Questions: Find the equation of the least-squares line for the stride length and speed of camels given in the table below. Stride Length (m) 2.5 3.0 3.2 3.4 3.5 3.8 4.0 4.2 Speed (m / s) 2.3 3.9 4.1 5.0 5.5 6.2 7.1 7.6 Use the equation of the least-squares line to predict the average speed (in meters per second) of a camel with a stride length of 3.5 meters. Round your results to the nearest tenth of a meter per second.

Solution Steps

The means of the stride length \( \bar{x} \) and speed \( \bar{y} \) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 3.45 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 5.2125 \]

The correlation coefficient \( r \) is found to be:

\[ r = 0.9959 \]

The numerator for the slope \( \beta \) is calculated as:

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 150.7 - 8 \cdot 3.45 \cdot 5.2125 = 6.835 \]

The denominator for \( \beta \) is:

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 97.38 - 8 \cdot 3.45^2 = 2.16 \]

Thus, the slope \( \beta \) is:

\[ \beta = \frac{6.835}{2.16} = 3.1644 \]

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 5.2125 - 3.1644 \cdot 3.45 = -5.7045 \]

The equation of the least-squares line is given by:

\[ y = -5.7045 + 3.1644x \]

To predict the average speed for a camel with a stride length of \( 3.5 \) meters, we substitute \( x = 3.5 \) into the equation:

\[ y = -5.7045 + 3.1644 \cdot 3.5 = 5.4 \]

Final Answer