Q. 3.132 - Elementary Statistics

Question

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).

Step-by-Step Solution

Answer

Part(a) Actual percentage of observation is greater than percentage of observation by Chebyshev's rule.

Part(b) Actual percentage of observation is greater than percentage of observation by Chebyshev's rule.

Part(c)  From part(a) and part (b) we can interpret  that actual percentage will usually be higher.

Part(a) Step 1 : Given information

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 $k=2$

Part(a) Step 2 : Simplify

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

Part(b) Step 1 : Given information

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 $k=3$

Part(b) Step 2 : Simplify

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 

Part(c) Step 1 : Given information

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

Part(c) Step 2 : Simplify

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.