Question

1. a bathtub factory makes a certain size and style of bathtub and the monthly demand for that model of tub at various prices is noted below. price in $ orders per month 275 130 285 116 300 104 325 92 350 80 375 70 400 62 415 52 rounded to the nearest hundredth, what is the equation of the line of best fit?

Explanation:

Step1: Input data into calculator

Let \(x\) be the price and \(y\) be the orders per month. Input the pairs \((275,130)\), \((285,116)\), \((300,104)\), \((325,92)\), \((350,80)\), \((375,70)\), \((400,62)\), \((415,52)\) into a graphing calculator or statistical software.

Step2: Calculate regression equation

Using the linear regression function (usually labeled as \(LinReg\) or \(ax + b\) in calculators), we find the slope \(a\) and the \(y\)-intercept \(b\).
The formula for the slope \(a=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\) and \(b=\frac{\sum y - a\sum x}{n}\), where \(n = 8\) (the number of data points).
After calculation (using calculator - based computation for simplicity as manual calculation of these sums \(\sum x\), \(\sum y\), \(\sum(xy)\), \(\sum(x^{2})\) is very time - consuming):
\(a\approx - 0.31\) and \(b\approx216.57\)

Answer:

\(y=-0.31x + 216.57\)