Consider the following sample of exam scores, arranged in increasing order.

The sample mean and sample standard deviation of these exam scores are $85$and $16.1$, respectively.
a. Compare the percentage of the observations that actually lie within two standard deviations to either side of the mean with that given by Chebyshev's rule with$k=2$
b. Repeat part (a) with $k=3$
c. Interpret your results from parts (a) and (b).

We are given that sample has $30$observations.

Mean $\overline{x}=85$

Standard deviation $s=16.1$.

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 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

$1-\frac{1}{k^2}=1-\frac{1}{2^2}=0.75=75\%$

$75\%$observations lie within two standard deviation to either side of mean that is between $52.8$and $117.2$.

From given data we can say only one exam score not lies between given interval.

Therefore, actually $96.7\%$ observations lies within two standard deviation to either side of mean

We are given that sample has $30$observations.

Mean$\overline{x}=85$

Standard deviation $s=16.1$.

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 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:

$1-\frac{1}{k^2}=1-\frac{1}{3^2}\approx 0.889=89\%$

$89\%$observations lie within three standard deviation to either side of mean that is between $36.7$and $133.3$

From given data we can say all exam score lies between given interval.

Therefore, actually $100\%$observations lies within two standard deviation to either side of mean

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

From given data we can Interprate Chebyshev's rule gives only a minimum for the percentage of observations that lie within a specified number of standard deviations to either side of the mean that is the actual percentage will usually be higher.