Q. 3.147 - Elementary Statistics

Question

A data set consists of $2m^2-1$ zeros, one $-m$, and one $m$.
a. Compute $\tilde{x}$ and $s$ for this data set.
b. How many standard deviations from the mean is the observation m ?
c. Assuming that $m\ge 4$, what percentage of the observations lie within three standard deviations to either side of the mean?

Step-by-Step Solution

Answer

a. $\stackrel{-}{x}=0 and s=1$

b. The observation m is considering as the m standard deviation from the mean. 

c.The required percentage is $\left(\frac{2m^2-1}{2m^2+1}\right)\times 100$

Part (a). Given information

Given that A data set consists of 2m² -1 zeros, one -m, and one m.

Part (a). compute the value of x - and s

$$\begin{gathered} \stackrel{-}{x}=\frac{\sum x_i}{n} \\ =\frac{\left(2m^2-1\right)\left(0\right)+\left(-m\right)+\left(m\right)}{2m^2-1+2} \\ =\frac{0}{2m^2+1} \\ =0 \end{gathered}$$

The value of $\stackrel{-}{x}$

Compute the value of s.

The formula for sample standard deviation is given below

$s=\sqrt{\frac{\sum {\left(x_i-{\displaystyle \stackrel{-}{x}}\right)}^2}{n-1}}$

Obtain $\sum {\left(x_i-\stackrel{-}{x}\right)}^2$

Substitute $2m^2$ for $\sum {\left(x-\stackrel{-}{x}\right)}^2$ and $2m^2+1$ for n in the sample standard deviation formula

$$\begin{gathered} s=\sqrt{\frac{2m^2}{(2m^2+1)-1}} \\ =\sqrt{\frac{2m^2}{2m^2}} \\ =1 \end{gathered}$$

Therefore,the value of s is 1

Part (b). Given information

Given that a data set consists of 2m² -1 zeros, one -m, and one m.

Part (b) Step 2. How many standard deviations from the mean is the observation m ?

Find the number of standard deviation from the mean.
From part (a), it is clear that the sample standard deviation is 1 and the sample mean is 0 . The number 5 is considering as the 5 standard deviation from the mean. Here, the observation is m. hence, the observation m is considering as the m standard deviation from the mean. 

Part (c) Step 1. Given information

Given that A data set consists of 2m² -1 zeros, one -m, and one m.

Part (c) Step 2. Assuming that m ≥ 4 , what percentage of the observations lie within three standard deviations to either side of the mean?

Find the percentage of the observations lie within three standard deviations to either side of the mean when $m\ge 4$.
The values of m and -m would be at least four standard deviations from the mean when $m\ge 4$. The remaining observations of the data set are zero, which is same as the sample mean. Thus, the observations lie within three standard deviations from the mean except two observations is zero. Therefore, the required percentage is $\left(\frac{2m^2-1}{2m^2+1}\right)\times 100$