Question

Given the measurements.

$\begin{array}{l}\mathbf{x}\mathbf{:}\mathbf{98}\mathbf{,}\mathbf{101}\mathbf{,}\mathbf{102}\mathbf{,}\mathbf{100}\mathbf{,}\mathbf{99}\\ \mathbf{y}\mathbf{:}\mathbf{21}\mathbf{.2}\mathbf{,}\mathbf{20}\mathbf{.8}\mathbf{,}\mathbf{18}\mathbf{.1}\mathbf{,}\mathbf{20}\mathbf{.3}\mathbf{,}\mathbf{19}\mathbf{.6}\mathbf{,}\mathbf{20}\mathbf{.4}\mathbf{,}\mathbf{19}\mathbf{.5}\mathbf{,}\mathbf{20}\mathbf{.1}\end{array}$Find the mean value and the probable error of $\mathit{x}\mathbf{-}\mathit{y}\mathbf{,}\mathit{x}\mathbf{/}\mathit{y}\mathbf{,}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}$and .$\mathit{y}\mathbf{ln}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

Short answer

Required solutions are,

$\begin{array}{c}\overline{x}=100,\\ r_x=0.477\\ \overline{y}=20,\\ r_y=0.23\end{array}$

$\begin{array}{c}\overline{w}=80,\\ r_w=0.35\\ \overline{x/y}=5,\\ r_{xy}=0.063\end{array}$

$\begin{array}{c}\overline{x^2{y}^3}=8.0\times {10}^7,\\ r=2.1\times {10}^6\\ \overline{y\ln x}=92,\\ r=1.07\end{array}$

Step-by-step solution

Given Information

Measurements are mentioned below:

$\begin{array}{l}x:98,101,102,100,99\\ y:21.2,20.8,18.1,20.3,19.6,20.4,19.5,20.1\end{array}$

Definition of Probable error. 

The amount by which a sample's arithmetic mean is predicted to vary solely due to chance.

Calculate mean and probable error value of x. 

Calculate expected value.

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\\ =\frac{98+99+100+101+102}{5}\\ =100\end{array}$

Calculate variance.

$\begin{array}{c}\mathrm{Var}\left(x\right)=\frac{1}{n_x-1}\sum _{i=1}^n{(x_i-\overline{x})}^2\\ =\frac{1}{4}[2\times {\left(2\right)}^2+2\times {\left(1\right)}^2]\\ =2.5\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{mx}=\sqrt{\frac{\mathrm{Var}\left(x\right)}{n}}\\ =\sqrt{\frac{5}{2\times 5}}\\ =0.707\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_x=0.6745\left(0.707\right)\\ =0.477\end{array}$

Calculate mean and probable error value of.y 

Calculate expected value.

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum _{i=1}^n{y}_i}}{n}\\ =\frac{21.2+20.8+18.1+20.3+19.6+20.4+19.5+20.1}{8}\\ =20\end{array}$

Calculate variance.

$\begin{array}{c}{\sigma }_y^2=\frac{1}{n_y-1}\sum _{i=1}^n{(y_i-\overline{y})}^2\\ =\frac{1}{7}[{\left(1.2\right)}^2+{\left(0.8\right)}^2+{\left(1.9\right)}^2+{\left(0.3\right)}^2+2{\left(0.4\right)}^2+{\left(0.5\right)}^2+{\left(0.1\right)}^2]\\ =\frac{159}{175}\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{my}=\sqrt{\frac{\mathrm{Var}\left(y\right)}{n_y}}\\ =\sqrt{\frac{159}{175\times 8}}\\ =0.345\end{array}$