An oceanographer measured the length, in meters, of a deepwater wave and its speed, in meters per second. The results are shown in the following table.
Wave Length (m) Speed (m/s)
100 14.4
125 13.6
130 15.9
175 16.8
210 21.7
350 24.4
400 24.8
(a)
Find the equation of a linear regression line for the data where wave length is the independent variable, x, and speed is the dependent variable. (Round your numerical values to two decimal places.)
To find the equation of the linear regression line, we first need to calculate the mean of the wave length and speed values:
Mean of wave length (x):
(100 + 125 + 130 + 175 + 210 + 350 + 400) / 7 = 196.43
Mean of speed (y):
(14.4 + 13.6 + 15.9 + 16.8 + 21.7 + 24.4 + 24.8) / 7 = 18.56
Next, we calculate the sum of the products of the wave length and speed values:
Σxy = (100 * 14.4) + (125 * 13.6) + (130 * 15.9) + (175 * 16.8) + (210 * 21.7) + (350 * 24.4) + (400 * 24.8)
Σxy = 1440 + 1700 + 2067 + 2940 + 4557 + 8530 + 9920
Σxy = 31214
Then, we calculate the sum of the squares of the wave length values:
Σx^2 = 100^2 + 125^2 + 130^2 + 175^2 + 210^2 + 350^2 + 400^2
Σx^2 = 10000 + 15625 + 16900 + 30625 + 44100 + 122500 + 160000
Σx^2 = 388750
Now, we can plug these values into the formula for the slope of the regression line:
slope (m) = (Σxy - (Σx * Σy / n)) / (Σx^2 - (Σx)^2 / n)
slope (m) = (31214 - (196.43 * 18.56 * 7)) / (388750 - (196.43)^2 / 7)
slope (m) = (31214 - (2045.37 * 7)) / (388750 - 38580.95 / 7)
slope (m) = (31214 - 14317.59) / (388750 - 5511.42)
slope (m) = 16896.41 / 383238.58
slope (m) ≈ 0.04412
Next, we calculate the y-intercept (b) of the regression line:
b = Σy / n - m * Σx / n
b = 18.56 - 0.04412 * 196.43
b = 18.56 - 8.67
b ≈ 9.89
Therefore, the equation of the linear regression line is:
speed = 0.04 * wave length + 9.89