Q. 3.131 - Elementary Statistics

Question

The data set has $150$ observations and has mean $35$ and standard deviation $4$. Approximately how many observations lie between $31$ and $39$ ?

Step-by-Step Solution

Answer

Zero observations lies between $31$and $39$.

Step 1 : Given information

We are given that the data has $150$ observations and

Mean $\overline{x}=35$

Standard deviation $s=4$

We have to find number of observations between $31$and $39$

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, according to given data

$$\begin{gathered} \overline{x}-k\left(s\right)=31 \\ 35-k\left(4\right)=31 \\ k=1 \\ \overline{x}+k\left(s\right)=39 \\ 35+k\left(4\right)=39 \\ k=1 \end{gathered}$$

Therefore, number of observations between $31$and $39$ are $1-\frac{1}{k^2}=1-1=0$