Sure, let's refine the calculations and round to three decimal places.

### Step 2: Calculate the Linear Correlation Coefficient \( r \)

Using the formula again and ensuring all intermediate steps are rounded appropriately:

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

Given:
\[
\begin{align*}
n &= 6 \\
\sum x &= 51.1 \\
\sum y &= 1150 \\
\sum xy &= 9976.7 \\
\sum x^2 &= 440.13 \\
\sum y^2 &= 229904 \\
\end{align*}
\]

Plugging in the values:

\[
r = \frac{6 \times 9976.7 - 51.1 \times 1150}{\sqrt{[6 \times 440.13 - 51.1^2][6 \times 229904 - 1150^2]}}
\]

\[
r = \frac{59860.2 - 58765}{\sqrt{[2640.78 - 2611.21][1379424 - 1322500]}}
\]

\[
r = \frac{1095.2}{\sqrt{29.57 \times 56924}}
\]

\[
r = \frac{1095.2}{\sqrt{1682411.48}}
\]

\[
r = \frac{1095.2}{1297.84}
\]

\[
r = 0.8441
\]

### Step 3: Determine the Critical Value for \( r \)

Given \(\alpha = 0.01\) and \( n = 6 \), the critical value from the table is 0.917.

### Step 4: Conclusion

The calculated correlation coefficient \( r = 0.844 \) is less than the critical value of 0.917. Therefore, we do not reject the null hypothesis \( H_0: \rho = 0 \).

### Conclusion
There is not sufficient evidence at the \(\alpha = 0.01\) level to conclude that there is a linear correlation between the overhead widths of seals from photographs and their weights.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:

1. How would the analysis change if the sample size increased?
2. What are the assumptions of Pearson's correlation coefficient?
3. How do outliers affect the correlation coefficient?
4. Can we use a different significance level to test the correlation?
5. How is the scatterplot useful in visualizing the data relationship?
6. What is the difference between correlation and causation?
7. Are there other methods to measure the relationship between two variables?
8. How can we interpret a correlation coefficient close to 1 or -1?

**Tip:** Always ensure to round intermediate calculations properly before arriving at the final result for accurate precision.

Learn how to calculate the linear correlation coefficient step-by-step with an example problem. This problem involves using Pearson's correlation coefficient formula to determine the relationship between overhead widths of seals from photographs and their weights. Detailed calculations and interpretation provided. Suitable for advanced high school and college-level students.

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