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Let's solve the problem step-by-step.

### Part (a): Determine the correlation coefficient

The data provided is:

| Movie | Minutes Shirtless (x) | Opening Weekend Gross (y) |
|-------|-----------------------|---------------------------|
| A     | 0                     | 6.2                       |
| B     | 0.6                   | 8.6                       |
| C     | 1.5                   | 16.6                      |
| D     | 2.1                   | 18.8                      |
| E     | 14.1                  | 22.4                      |

We'll use the formula for the correlation coefficient \( r \):
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]

Where \( n \) is the number of data points.

First, we calculate the sums and products needed:
\[ \sum x = 0 + 0.6 + 1.5 + 2.1 + 14.1 = 18.3 \]
\[ \sum y = 6.2 + 8.6 + 16.6 + 18.8 + 22.4 = 72.6 \]
\[ \sum xy = (0 \times 6.2) + (0.6 \times 8.6) + (1.5 \times 16.6) + (2.1 \times 18.8) + (14.1 \times 22.4) \]
\[ = 0 + 5.16 + 24.9 + 39.48 + 316.64 = 386.18 \]
\[ \sum x^2 = 0^2 + 0.6^2 + 1.5^2 + 2.1^2 + 14.1^2 \]
\[ = 0 + 0.36 + 2.25 + 4.41 + 198.81 = 205.83 \]
\[ \sum y^2 = 6.2^2 + 8.6^2 + 16.6^2 + 18.8^2 + 22.4^2 \]
\[ = 38.44 + 73.96 + 275.56 + 353.44 + 501.76 = 1243.16 \]

Now plug these values into the formula:
\[ r = \frac{5(386.18) - (18.3)(72.6)}{\sqrt{[5(205.83) - (18.3)^2][5(1243.16) - (72.6)^2]}} \]
\[ r = \frac{1930.9 - 1328.58}{\sqrt{[1029.15 - 334.89][6215.8 - 5271.76]}} \]
\[ r = \frac{602.32}{\sqrt{694.26 \times 944.04}} \]
\[ r = \frac{602.32}{\sqrt{655609.4744}} \]
\[ r = \frac{602.32}{809.67} \]
\[ r \approx 0.74 \]

The correlation coefficient \( r \) is approximately 0.74.

### Part (b): Find the equation of the regression line

The equation of the regression line is \( y = mx + b \), where:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{\sum y - m(\sum x)}{n} \]

Using the values calculated above:
\[ m = \frac{5(386.18) - (18.3)(72.6)}{5(205.83) - (18.3)^2} \]
\[ m = \frac{1930.9 - 1328.58}{1029.15 - 334.89} \]
\[ m = \frac{602.32}{694.26} \]
\[ m \approx 0.87 \]

Now calculate \( b \):
\[ b = \frac{72.6 - 0.87(18.3)}{5} \]
\[ b = \frac{72.6 - 15.921}{5} \]
\[ b = \frac{56.679}{5} \]
\[ b \approx 11.34 \]

Thus, the equation of the regression line is:
\[ y = 0.87x + 11.34 \]

Do you want more details or have any questions?

### Questions for you to ask:
1. How do you interpret the correlation coefficient value in this context?
2. Can you explain the steps to derive the slope \( m \) of the regression line?
3. How would you predict the opening weekend gross if the actor appeared shirtless for 5 minutes?
4. What does the y-intercept \( b \) represent in the regression equation?
5. How can you assess the strength and direction of a relationship using the correlation coefficient?

### Tip:
Always check your calculations step by step to avoid any mistakes, especially when dealing with multiple sums and products in statistical formulas.

Learn how to calculate the correlation coefficient and regression line for a given dataset of movie data. Step-by-step solution includes formulas for correlation coefficient and regression line equation. Suitable for higher education level.

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