91Ó°ÊÓ

We are given that sample has $33$observations.

Mean $\overline{x}=11.9$

Standard deviation $s=6.5$.

We have to find number of observations when$k=2$

As we know,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 observations lies within $k$standard deviation to either side of mean ,that is , between $\overline{x}-k\left(s\right)$and $\overline{x}+k\left(s\right)$

Now, according to given data

$$\begin{gathered} \overline{x}-k\left(s\right)=11.9-2\left(6.5\right)=-1.1 \\ \overline{x}+k\left(s\right)=11.9+2\left(6.5\right)=24.9 \\ 1-\frac{1}{k^2}=1-\frac{1}{2^2}=0.75=75\% \end{gathered}$$

Observations that are two standard deviations to either side of the mean, according to Chebyshev's rule are $75\%$
However, based on the table, we can deduce that only two observations do not fall within the prescribed interval, implying that the observations are those that are within two standard deviations of the mean.

We are given that sample has $33$observations.

Mean$\overline{x}=11.9$

Standard deviation $s=6.5$.

We have to find number of observations when$k=3$

As we know,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 observations lies within $k$standard deviation to either side of mean ,that is , between $-7.6$and $31.4$

Now, according to given data

$$\begin{gathered} \overline{x}-k\left(s\right)=11.9-3\left(6.5\right)=-7.6 \\ \overline{x}+k\left(s\right)=11.9+3\left(6.5\right)=31.4 \\ 1-\frac{1}{k^2}=1-\frac{1}{3^2}≈89\% \end{gathered}$$

Observations that are three standard deviations to either side of the mean, according to Chebyshev's rule are $75\%$.

However, we may deduce from the table that all observations fall inside the provided interval, implying that all observations are within two standard deviations of the mean.

We are given percentage of observations when $k=2$ and $k=3$

Chebyshev's rule only specifies a minimum for the percentage of observations that lie within a specific number of standard deviations to either side of the mean from provided data, implying that the actual percentage will usually be higher.