A data set has a mean of 1,200 and a standard deviation of 80.
a. Using Chebyshev's theorem, what percentage of the observations fall between 880 and 1,520? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)
b. Using Chebyshev's theorem, what percentage of the observations fall between 1,040 and 1,360? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)
Solution
Given that mean $\mu=1200$ and standard deviation $\sigma=80$.
(a) The probability that the observations fall between 880 and 1,520 is
$$ \begin{aligned} P(880 < X < 1520) &= P(\frac{880-1200}{80} < \frac{X-\mu}{\sigma} < \frac{1520-1200}{80})\\ &= P(-4 < \frac{x-\mu}{\sigma} < 4)\\ &= P(\big|\frac{x-\mu}{\sigma} \big| < 4)\\ &\geq 1-(\frac{1}{4})^2\\ &\geq 0.9375\\ &\geq 0.94. \end{aligned} $$
94 percentage of the observations fall between 880 and 1,520.
(b) The probability that the observations fall between 1,040 and 1,360 is
$$ \begin{aligned} P(1040 < X < 1360) &= P(\frac{1040-1200}{80} < \frac{X-\mu}{\sigma} < \frac{1360-1200}{80})\\ &= P(-2 < \frac{x-\mu}{\sigma} < 2)\\ &= P(\big|\frac{x-\mu}{\sigma} \big| < 2)\\ &\geq 1-(\frac{1}{2})^2\\ &\geq 0.75 \end{aligned} $$
75 percentage of the observations fall between 1040 and 1360.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
