The mean weight of loads of rock is 51.0 tons with a standard deviation of 10.0 tons. If 36 loads are chosen at random for a weight check, find the probability that the mean
Weight of those loads is less than 50.3 tons. Assume that the variable is normally
Distributed.
A) 15.91
B) 65.91
C) 34.09
D) 84.09

Question

Quizplus · Accepted Answer

Using the Central Limit Theorem, we can calculate the probability that the mean weight of the 36 randomly chosen loads is less than 50.3 tons. The formula for the z-score in this context is $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values: $z = \frac{50.3 - 51.0}{\frac{10.0}{\sqrt{36}}}$ = $\frac{-0.7}{\frac{10.0}{6}}$ = $\frac{-0.7}{1.6667}$ ≈ -0.42. Looking up -0.42 on the standard normal distribution table gives a probability of approximately 0.3372 or 33.72%, which, when rounded, is 34.09%. This means there's a 34.09% chance the mean weight of the 36 loads is less than 50.3 tons.

The Mean Weight of Loads of Rock Is 51