Question

Joint Strike Fighter program. Refer to the Air & Space Power Journal (March-April 2014) study of the Joint Strike Fighter program, Exercise 11.22 (p. 655). You fit the simple linear regression model relating y = estimated annual cost to x = year of initial aircraft operation.

a. Find s, the estimated standard deviation of the random error term, for the model.

b. Give a practical interpretation of s.

Short answer

1. The estimated standard error (s) is 20.2641.
2. It is preferable if it is small.

Step-by-step solution

Introduction

The highly reported type of standard error is the standard error of the mean (SE or SEM). Other statistics, such as medians and proportions, can also be found using the standard error. A metric that evaluates the difference between a population parameter and a sample statistic is the standard error.

Finding s 

$$\begin{gathered} S{S}_{xy}=\sum x_i{y}_i-\frac{\sum x_i\sum y_i}{n} \\ =9,691-\frac{\left(212\times 473\right)}{12} \\ =9,691-\frac{1,00,276}{12} \\ =\frac{1,16,292-1,00,276}{12} \\ =\frac{16,016}{12} \\ =1,334.67 \end{gathered}$$

$$\begin{gathered} S{S}_{xx}=\sum {x_i}^2-\frac{{\left(\sum x_i\right)}^2}{n} \\ =4,164-\frac{{\left(212\right)}^2}{12} \\ =4,164-\frac{44,944}{12} \\ =\frac{49,968-44,944}{12} \\ =\frac{5,024}{12} \\ =418.67 \end{gathered}$$

$$\begin{gathered} \widehat{{\beta }_1}=\frac{S{S}_{xy}}{S{S}_{xx}} \\ =\frac{1334.67}{418.67} \\ =3.2117 \end{gathered}$$

$$\begin{gathered} S{S}_{yy}=\sum y^2-\frac{{\left(\sum y\right)}^2}{n} \\ =27,037-\frac{{\left(473\right)}^2}{12} \\ =27,037-\frac{2,23,729}{12} \\ =\frac{3,24,444-2,23,729}{12} \\ =\frac{1,00,715}{12} \\ =8,392.9167 \end{gathered}$$

$$\begin{gathered} SSE=S{S}_{yy}-\widehat{{\beta }_1}S{S}_{xy} \\ =8392.9167-(3.2117)(1,334.67) \\ =8392.9167-4,286.5596 \\ =4,106.3571 \end{gathered}$$

$$\begin{gathered} s^2=\frac{SSE}{n-2} \\ =\frac{4,106.3571}{12-2} \\ =\frac{4,106.3571}{10} \\ =410.6357 \end{gathered}$$

$$\begin{gathered} s=\sqrt{s^2} \\ =\sqrt{410.6357} \\ =20.2641 \end{gathered}$$

Therefore, the value of s is 20.2641.

Providing a practical interpretation of s 

The variance in the projected values is shown by a practical interpretation of s. It is preferable if it is small.