To compute the correlation coefficient with and without the outlier, let's outline the process and verify the steps. The correlation coefficient $r$ is calculated using:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}$

Case 1: With the outlier

Given:

• $r_\text{with} = 0.021$ was stated as incorrect.
• The incorrectness could arise from a miscalculation or rounding issue.

---

Case 2: Without the outlier

Given:

• $r_\text{without} = -0.991$

This suggests a strong negative linear correlation when the outlier is excluded. If you have the data, I can recheck the calculations for accuracy.

---

Let me know if you'd like assistance to:

1. Recalculate these values.
2. Understand the impact of the outlier on $r$.
3. Verify $r_\text{with}$ and $r_\text{without}$ computationally.
4. Explore the definition and application of the correlation coefficient.
5. Analyze outliers' impact on statistical models.

Tip:

Outliers can significantly distort correlation coefficients. Always plot the data to visually assess relationships.