Q. 3.147 - Elementary Statistics

Question

A data set contains $2m^2-1$ zeroes, one $-m$ and one $m$.

a. Compute $\hat{x}$ 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. $\hat{x}=0$ and $s=1$

b. the observation $m$ is $m$ standard deviations away from the mean.

c. According to chebyshev's rule, the required percentage is $88.89\%$.

Part(a) Step 1: Given information

We have been given a data set comprises of $2m^2-1$ zeroes, one $-m$ and one $m$.

We need to find out the mean, that is, $\hat{x}$ and standard deviation, that is, $s$.

Part(a) Step 2: Simplify

Frequency table:-

$x_i$ (Data value)	$f_i$ (Frequency)
$0$	$2m^2-1$
$-m$	$1$
$m$	$1$

Mean,

$$\begin{gathered} \hat{x}=\frac{{\displaystyle \sum _{i=1}^n}{x}_i{f}_i}{{\displaystyle \sum _{i=1}^n}{f}_i} \\ \hat{x}=\frac{0(2m^2-1)+(-m)1+m\times 1}{2m^2-1+1+1}=0 \end{gathered}$$

Standard deviation,

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \sum _{i=1}^n}{f}_i{(x_i-\hat{x})}^2}{{\displaystyle \sum _{i=1}^n}{f}_i-1}} \\ s=\sqrt{\frac{(2m^2-1){\left(0\right)}^2+1{(-m-0)}^2+1{(m-0)}^2}{2m^2-1+1+1-1}} \\ s=\sqrt{\frac{2m^2}{2m^2}}=1 \end{gathered}$$

Part(b) Step 1: Given information

We need to find out how many standard deviations from the mean is the observation $m$.

Part(b) Step 2: Simplify

Chebyshev's rule states that for any quantitative data collection with a real number $k$ higher than or equal to $1$, at least $1-\frac{1}{k^2}$ of the observations are within standard deviations of the mean, that is, $\hat{x}-ks$ and $\hat{x}+ks$.

Therefore,

$\hat{x}+ks=$upper limit

$$\begin{gathered} \hat{x}+ks=m \\ 0+k\left(1\right)=m \\ k=m \end{gathered}$$

As a result, the observation $m$ is $m$ standard deviations away from the mean.

Part(c) Step 1: Given information

We have been given to assume that $m\ge 4$.

We need to find out what percentage of the observations lie within three standard deviations to either side of the mean.

Part(c) Step 2: Simplify

Chebyshev's rule states that for any quantitative data collection with a real number $k$ higher than or equal to $1$, at least $1-\frac{1}{k^2}$ of the observations are within standard deviations of the mean, that is, $\hat{x}-ks$ and $\hat{x}+ks$.

When $m\ge 4$, the values of $m$ and $-m$ are at least four standard deviations from the the mean. The data set's remaining observations are all $0$, which is same as sample mean calculated. Hence, all the observations that are within three standard deviations of the mean are zero.

According to chebyshev's rule, the required percentage is:-

$(1-\frac{1}{k^2})\times 100=(1-\frac{1}{3^2})\times 100=\left(\frac{8}{9}\right)\times 100=88.89\%$