The Olsen Agricultural Company has determined that the weight of hay bales is normally distributed with a mean equal to 80 pounds and a standard deviation equal to 8 pounds. Based on this, what is the probability that the mean weight of the bales in a sample of n = 64 bales will be between 78 and 82 pounds?
A) 0.4772
B) 0.0228
C) 0.6346
D) 0.9544

Question

Quizplus · Accepted Answer

Given that the weight of hay bales is normally distributed with a mean ($\mu$) of 80 pounds and a standard deviation ($\sigma$) of 8 pounds, and considering a sample size (n) of 64 bales, we want to find the probability that the sample mean ($\bar{x}$) is between 78 and 82 pounds.To find this probability, we use the Z-score formula for a sample mean: $$Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$$First, calculate the standard error ($\frac{\sigma}{\sqrt{n}}$): $\frac{8}{\sqrt{64}} = \frac{8}{8} = 1$ pound.Then, calculate the Z-scores for 78 and 82 pounds:- For 78 pounds: $Z = \frac{78 - 80}{1} = -2$- For 82 pounds: $Z = \frac{82 - 80}{1} = 2$Using the Z-table or a standard normal distribution calculator, the probability of a Z-score being between -2 and 2 is approximately 0.9544. This means there's about a 95.44% chance that the mean weight of the bales in a sample of 64 bales will be between 78 and 82 pounds.

The Olsen Agricultural Company Has Determined That the Weight of Hay