Q. 3.108 - Elementary Statistics

Question

A quantitative data set has mean$10$ and standard deviation $3$.

Fill in the following blanks:

a. At least $75\%$ of the observations lie between ____ and ____.

b. At least ____$\%$ of the observations lie between $1$ and $19$.

Step-by-Step Solution

Answer

a. At least $75\%$of the observations lie between $4$ and $16$. 

b. At least $88.89\%$ of the observations lie between $1$ and $19$.

Part(a) Step 1: Given information

We have been given that the data set has mean,  $\hat{x}=10$ and standard deviation, $s=3$.

We need to find out upper observation limit and lower observation limit.

Part(a) 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 $\hat{x}+ks$.

By Chebyshev's rule,

$$\begin{gathered} 1-\frac{1}{k^2}=75\% \\ 1-\frac{1}{k^2}=\frac{75}{100} \\ 1-\frac{1}{k^2}=\frac{3}{4} \\ \frac{1}{k^2}=1-\frac{3}{4} \\ \frac{1}{k^2}=\frac{1}{4} \\ {k}^2=4 \\ k=2 \end{gathered}$$

Neglecting negative value as $k$ should be greater than $1$.

Now,

Upper limit$=\hat{x}+k\left(s\right)=10+2\times 3=16$ 

Lower limit $=\hat{x}-k\left(s\right)=10-2\times 3=4$

Part(b) Step 1: Given information

We have been given that the data set has mean, $\hat{x}=10$ and standard deviation, $s=3$..

We need to find out what percent of the data set lies between $1$ and $19$.

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 $\hat{x}+ks$.

By  Chebyshev's rule,

Lower limit $=\overbrace{x}-k\left(s\right)$

$$\begin{gathered} 1=10-k\times 3 \\ 3k=9 \\ k=3 \end{gathered}$$

Percentage$=1-\frac{1}{k^2}=1-\frac{1}{9}=\frac{8}{9}=88.89\%$