first calculate the covariance $ \text{Cov}(X,Y) $ and the standard deviations $ \sigma_X $ and $ \sigma_Y $.

Given: \[ \text{Cov}(X,Y) = -5.0 \] \[ \sigma_X = 1.5811 \] \[ \sigma_Y = 3.1623 \]

Step 2: Calculate the Correlation Coefficient

The correlation coefficient $ r $ is calculated using the formula: \[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \] Substituting the values: \[ r = \frac{-5.0}{1.5811 \times 3.1623} = -1.0 \]

Step 3: Interpret the Correlation Coefficient

The correlation coefficient $ r = -1.0 $ indicates a perfect negative association between the variables $ X $ and $ Y $. This means that as $ X $ increases, $ Y $ decreases in a perfectly linear manner.

Step 4: Conclusion on Association

The negative association implies that above-average values of $ X $ are generally associated with below-average values of $ Y $. However, it does not mean that every individual data point will fit this trend perfectly.

Final Answer

The correct interpretation is: No, because association does not mean that every point fits the trend. The negative association only means that above-average values of $ X $ are generally associated with below-average values of $ Y $.

Thus, the answer is: $\boxed{C}$

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