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.

Question

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.)

Answer 1

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