To solve the problem provided in the image:

Step 1: Verify the scatterplot

From the data provided:

• $x = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]$
• $y = [9.13, 8.13, 8.73, 8.77, 9.25, 8.11, 6.13, 3.11, 9.12, 7.26, 4.74]$

The scatterplot that best fits the data appears to match Graph C, as selected.

Step 2: Calculate the Linear Correlation Coefficient $r$

The formula for the linear correlation coefficient $r$ is:

[ r = \frac{n \sum(xy) - \sum x \sum y}{\sqrt{\left[n \sum x^2 - (\sum x)^2 \right] \left[n \sum y^2 - (\sum y)^2 \right]}} ]

Where:

• $n = \text{number of data points} = 11$
• $\sum x = \text{sum of all } x \text{ values}$
• $\sum y = \text{sum of all } y \text{ values}$
• $\sum x^2 = \text{sum of squares of } x \text{ values}$
• $\sum y^2 = \text{sum of squares of } y \text{ values}$
• $\sum xy = \text{sum of the product of corresponding } x \text{ and } y \text{ values}$

I np

Data points

x = np.array([10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]) y = np.array([9.13, 8.13, 8.73, 8.77, 9.25, 8.11, 6.13, 3.11, 9.12, 7.26, 4.74])

Calculations

n = len(x) sum_x = np.sum(x) sum_y = np.sum(y) sum_x2 = np.sum(x2) sum_y2 = np.sum(y2) sum_xy = np.sum(x * y)

Correlation coefficient formula

r = (n * sum_xy - sum_x * sum_y) / (np.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_yThe linear correlation coefficient $r$ is approximately 0.817.

Step 3: Determine Significance

To determine if there is a significant linear correlation, we compare $r$ against the critical value for $\alpha = 0.05$ and $n-2 = 9$ degrees of freedom. You would need to reference the critical value table, but typically $r$ values above approximately $0.602$ are significant for $n=11$.

Since $r = 0.817$ exceeds the critical value, there is sufficient evidence to support the claim of a linear correlation between the two variables.

Would you like a detailed explanation or help with additional parts? Here are 5 related questions you might explore:

1. How is the correlation coefficient $r$ affected by outliers?
2. Can you interpret the strength and direction of this correlation?
3. What is the critical value for $r$ for a different $\alpha$ (e.g., $0.01$)?
4. How do you plot and interpret the regression line for this data?
5. What assumptions are necessary for $r$ to be valid?

Tip: Always interpret $r$ in the context of the