The mean amount spent by all customers at an organic food market is estimated to be \$48, and the standard deviation is estimated to be \$12. According to Chebyshev's Theorem, at least what percentage of customers would be expected to spend between \$27 and \$69?

#### Solution

Given that $\mu=48$ and $\sigma = 12$.

Using the Chebyshev's Theorem,

` $$ P(|X-\mu| < k)\geq 1-\frac{\sigma^2}{k^2} $$ `

The probability that customers would be expected to spend between \$27 and \$69

` $$ \begin{aligned} P(27 < X < 69)&=P(27-48 < X-48 < 69-48)\\ &= P(-21 < X-48 < 21)\\ &=P(|X-48| < 21)\\ &\geq (1-\frac{12^2}{21^2})\\ &\geq 0.6734694 \end{aligned} $$ `

At least 67.45 percentage of customers would be expected to spend between \$27 and \$69.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators