Question

interpolating data
heather is training for a long - distance run. her data points listed below represent the days of practice, x, and the number of miles run, y.
(1, 2.5), (2, 4.2), (4, 5.6), (6, 7), (8, 8.1), (10, 11)
use the equation to interpolate the value and estimate the distance that she could have run on day 3. round to the nearest tenth of a mile.
day 3 ≈ miles

Explanation:

Step1: Find the equation of the line of best fit

Using a calculator or software to perform linear regression on the data points \((1,2.5),(2,4.2),(4,5.6),(6,7),(8,8.1),(10,11)\). The linear regression formula is \(y = ax + b\). After calculation (using the formula \(a=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\) and \(b=\frac{\sum y - a\sum x}{n}\) where \(n = 6\)), we get \(y=0.9x + 1.6\)

Step2: Substitute \(x = 3\) into the equation

Substitute \(x = 3\) into \(y=0.9x + 1.6\). Then \(y=0.9\times3+1.6\)

$$y=2.7 + 1.6=4.3$$

Answer:

\(4.3\)