Correlation Coefficient

(a) To find the least-squares regression line, we will use the following formula:

y = a + bx

where

• y is the distance the ball traveled (in feet)
• x is the speed at which the ball was hit (in miles per hour)
• a is the y-intercept
• b is the slope

We can calculate the values of a and b using the following formulas:

a = Σy – bΣx/n

b = Σxy – ΣxΣy/n(n – 1)

where

• Σ represents the sum of
• n is the number of data points

Substituting the values from the table, we get the following:

a = Σy – bΣx/n = 3931 – b(1030)/12 = 287 – 85b

b = Σxy – ΣxΣy/n(n – 1) = (3931 * 1030) – (1030 * 287)/12 * 11 = 410470 – 30481/132 = 3116

Therefore, the least-squares regression line is:

y = 287 + 3116x

(b) The slope of the least-squares regression line, b, is 3116. This means that for every 1 mile per hour increase in the speed at which the ball is hit, the distance it travels is expected to increase by 3116 feet.

(c) The y-intercept of the least-squares regression line, a, is 287. This means that if the speed at which the ball is hit is 0 miles per hour, the expected distance it travels is 287 feet.

(d) The correlation coefficient between the speed at which the ball is hit and the distance it travels is 0.878. This is a strong positive correlation, which means that there is a strong positive relationship between the two variables.

(e) The coefficient of determination, R^2, is 0.765. This means that 76.5% of the variation in the distance the ball travels can be explained by the variation in the speed at which the ball is hit.

(f) To predict the distance a home run will travel if it is hit with a speed of 105 mph, we can substitute this value into the least-squares regression line:

y = 287 + 3116(105) = 4353

Therefore, the predicted distance the home run will travel is 4353 feet.