To calculate the correlation coefficient ($r$), we use the formula:

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

Step-by-step Calculation:

1. Identify the variables:

   • $x$: Number of absences $[5, 8, 9, 10, 11]$
   • $y$: Grades $[90, 78, 74, 62, 58]$

2. Calculate the required sums:

   • $\sum{x} = 5 + 8 + 9 + 10 + 11 = 43$
   • $\sum{y} = 90 + 78 + 74 + 62 + 58 = 362$
   • $\sum{x^2} = 5^2 + 8^2 + 9^2 + 10^2 + 11^2 = 25 + 64 + 81 + 100 + 121 = 391$
   • $\sum{y^2} = 90^2 + 78^2 + 74^2 + 62^2 + 58^2 = 8100 + 6084 + 5476 + 3844 + 3364 = 26868$
   • $\sum{xy} = (5 \cdot 90) + (8 \cdot 78) + (9 \cdot 74) + (10 \cdot 62) + (11 \cdot 58) = 450 + 624 + 666 + 620 + 638 = 2998$

3. Substitute into the formula:

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

Where $n = 5$:

$r = \frac{5(2998) - (43)(362)}{\sqrt{(5 \cdot 391 - 43^2)(5 \cdot 26868 - 362^2)}}$

4. Simplify the terms:
   • $5 \cdot 2998 = 14990$
   • $43 \cdot 362 = 15566$
   • $5 \cdot 391 = 1955$, $43^2 = 1849$, so $1955 - 1849 = 106$
   • $5 \cdot 26868 = 134340$, $362^2 = 131044$, so $134340 - 131044 = 3296$

$r = \frac{14990 - 15566}{\sqrt{106 \cdot 3296}}$

5. Further simplification:
   • $14990 - 15566 = -576$
   • $\sqrt{106 \cdot 3296} = \sqrt{349376} = 591.09$

$r = \frac{-576}{591.09} \approx -0.974$

Final Answer:

The correlation coefficient ($r$) is -0.974.

Would you like a detailed explanation of the process or have further questions?

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Related Questions:

1. What does a correlation coefficient of -0.974 imply about the data?
2. How would the relationship change if grades were normalized?
3. What is the line of best fit for this dataset?
4. How would adding more data points impact the value of $r$?
5. What are the limitations of using the correlation coefficient in this context?

Tip: Always check if your data shows a linear trend before using $r$ to ensure meaningful interpretation!