Question

A quantitative data set has mean $25$and standard deviation $5$.Fill in the following blanks:

(a)At least $89\%$of the observation lie between ____ and____ .

(b)At least____ $\%$of the observation lie between$15$and$35$.

Short answer

(a) $89\%$of the observations lie between $10$and $40$.

(b)$75\%$of the observation lie between$15$and$35$.

Step-by-step solution

Part(a) Step1: Given information.

We have been given that,

Mean= $25$, Standard deviation = $5$

We need to find the values where in between $89\%$of the observations lies.

Part(a) Step2: Simplify.

We know that,

Chebyshev's Rule: At least $100\left(1-\frac{1}{K^2}\right)\%$of the data values is written $K$standard deviation from the mean $\left(K>1\right)$.

By using Chebyshev's Rule with $k=3$, we know that at least

$$\begin{gathered} =100\left(1-\frac{1}{K^2}\right)\% \\ =100\left(1-\frac{1}{3^2}\right)\% \\ =100\left(1-\frac{1}{9}\right)\% \\ =100\times \frac{8}{9}\%=89\% \end{gathered}$$

is within $3$standard deviation from the mean.

Determine the values that are $3$standard deviation from the mean:

$$\begin{gathered} Mean-3S\tan darddeviation=25-3\times 5=25-15=10 \\ Mean+3S\tan darddeviation=25+3\times 5=25+15=40 \end{gathered}$$

Therefore, at least $89\%$of the observations lie between $10$and $40$.

Part(b) Step1: Given information.

We have been given that,

Range of the observations is $15$and $35$.

We need to find the $\%$of the observations that lie between given two values.

Part(b) Step2: Simplify.

From the question,$15$and $35$are $2$standard deviation from the mean.

Using Chebyshev's Rule with $k=2$, we know that at least

$$\begin{gathered} =100\left(1-\frac{1}{K^2}\right)\% \\ =100\left(1-\frac{1}{2^2}\right)\% \\ =100\left(1-\frac{1}{4}\right)\% \\ =100\times \frac{3}{4}\% \\ =75\% \end{gathered}$$

is within $2$standard deviation from the mean.