Question

The mean and the standard deviation of weights of ponies in a large animal shelter are 20 kg and 3 kg, respectively. A pony is selected at random. Using Chebyshev's inequality, find the lower bound of the probability that its weight lies between 14 kg and 26 kg.

• A. $\dfrac{3}{4}$
• B. $\dfrac{1}{4}$
• C. 0
• D. 1

Hint:

Chebyshev's inequality gives a minimum probability bound for values within $k$ standard deviations from the mean, regardless of the distribution.

Answer 1

The Correct Option is A

Step 1: Identify parameters.
Mean $\mu = 20$, standard deviation $\sigma = 3$. The range is 14 to 26, i.e., $\pm 6$ from the mean.

Step 2: Apply Chebyshev's inequality.
\[ P(|X - \mu| < k\sigma) \ge 1 - \frac{1}{k^2}. \] Here, $k = \frac{6}{3} = 2$. So, \[ P(|X - 20| < 6) \ge 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}. \]

Step 3: Conclusion.
Hence, the lower bound of the probability that the selected pony's weight is between 14 kg and 26 kg is $\dfrac{3}{4}$.