A certain type of plywood consists of five layers. The thicknesses of the layers are independent and normally distributed with mean 5.0 mm and standard deviation 0.15 mm.

Find the mean thickness of the plywood.

Find the standard deviation of the thickness of the plywood.

Find the probability that the plywood is less than 24 mm thick. Round the answer to four decimal places.

A) The mean thickness of each layer is 5.0 mm, and there are 5 layers, so the mean thickness of the plywood is (5.0 mm / layer) * (5 layers) = 25 mm. This holds as long as the thicknesses of the layers are independent of each other, and are all normally distributed.

b) We cannot simply add up standard deviations, but we can add up variance. If a single layer has SD = 0.15 mm, then the variance is (SD) ^2 = 0.0225. If we add up the variance for the 5 layers, this is (0.0225) * 5 = 0.1125. Then we take the square root to get the SD of the plywood, which is SD = sqrt (0.1125) = 0.3354 mm.

c) We first get the z-score for x = 24 mm, using the formula z = (x - mean) / SD. For this problem, the x = 24 value corresponds to z = (24 - 25) / 0.3354 = - 2.98. Based on z-tables, the probability that z < - 2.98 is 0.0014, and this is the probability that a piece of plywood has thickness below 24 mm.