Chapter 15: Q9P (page 775)

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

Short Answer

Expert verified

Required solutions are,

$\begin{matrix} \bar{x} = 100, \\ r_{x} = 0.477 \\ \bar{y} = 20, \\ r_{y} = 0.23 \end{matrix}$

$\begin{matrix} \bar{w} = 80, \\ r_{w} = 0.35 \\ \bar{x / y} = 5, \\ r_{x y} = 0.063 \end{matrix}$

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

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.

$\begin{matrix} \bar{x} = \frac{\sum_{i = 1}^{n} x_{i}}{n} \\ = \frac{98 + 99 + 100 + 101 + 102}{5} \\ = 100 \end{matrix}$

Calculate variance.

$\begin{matrix} Var (x) = \frac{1}{n_{x} - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} \\ = \frac{1}{4} [2 \times {(2)}^{2} + 2 \times {(1)}^{2}] \\ = 2.5 \end{matrix}$

Calculate error of values.

$\begin{matrix} σ_{m x} = \sqrt{\frac{Var (x)}{n}} \\ = \sqrt{\frac{5}{2 \times 5}} \\ = 0.707 \end{matrix}$

Calculate probable error.

$\begin{matrix} r_{x} = 0.6745 (0.707) \\ = 0.477 \end{matrix}$

Calculate mean and probable error value of.y

Calculate expected value.

$\begin{matrix} \bar{y} = \frac{\sum_{i = 1}^{n} y_{i}}{n} \\ = \frac{21.2 + 20.8 + 18.1 + 20.3 + 19.6 + 20.4 + 19.5 + 20.1}{8} \\ = 20 \end{matrix}$

Calculate variance.

$\begin{matrix} σ_{y}^{2} = \frac{1}{n_{y} - 1} \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2} \\ = \frac{1}{7} [{(1.2)}^{2} + {(0.8)}^{2} + {(1.9)}^{2} + {(0.3)}^{2} + 2 {(0.4)}^{2} + {(0.5)}^{2} + {(0.1)}^{2}] \\ = \frac{159}{175} \end{matrix}$

Calculate error of values.

$\begin{matrix} σ_{m y} = \sqrt{\frac{Var (y)}{n_{y}}} \\ = \sqrt{\frac{159}{175 \times 8}} \\ = 0.345 \end{matrix}$

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

Short Answer

Step by step solution

Given Information

Definition of Probable error.

Calculate mean and probable error value of x.

Calculate mean and probable error value of.y

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Given the measurements.x:98,101,102,100,99y:21.2,20.8,18.1,20.3,19.6,20.4,19.5,20.1Find the mean value and the probable error of x−y, x/y, x2y3and .yln(x)

Short Answer

Step by step solution

Given Information

Definition of Probable error.

Calculate mean and probable error value of x.

Calculate mean and probable error value of.y

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Medical Physics

Linear Momentum

Energy Physics

Electromagnetism

Thermodynamics

Study anywhere. Anytime. Across all devices.

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