Questions: Wave Length (m),Speed (m/s),100,14.6 125,13.6 130,16.1 175,17 210,21.8 350,24.3 400,24.9 Using the equation from part (a), estimate the speed (in meters per second) of a wave that is 200 m long. (Round your answer to one decimal place.) v = Find the equation of a linear regression line for the data above. (Round your answer to three decimal places.) y = mx + b

Solution Steps

First, we organize the given data into two lists: one for wave lengths \( x \) and one for speeds \( y \).

\[ \begin{align_} x: & \quad 100, 125, 130, 175, 210, 350, 400 \\ y: & \quad 14.6, 13.6, 16.1, 17, 21.8, 24.3, 24.9 \end{align_} \]

To find the equation of the linear regression line \( y = mx + b \), we need to calculate the slope \( m \) and the y-intercept \( b \).

The formulas for \( m \) and \( b \) are:

\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]

\[ b = \frac{(\sum y) - m(\sum x)}{n} \]

where \( n \) is the number of data points.

Calculate the necessary sums:

\[ \begin{align_} \sum x &= 100 + 125 + 130 + 175 + 210 + 350 + 400 = 1490 \\ \sum y &= 14.6 + 13.6 + 16.1 + 17 + 21.8 + 24.3 + 24.9 = 132.3 \\ \sum xy &= (100 \times 14.6) + (125 \times 13.6) + (130 \times 16.1) + (175 \times 17) + (210 \times 21.8) + (350 \times 24.3) + (400 \times 24.9) = 30395 \\ \sum x^2 &= 100^2 + 125^2 + 130^2 + 175^2 + 210^2 + 350^2 + 400^2 = 366750 \end{align_} \]

Substitute these into the formulas for \( m \) and \( b \):

\[ m = \frac{7(30395) - (1490)(132.3)}{7(366750) - (1490)^2} = \frac{212765 - 197187}{2567250 - 2220100} = \frac{15578}{347150} \approx 0.0449 \]

\[ b = \frac{132.3 - 0.0449 \times 1490}{7} = \frac{132.3 - 66.901}{7} = \frac{65.399}{7} \approx 9.3427 \]

Thus, the equation of the linear regression line is:

\[ y = 0.0449x + 9.343 \]

Using the linear regression equation, estimate the speed for a wave length of 200 m:

\[ v = 0.0449 \times 200 + 9.343 = 8.98 + 9.343 = 18.323 \]

Rounding to one decimal place, the estimated speed is 18.3 m/s.

Final Answer