Question

The diameters of bolts produced by a certain machine are normally distributed with a mean of 1.20 inches and a standard deviation of 0.01 inches. What proportion of bolts will have a diameter greater than 1.211 inches

Answer 1

0.1357 = 13.57% of bolts will have a diameter greater than 1.211 inches

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean of 1.20 inches and a standard deviation of 0.01 inches.

This means that
`\mu = 1.20, \sigma = 0.01`

What proportion of bolts will have a diameter greater than 1.211 inches?

This is 1 subtracted by the p-value of Z when X = 1.211. So

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

`Z = (1.211 - 1.20)/(0.01)`

`Z = 1.1`

`Z = 1.1` has a p-value of 0.8643.

1 - 0.8643 = 0.1357

0.1357 = 13.57% of bolts will have a diameter greater than 1.211 inches