Question

a manufacturer of dinner plates make a certain type of ceramic plate that when packaged for delivery has a vertical stacking height for each plate that is normally distributed with a mean of 2cm and standard deviation of 0.1cm. the manufacturer ships the plates vertically in stacks of 16 plates. to cut down on shipping costs, the manufacturer wants to pick a size of shipping box that best fits the stack of 16 plates. (a) what is the probability that height of the 16 vertically stacked plates will be less than 32.5cm? (b) how tall should the shipping box for these plates be such that the height of the 16 vertically stacked plates will exceed that size with a probability of 1%? (c) how tall should the shipping box for these plates be such that the height of the 16 vertically stacked plates will exceed that size with a probability of 0.1%?

Answer 1

The question involves calculating probabilities considering the normal distribution of plate stacking heights. For (a), we determine the probability that a stack is below a certain height using the z-score. For (b) and (c), we find the box height that would be exceeded with probabilities of 1% and 0.1%, respectively, by using appropriate z-scores.

Step-by-step explanation:

The student's question involves a scenario where the height of dinner plates stacked vertically is normally distributed with a mean of 2cm and a standard deviation of 0.1cm. When considering a stack of 16 plates, the sampling distribution of the stacking height also follows a normal distribution.

Here the mean of the sampling distribution would be 16 times the mean of a single plate, and the standard deviation would be the standard deviation of a single plate divided by the square root of the number of plates, which is the central limit theorem in action. To find the probability that the height of the stack is less than 32.5cm, we would need to first calculate the z-score for 32.5cm, and then look up this z-score in a standard normal (Z) table or use a calculator to determine the probability.

For questions (b) and (c), where the manufacturer wants to find the shipping box height so that the probability of the stack of plates exceeding the box height is 1% and 0.1%, respectively, we would need to use the Z-table to find the z-scores corresponding to the upper 1% and 0.1% of the normal distribution, and then use those z-scores to calculate the respective heights of the boxes.