Question

consider sample data with x = 25 and s = 5. (a) compute the coefficient of variation. % (b) compute a 75% chebyshev interval around the sample mean. lower limit upper limit

Explanation:

Step1: Recall coefficient - of - variation formula

The coefficient of variation (CV) is given by the formula $CV=\frac{s}{\bar{x}}\times100\%$, where $\bar{x}$ is the sample mean and $s$ is the sample standard deviation. Given $\bar{x} = 25$ and $s = 5$.
\[CV=\frac{5}{25}\times100\%\]

Step2: Calculate the coefficient of variation

\[\frac{5}{25}\times100\%=0.2\times100\% = 20\%\]

Step3: Recall Chebyshev's theorem formula

Chebyshev's theorem states that at least $1-\frac{1}{k^{2}}$ of the data lies within $k$ standard - deviations of the mean. We want to find $k$ such that $1-\frac{1}{k^{2}}=0.75$.
\[1-\frac{1}{k^{2}} = 0.75\]
\[\frac{1}{k^{2}}=1 - 0.75=0.25\]
\[k^{2}=\frac{1}{0.25}=4\]
\[k = 2\]

Step4: Calculate the lower and upper limits

The lower limit is $\bar{x}-ks$ and the upper limit is $\bar{x}+ks$.
Lower limit: $\bar{x}-ks=25-2\times5=25 - 10 = 15$
Upper limit: $\bar{x}+ks=25 + 2\times5=25+10 = 35$

Answer:

(a) 20%
(b) Lower Limit: 15, Upper Limit: 35