Question

Given the sample 4, −2, −6, 19, 6, add one more sample value that will neither change the mean nor the variance. Round to two decimal places as necessary. If this is not possible, indicate "Cannot create sample".

Answer 1

Answer:

Cannot create sample

Step-by-step explanation:

We are given the following in the question:

4, -2, -6, 19, 6

Formula:

$\text{Variance} = \displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}$  

where $x_i$ are data points, $\bar{x}$ is the mean and n is the number of observations.  

$Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}$

$Mean =\displaystyle\frac{21}{5} = 4.2$

Sum of squares of differences = 0.04 + 38.44 + 104.04 + 219.04 + 3.24 = 364.8

$s^2 = \dfrac{364.8}{4}} = 91.2$

If we add an observation equal to the mean of given sample, then the mean and of new sample will not change.

New sample:

4, -2, -6, 19, 6, 4.2

$Mean =\displaystyle\frac{25.2}{6} = 4.2$

Sum of squares of differences = 0.04 + 38.44 + 104.04 + 219.04 + 3.24 + 0 = 364.8

$s^2 = \dfrac{364.8}{5}} = 72.96$

Thus, the variance of new sample changes.

Thus, it is not possible to create a sample.