Chapter 4, Problem 4.AE

Interpretation Introduction

Interpretation:

The mean, standard deviation, standard uncertainty, range and $90\%$ confidence interval for the mean has to be calculated.

Also whether the number $97.9$ has to be discarded or retained has to be decided from the Grubbs test.

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 deviation of the mean of sets of n values is:

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

Range:

The difference between the highest value and the lowest value in a set of data gives the range.

Confidence Intervals:

The confidence interval is given by the equation:

$\begin{array}{l}\text{Confidence interval = }\overline{x}\pm \frac{ts}{\sqrt{n}}\\ =\overline{x}\pm t{u}_x(\text{since standard uncertainty(}{u}_x)=s/\sqrt{n}\text{) }\end{array}$

Where,

$\overline{x}$ is mean

$s$ is standard deviation

$t$ is Student’s t

n is number of measurements

$u_x$ is standard uncertainty

Grubbs Test for an Outlier:

A datum that is far from other points is known as an outlier. The consideration or rejection of that datum from the entire points is found through Grubbs test.

The Grubbs statistic G, is given as:

${\text{G}}_{\text{calculated}}=\frac{\left|\text{questionable value}-\overline{x}\right|}{s}$

Where,

$\overline{x}$ is the mean of entire data set

$s$ is the standard deviation of entire data set

If ${\text{G}}_{\text{calculated}}>{\text{G}}_{\text{table}}$, then the questionable point should be discarded.

If ${\text{G}}_{\text{calculated}}{\text{< G}}_{\text{table}}$, then the questionable point should be retained.

Answer to Problem 4.AE

The mean is $108.6$

The standard deviation is $7.1$

The standard uncertainty is $3.1$

The range is $18.1$

The $90\%$ confidence interval for the mean is $6.8$

Using Grubbs test, the number $97.9$ should be retained.

Explanation of Solution

Given data:

The given numbers are $116.0,\text{ 97}\text{.9, 114}\text{.2, 106}\text{.8 and 108}\text{.3}$

To Calculate: The mean, standard deviation, standard uncertainty, range and $90\%$ confidence interval for the mean

Mean:

The mean is calculated as follows,

$\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum _i{x}_i}}{n}\\ =\frac{116.0+97.9+114.2+106.8+108.3}{5}\\ =108.64\\ =108.6(\text{rounded off)}\end{array}$

Standard deviation:

The standard deviation is calculated as follows,

$\begin{array}{l}s=\sqrt{\frac{{\displaystyle \sum _i{(x_i-\overline{x})}^2}}{n-1}}\\ =\sqrt{\frac{{(108.6-116.0)}^2+\mathrm{..............}+{(108.6-108.3)}^2}{(5-1)}}\\ =7.14\\ =7.1(\text{rounded off)}\end{array}$

Standard uncertainty:

The standard uncertainty is calculated as follows,

$\begin{array}{l}{\sigma }_n=\frac{\sigma }{\sqrt{n}}\\ =\frac{7.14}{\sqrt{5}}\\ =3.19\\ =3.1(\text{rounded off)}\end{array}$

Range:

The range is calculated as follows,

$\begin{array}{l}\text{Range}\text{=116}\text{.0 }-\text{97}\text{.9}\\ =18.1\end{array}$

$90\%$ Confidence interval:

The $90\%$ confidence interval is calculated as follows,

$\begin{array}{l}\text{Confidence interval = }\overline{x}\pm \frac{ts}{\sqrt{n}}\\ =108.64\pm \frac{(2.132)(7.14)}{\sqrt{5}}\\ \text{ =}108.64\pm 6.81\\ \text{ =}108.6\pm 6.8(\text{rounded off)}\end{array}$

To Decide: Whether the number $97.9$ has to be discarded or retained

Grubbs Test:

The questionable number $97.9$

$\begin{array}{l}{\text{G}}_{\text{calculated}}=\frac{\left|\text{questionable value}-\overline{x}\right|}{s}\\ =\frac{\left|97.9-108.64\right|}{7.14}\\ =1.50\end{array}$

The ${\text{G}}_{\text{table}}$ value for the five number of measurements is $1.672$

${\text{G}}_{\text{calculate}}<{\text{G}}_{\text{table}}$

The questionable value has to be retained.

Conclusion

The mean is calculated as $108.6$

The standard deviation is calculated as $7.1$

The standard uncertainty is calculated as $3.1$

The range is calculated as $18.1$

The $90\%$ confidence interval for the mean is calculated as $6.8$

Using Grubbs test, the number $97.9$ is found out to be retained.