Chapter 4, Problem 4.EE

(a)

Interpretation Introduction

Interpretation:

The mean, standard deviation and standard uncertainty for each of the given analysis has to be calculated.

Concept Introduction:

Mean:

The Mean or the arithmetic mean ( $\overline{x}$) is given as the summation of measured values divided by the number of measurements (n). Mean is also called as average.

$\overline{x}=\frac{{\displaystyle \sum _i{x}_i}}{n}$

Standard deviation:

Standard deviation measures how closely the data are clustered about the mean.

The standard deviation (s) is given by the formula:

$s=\sqrt{\frac{{\displaystyle \sum _i{(x_i-\overline{x})}^2}}{n-1}}$

Standard deviation of the mean: (Standard uncertainty)

Standard deviation of the mean of sets of n values is:

${\sigma }_n=\frac{\sigma }{\sqrt{n}}$

To Compute: The mean and standard deviation of a given column of numbers in two different ways using spreadsheet.

(b)

Interpretation Introduction

Interpretation:

Whether the standard deviation differ significantly at the $95\%$ confidence level has to be given.

Concept Introduction:

Comparison of Standard Deviation with F Test:

For the comparison of mean values of two sets of measurements, we decide whether the standard deviations of the two sets are “statistically different”. This is done by F test.

The F test with quotient F is given as:

${\text{F}}_{\text{calculated}}=\frac{s_1^2}{s_2^2}$

Where, $s_1$ and $s_2$ are standard deviations for the set of measurements using original instrument and substitute instrument.

If ${\text{F}}_{\text{calculated}}>{\text{F}}_{\text{table}}$ , then the difference is significant.

Degrees of Freedom:

For a n set of measurement, the degrees of freedom will be $(n-1)$

To Give: Whether the standard deviation differ significantly at the $95\%$ confidence level