Question

the Boeing 787-8 Dreamliner is designed to be 186 feet in length. The manufacturing process for assembling the 787-8 follows a bell shaped symmetric distribution with mean 185.4 feet and standard deviation 0. 5 feet. What proportion of the 787-8 airplanes are between 185' and 187'

Answer 1

78.74% of the 787-8 airplanes are between 185' and 187'.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 185.4, \sigma = 0.5`

What proportion of the 787-8 airplanes are between 185' and 187'?

This is the pvalue of Z when X = 187 subtracted by the pvalue of Z when X = 185. So

X = 187

`Z = (X - \mu)/(\sigma)`

`Z = (187 - 185.4)/(0.5)`

`Z = 3.2`

`Z = 3.2` has a pvalue of 0.9993.

X = 185

`Z = (X - \mu)/(\sigma)`

`Z = (185 - 185.4)/(0.5)`

`Z = -0.8`

`Z = -0.8` has a pvalue of 0.2119.

0.9993 - 0.2119 = 0.7874

78.74% of the 787-8 airplanes are between 185' and 187'.