Chapter B.2, Problem 39E

To determine

To find: The mean, standard deviation and variance using the provided data using the graphing utility. Also, compute the percentage of data that is falling within 2 standard deviation of the mean.

Answer to Problem 39E

The mean, standard deviation and variance are 613.4583, 437.780 and 191651.3284 respectively.

Explanation of Solution

Given:

The data set is:

384
362
344
360
384
384
388
331
294
279
279
271
310
363
410
445
603
695
872
972
1225
1568
1669
1531

Calculation:

1. Turn on the calculator.
2. Click on STAT > Edit > Enter.
3. Enter the data of x in L1 and data of y in L2.
4. Click om STAT > CALC > 1-Var Stats > ENTER.
5. Hit 2nd and 1 to choose L1.
6. Hit Enter.

The obtained output is:

  

From the obtained output, the mean is 613.4583, the standard deviation is 437.780.

Using the standard deviation, the variance could be calculated as:

  $\begin{array}{c}{\sigma }_x{}^2={\left({\sigma }_x\right)}^2\\ ={\left(437.780\right)}^2\\ =191651.3284\end{array}$

Thus, the variance is 191651.3284.

The interval which is containing the data that is falling within 2 standard deviation of the mean could be computed as:

  $\begin{array}{c}\text{Interval = Mean}\pm \text{2}\times \text{Standard deviation}\\ =613.4583+2\times 437.7800\\ =\left(262.1017,1489.01\right)\end{array}$

There are 21 data points that lie between the computed interval.

The percentage of data that is falling within 2 standard deviation of the mean could be computed as:

  $\begin{array}{c}\text{Percentage = }\frac{\left(\begin{array}{l}\text{Number of data points that lie }\\ \text{between the computed interval}\end{array}\right)}{\text{Total data points}}\times 100\\ =\frac{21}{24}\times 100\\ =87.5\%\end{array}$

Thus, the required percentage is 87.5%