Q. 3.130 - Elementary Statistics

Question

The data set has $200$ observations and has mean $20$ and standard deviation $4$. Approximately how many observations lie between $16$and $24$.

Step-by-Step Solution

Answer

Zero observations lie between $16$and $24$.

Step 1 : Given information

We are given that the data has $200$observations.

Mean $\overline{x}=20$

Standard deviation $s=4$

We have to find how many observations lies between $16$and $24$

Step 2 : Simplify

According to Chebyshev's Rule for any quantitative data set and any real number $k$ greater than or equal to $1$ , at least $1-\frac{1}{k^2}$ of the observations lie within $k$ standard deviations to either side of the mean, that is, between $\overline{x}-k\left(s\right)$ and $\overline{x}+k\left(s\right)$.

Now, 

$$\begin{gathered} \overline{x}-k\left(s\right)=16 \\ 20-k\left(4\right)=16 \\ k=1 \\ \overline{x}+k\left(s\right)=24 \\ 20+k\left(4\right)=24 \\ k=1 \end{gathered}$$

Therefore, number of observations between $16$ and $24$will be $1-\frac{1}{k^2}=1-\frac{1}{1}=0$