Question

1. movie film is contained on a circular reel. the table shows approximately how long a film will run, based on the diameter of the film reel.

regression equation:
use your equation to determine about how long a film would run if its diameter is 15 inches?
film run times (for 16mm film)

Explanation:

Step1: Find the regression equation

Using a calculator or software to perform linear regression on the data points \((x,y)\) where \(x\) is the diameter (in.) and \(y\) is the film run time (min).
Let \(x_1 = 5,y_1=5.5\); \(x_2 = 7,y_2 = 11.12\); \(x_3=9.25,y_3 = 16.67\); \(x_4 = 10.5,y_4=22.22\); \(x_5=12.25,y_5 = 33.33\); \(x_6=13.75,y_6=44.45\)

The formula for the slope \(m\) of the regression line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\overline{y}-m\overline{x}\) where \(n = 6\), \(\overline{x}=\frac{\sum_{i=1}^{n}x_i}{n}\), \(\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\)

\(\sum_{i=1}^{6}x_i=5 + 7+9.25+10.5+12.25+13.75=57.75\)

\(\sum_{i=1}^{6}y_i=5.5 + 11.12+16.67+22.22+33.33+44.45=133.29\)

\(\sum_{i=1}^{6}x_iy_i=(5\times5.5)+(7\times11.12)+(9.25\times16.67)+(10.5\times22.22)+(12.25\times33.33)+(13.75\times44.45)\)
\(=27.5+77.84+154.2+233.31+408.29+611.19=1512.33\)

\(\sum_{i=1}^{6}x_i^{2}=5^{2}+7^{2}+9.25^{2}+10.5^{2}+12.25^{2}+13.75^{2}\)
\(=25 + 49+85.56+110.25+150.06+189.06=598.93\)

\(m=\frac{6\times1512.33-57.75\times133.29}{6\times598.93-(57.75)^{2}}\)
\(=\frac{9073.98 - 7700.2}{3593.58-3335.06}\)
\(=\frac{1373.78}{258.52}\approx5.31\)

\(\overline{x}=\frac{57.75}{6}=9.625\), \(\overline{y}=\frac{133.29}{6}=22.215\)

\(b=22.215-5.31\times9.625=22.215 - 51.12 = - 28.905\)

The regression equation is \(y = 5.31x-28.905\)

Step2: Predict the run - time for \(x = 15\)

Substitute \(x = 15\) into the regression equation \(y=5.31x - 28.905\)

\(y=5.31\times15-28.905\)
\(y = 79.65-28.905\)
\(y=50.745\approx50.75\)

Answer:

The regression equation is \(y = 5.31x-28.905\). A film with a diameter of \(15\) inches would run for approximately \(50.75\) minutes.