To calculate the correlation coefficient $r$ for the (normal score, $x$) pairs, we can follow these steps:

Formula:

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

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

Where:

• $n$ is the number of pairs.
• $x$ represents the normal scores.
• $y$ represents the fat content.
• $\sum xy$ is the sum of the products of the corresponding $x$ and $y$ values.
• $\sum x^2$ is the sum of the squares of the $x$-values.
• $\sum y^2$ is the sum of the squares of the $y$-values.

Data from the table:

• $x$ (fat contents): 6, 10, 11, 14, 16, 16, 17
• Normal scores: -1.364, -0.758, -0.353, 0, 0.353, 0.758, 1.364

Step-by-step solution:

1. Multiply each pair of $x$ and $y$ values.
2. Find the sums of $x$, $y$, $x^2$, and $y^2$.
3. Plug these values into the correlation coefficient formula.

Let me now calculate these values.The correlation coefficient $r$ for the (normal score, fat content) pairs is approximately $0.9573$.

This indicates a strong positive correlation between the normal scores and the fat content in grams.

Would you like further details on the calculations, or do you have any questions?

Here are 5 related questions:

1. What does a correlation coefficient close to 1 signify in this context?
2. How would you interpret the strength of the relationship between fat content and normal scores?
3. Can you explain how changing one of the data points would affect the correlation?
4. What other statistical methods could be used to analyze this data?
5. How could outliers in the data affect the correlation coefficient?

Tip: Always ensure your data is linearly related before interpreting the Pearson correlation coefficient, as it only captures linear relationships.