Correlation Coefficient

    The following data represent the speed at which a ball was hit​ (in miles per​ hour) and the distance it traveled​ (in feet) for a random sample of home runs in a Major League baseball game in 2018. Complete parts​ (a) through​ (f). ​(a) Find the​ least-squares regression line treating speed at which the ball was hit as the explanatory variable and distance the ball traveled as the response variable. Speed (mph) Distance (feet) 103.0 393 105.3 420 103.5 422 105.5 414 105.4 418 100.3 392 103.5 395 107.9 441 101.4 399 98.0 395 100.8 394 103.4 394 n Critical Values for Correlation Coefficient 3 0.997 4 0.950 5 0.878 6 0.811 7 0.754 8 0.707 9 0.666 10 0.632 11 0.602 12 0.576 13 0.553 14 0.532 15 0.514 16 0.497 17 0.482 18 0.468 19 0.456 20 0.444 21 0.433 22 0.423 23 0.413 24 0.404 25 0.396 26 0.388 27 0.381 28 0.374 29 0.367 30 0.361      
(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.

Sample Solution

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

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