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.