91Ó°ÊÓ

Measurements are mentioned below:

$\begin{array}{l}x:5.1,4.9,5.0,5.2,4.9,5.0,4.8,5.1\\ y:1.03,1.05,0.96,1.00,1.02,0.95,0.99,1.01,1.00,0.99\end{array}$

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

Calculate expected value.

$\begin{array}{c}\stackrel{¯}{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{∑}}{x}_i}}{n}\\ =\frac{2×5.1+2×4.9+2×5.0+5.2+4.8}{8}\\ =5\end{array}$

Calculate variance.

$\begin{array}{c}\mathrm{Var}\left(x\right)=\frac{1}{n_x−1}\underset{i=1}{\overset{n}{∑}}{(x_i−\stackrel{¯}{x})}^2\\ =\frac{1}{7}[4×{\left(0.1\right)}^2+2×{\left(0.2\right)}^2]\\ =\frac{3}{175}\end{array}$

Calculate standard deviation.

$\begin{array}{c}{\mathrm{σ}}_x=\sqrt{\mathrm{Var}\left(x\right)}\\ =\sqrt{\frac{3}{175}}\\ =0.13\end{array}$

Calculate error of values.

$\begin{array}{c}{\mathrm{σ}}_{mx}=\sqrt{\frac{\mathrm{Var}\left(x\right)}{n}}\\ =\sqrt{\frac{3}{175×8}}\text{ â¶Ä‰}\\ =0.046\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_x=0.6745\left(0.046\right)\\ =0.031\end{array}$

Calculate expected value.

$\begin{array}{c}\stackrel{¯}{y}=\frac{{\displaystyle \underset{i=1}{\overset{n}{∑}}{y}_i}}{n}\\ =\frac{2×1.00+2×0.99+1.03+1.05+1.02+0.95+1.01+0.96}{10}\\ =1\end{array}$

Calculate variance.

$\begin{array}{c}{\mathrm{σ}}_y^2=\frac{1}{n_y−1}\underset{i=1}{\overset{n}{∑}}{(y_i−\stackrel{¯}{y})}^2\\ =\frac{1}{9}[{\left(0.03\right)}^2+2{\left(0.05\right)}^2+3{\left(0.01\right)}^2+{\left(0.02\right)}^2+{\left(0.04\right)}^2]\\ =\frac{41}{45000}\end{array}$

Calculate standard deviation.

$\begin{array}{c}{\mathrm{σ}}_y=\sqrt{\mathrm{Var}\left(y\right)}\\ =\sqrt{\frac{41}{45000}}\\ =0.03\end{array}$

Calculate error of values.

$\begin{array}{c}{\mathrm{σ}}_{my}=\sqrt{\frac{\mathrm{Var}\left(y\right)}{n_y}}\\ =\sqrt{\frac{41}{45000×10}}\\ =0.0095\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_y=0.6745\left(0.0095\right)\\ =0.0064\end{array}$

Calculate expected value.

$\begin{array}{c}\stackrel{¯}{w}=\stackrel{¯}{x+y}\\ =\stackrel{¯}{x}+\stackrel{¯}{y}\\ =5+1\\ =6\end{array}$

Calculate error of values.

$\begin{array}{c}{\mathrm{σ}}_{mw}=\sqrt{{\mathrm{σ}}_{my}^2+{\mathrm{σ}}_{mw}^2}\\ =\sqrt{{\left(0.0095\right)}^2+{\left(0.046\right)}^2}\\ =0.047\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_{\text{w}}=0.6745\left(0.047\right)\\ =0.0317\end{array}$

Calculate expected value.

$\begin{array}{c}w=xy\\ E\left(w\right)={\mathrm{μ}}_x{\mathrm{μ}}_y\\ =5×1\\ =5\end{array}$

Calculate error of values.

$\begin{array}{c}{\mathrm{σ}}_{mw}={\left[\sqrt{{\left(\frac{∂w}{∂x}\right)}^2{\mathrm{σ}}_{mx}^2+{\left(\frac{∂w}{∂y}\right)}^2{\mathrm{σ}}_{my}^2}\right]}_{(x,y)=(x,y)}\\ {\mathrm{σ}}_{mw}=\sqrt{{\left(1\right)}^2{\left(0.046\right)}^2+{\left(5\right)}^2{\left(0.0095\right)}^2}\\ =0.066\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_{\text{w}}=\left(0.6745\right)\left(0.066\right)\\ =0.044\end{array}$

Calculate expected value.

$\begin{array}{c}f(x,y)=x^3\sin \left(y\right)\\ E\left(f\right)={\mathrm{μ}}_x^3\sin \left({\mathrm{μ}}_y\right)\\ =5^3\sin \left(1\right)\\ =105\end{array}$

Calculate error of values.

$\begin{array}{c}{\mathrm{σ}}_{mf}=\sqrt{{\left[3x^2\sin \left(y\right)\right]}^2{\mathrm{σ}}_{\mathrm{me}}^2+{\left[x^3\cos \left(y\right)\right]}^2{\mathrm{σ}}_{\text{my}}^2}\\ =\sqrt{{[3×25\sin (1\left)\right]}^2{\left[0.046\right]}^2+{[125×\cos (1\left)\right]}^2{\left(0.0995\right)}^2}\\ =3\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_f=\left(3\right)\left(0.6745\right)\\ =2\end{array}$

Calculate expected value.

$\begin{array}{c}E\left(z\right)=\ln \left({\mathrm{μ}}_s\right)\\ =\ln \left(5\right)\\ =1.609\end{array}$

Calculate error of values.

$\begin{array}{c}{\mathrm{σ}}_{ms}=\sqrt{\frac{1}{25}×{\left(0.046\right)}^2}\\ =\frac{23}{2500}\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_z=0.6745\left(\frac{23}{2500}\right)\\ =0.006\end{array}$

Hence, required expressions are,

$\begin{array}{c}\stackrel{¯}{x}=5,\\ r_x=0.031\\ \stackrel{¯}{y}=1,\\ r_y=0.0064\end{array}$

$\begin{array}{c}\stackrel{¯}{w}=6,\\ r_w=0.0317\\ \stackrel{¯}{xy}=5,\\ r_{xy}=0.044\end{array}$

$\begin{array}{c}\stackrel{¯}{x\sin \left(y\right)}=105,\\ r=2\\ \stackrel{¯}{\ln x}=1.609,\\ r=0.006\end{array}$