Answer to Question #99293 in Statistics and Probability for fatima1

Question

Problem 2: A bottling plant produces bottles of fruit juice in 32-ounce bottles. The machine that dispenses the juice is set to fill on average 32 ounces into a bottle. However the actual amount filled varies from bottle to bottle. A quality control engineer determined that the standard deviation of the amount dispensed in a bottle is 0.75 ounces. Use Chebyshev’s inequality to approximate the proportion of bottles that contain at least 33 ounces or at most 31 ounces of fruit juice.

Answer 1

We have $\mu=32, \sigma=0.75$

From the question, we need to find $P(|X-\mu|\ge1)$ that means we need to find the probability that bottle contains either one or more ounce more than the mean value 32 ounce or one or more ounce less than the mean value 32 ounce.

From Chebyshev's inequality, we know that

$P(|X-\mu|\ge k\sigma)\le(1/k^2)$

In our case, $k\sigma=1$

So, $k=(1/\sigma)=(1/0.75)$

So, $k=(4/3)$

Thus, $P(|X-\mu|\ge1)\le1/(4/3)^2$

$=(9/16)$

$=0.5625$

So, the required proportion will be $0.5625$