Chapter 2, Problem 11CRE

To determine

To explain: whether the given data is a Normal probability plot.

Answer to Problem 11CRE

The given data does not follow normal distribution.

Explanation of Solution

Given:

  

Calculation:

The histogram and normal probability plot of the given data are as follows:

  

From the histogram, it is seen that the data is more concentrated towards the right side. Hence, it is said that the data is not symmetric. Also, in the normal probability plot, all points are not close to straight line. Hence, it is said that the data is not normally distributed.

The descriptive statistics of the given data of lengths are given as follows:

  $\begin{array}{l}\overline{\begin{array}{ccccccccc}\text{Variable}& \text{ N}& \text{ Mean}& \text{ St}\text{.Dev}\text{.}& \text{Min}& {\text{ Q}}_{\text{1}}& \text{ M}& {\text{ Q}}_{\text{3}}& \text{ Max }\end{array}}\\ \underset{\_}{\begin{array}{ccccccccc}\text{length of thorax}& 49& 0.8004& 0.0782& 0.6400& 0.7600& 0.8000& 0.8600& 0.9400\end{array}}\end{array}$

The descriptive statistics given above indicate that the mean and median are very similar which is consistent with rough symmetry.

From the above descriptive statistics, the mean length of the given data, $\mu =0.8004$ The standard deviation of lengths, $\sigma =0.0782$

According to 68-95-99.7 rule, 68% of observations will be within one-sigma interval, 95% of observations will be within two-sigma limits and 99.7% of observations will be within 3-sigma limits.

First calculate $1-\sigma ,$ limits as follows:

  $\begin{array}{l}\mu \pm \sigma =0.8004\pm 0.0782\\ =\left(0.7222,\text{ 0}\text{.8786}\right)\end{array}$

The given data contains 28 observations which are within 0.7222 and 0.8786.

That is, $\frac{28}{49}=57.14\%$ observations are within $1-\sigma$ limits.

Calculate $2-\sigma$ limits as follows

  $\begin{array}{l}\mu \pm 2\sigma =0.8004\pm 2\times 0.0782\\ =\left(0.644,\text{ 0}\text{.9568}\right)\end{array}$

The given data contains 47 observations which are within 0.644 and 0.9568.

That is, $\frac{47}{49}=95.92\%$ observations are within $2-\sigma$ limits.

Calculate $3-\sigma$ limits as follows

  $\begin{array}{l}\mu \pm 3\sigma =0.8004\pm 3\times 0.0782\\ =\left(0.5658,\text{ 1}\text{.035}\right)\end{array}$

The given data contains all 49 observations which are within 0.5658 and 1.035.

That is, $\frac{49}{49}=100\%$ observations are within $3-\sigma$ limits.

As the above data, does not satisfy the empirical rule, it is concluded the given data does not follow normal distribution.

Conclusion:

Therefore, the given data does not follow normal distribution.