Question

Repair and replacement costs of water pipes. Refer to the IHS Journal of Hydraulic Engineering (September 2012) study of water pipes, Exercise 11.21 (p. 655). Refer, again, to the Minitab simple linear regression printout (p. 655) relating y = the ratio of repair to replacement cost of commercial pipe to x = the diameter (in millimeters) of the pipe.

a. Locate the value of s on the printout.

b. Give a practical interpretation of s.

Short answer

1. S is 14.7849.
2. It is preferable if it is small.

Step-by-step solution

Introduction

The standard error of the mean is the most commonly reported kind of standard error (SE or SEM). The standard error can also be used to find other statistics, such as medians or proportions. The standard error is a metric that measures the difference between a population parameter and a sample statistic.

Find s

\(\begin{aligned}S{S_{xy}} &= \sum {{x_i}} {y_i} - \frac{{\sum {{x_i}} \sum {{y_i}} }}{n}\\ &= 36,849.15 - \frac{{4205 x 106.94}}{{13}}\\ &= 36,849.15 - \frac{{449682.5}}{{13}}\\ &= \frac{{479038.95 - 34590.98}}{{13}}\end{aligned}\)

\(\begin{aligned} &= \frac{{444447.97}}{{13}}\\ &= 34188.30\end{aligned}\)

\(\begin{aligned} S{S_{xx}} &= {\sum {{x_i}} ^2} - \frac{{{{\left( {\sum {{x_i}} } \right)}^2}}}{n}\\ &= 1832025 - \frac{{{{\left( {4205} \right)}^2}}}{{13}}\\ &= 1832025 - \frac{{17682025}}{{13}}\\ &= \frac{{23816325 - 17682025}}{{13}}\end{aligned}\)

\(\begin{aligned} &= \frac{{6134300}}{{13}}\\ &= 471869.23\end{aligned}\)

\(\begin{aligned}\widehat {{\beta _1}} &= \frac{{S{S_{xy}}}}{{S{S_{xx}}}}\\ &= \frac{{34188.30}}{{471869.23}}\\ &= 0.07\end{aligned}\)

\(\begin{aligned} S{S_{yy}} &= \sum {y^2} - \frac{{{{\left( {\sum y} \right)}^2}}}{n}\\ &= 891.049 - \frac{{{{\left( {106.94} \right)}^2}}}{{13}}\\ &= 891.049 - \frac{{11436.1636}}{{13}}\\ &= \frac{{11583.637 - 11436.1636}}{{13}}\end{aligned}\)

\(\begin{aligned} &= \frac{{147.4734}}{{13}}\\ &= 11.3441\end{aligned}\)

\(\begin{aligned}SSE &= S{S_{yy}} - \widehat {{\beta _1}}S{S_{xy}}\\ &= 11.3441 - (0.07)(34188.30)\\ &= 11.3441 + 2393.181\\ &= 2404.5251\end{aligned}\)

\(\begin{aligned} {s^2} &= \frac{{SSE}}{{n - 2}}\\ &= \frac{{2404.5251}}{{13 - 2}}\\ &= \frac{{2404.5251}}{{11}}\\ &= 218.5932\end{aligned}\)

\(\begin{aligned}s &= \sqrt {{s^2}} \\ &= \sqrt {218.5932} \\ &= 14.7849\end{aligned}\)

Therefore, the value of s is 14.7849.

Provide a practical interpretation of s

A practical interpretation of s shows the variation in predicted values. It is preferable if it is small.