Uncertainty Analysis

Q.

The correlation coefficient between $x$and $y$ from the following data $\sum x=40$, $\sum y=50$, $\sum xy=220$, $\sum x^2=200$, $\sum y^2=262$, $n=10$is

1. $0.89$

2. $0.76$

3. $0.91$

4. $0.98$

Q. A resistor has a nominal value of $10\Omega \pm 0.1\%.$ A voltage is applied across the resistor and the power consumed in the resistor is calculated as
$P=EI$
If the measured values of E and I are:
$E=100V\pm 1\%andI=10A\pm 1\%$, then the uncertainity in the power determination is $\pm$ ____ %.

1. 1.414
2. 1
3. 0.414
4. 2.414

Q. Two resistors $R_1$ and $R_2$ are connected in series. The value of resistance are
$R_1=100.0\pm 0.1\Omega ,R_2=50\pm 0.03\Omega$
If the error in $R_{1}or{R}_2$ has to be considered as standard deviation then the standard deviation in series equivalent resistance is

1. $\pm 0.13\Omega$
2. $\pm 0.10\Omega$
3. $\pm 0.20\Omega$
4. $\pm 0.15\Omega$

Q.

What is variance in machine learning?

Q. A large number of $230\Omega$ resistors are obtained by combining $120\Omega$ resistors with a standard deviation of $4.0\Omega$ and $110\Omega$ resistors with a standard deviation of $3.0\Omega$. The standard deviation of the $230\Omega$ resistors thus formed will be

1. $3.5\Omega$
2. $5.0\Omega$
3. $3.0\Omega$
4. $12.0\Omega$

Q. A voltage $V_1$ is measured 100 times and its average and standard deviation are 100 V and 1.5 V respectively. A second voltage $V_2$, which is independent of $V_1,$ is measured 200 times and its average and standard deviation are 150 V and 2 V respectively. $V_3$ is computed as: $V_3=V_1+V_2.$ Then the standard deviation of $V_3$ in volt is

1. 2.5

Q. Two ammeters $A_1$ and $A_2$ measure the same current and provide readings $I_{1}and{I}_2,$ respectively. The ammeter errors can be characterized as independent zero mean Gaussian random variable of standard deviations ${\sigma }_{1}and{\sigma }_2,$ respectively. The value of the current is computed as:

$I=\mu I_1+(1-\mu )I_2$

The value of $\mu$ which gives the lowest standard deviation of I is

1. $\frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$
2. $\frac{{\sigma }_1^2}{{\sigma }_1^2+{\sigma }_2^2}$
3. $\frac{{\sigma }_2}{{\sigma }_1+{\sigma }_2}$
4. $\frac{{\sigma }_1}{{\sigma }_1+{\sigma }_2}$

Q. Two resistors with nominal resistance values $R_1$ $R_2$ have additive uncertainties $\Delta R_1$ and $\Delta R_2$ respectively. When these resistances are connected in parallel, the standard deviatioin of the error in the equivalent resistance $R$ is

1. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\Delta R_1\right\}}^2+{\left\{\frac{\partial R}{\partial R_2}\Delta R_2\right\}}^2}$
2. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_2}\Delta R_1\right\}}^2+{\left\{\frac{\partial R}{\partial R_1}\Delta R_2\right\}}^2}$
3. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_2+{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_1}$
4. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_1+{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_2}$

Q. The measurements of a source voltage are 5.9 V, 5.7 V and 6.1 V. The sample standard deviation of the reading is

1. 0.013
2. 0.04
3. 0.115
4. 0.2

Q. The volume of a cylinder is computed from measurements of its height (h) and diameter (d). A set of several measurements of height has an average value of 0.2 m and a standard deviation of 1%. The average value obtained for the diameter is 0.1 m and the standard deviation is 1%. Assuming the errors in the measurements of height and diameter are uncorrelated, the standard deviation of the computed volume is

1. 1.00%
2. 1.73%
3. 2.23%
4. 2.41%

Q. Two sensors have measurement errors that are Gaussian distributed with zero means and variances ${\sigma }_1^{2}and{\sigma }_2^2,$ respectively. The two sensor measurements $x_{1}and{x}_2$ are combined to form the weighted average $x=\alpha x_1+(1-\alpha )x_2,0\le \alpha \le 1.$ Assuming that the measurement errors of the two sensors are uncorrelated, the weighting factor $\alpha$ that yields the smallest error variance of x is

1. $\frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$
2. $\frac{{\sigma }_1^2}{{\sigma }_1^2+{\sigma }_2^2}$
3. $\frac{{\sigma }^2}{{\sigma }_1+{\sigma }_2}$
4. 0.5