Question

PLEASE HELP!!!

For a specific type of machine part being produced, the diameter is normally distributed, with a mean of 15.000 cm and a standard deviation of 0.030 cm. Machine parts with a diameter more than 2 standard deviations away from the mean are rejected.

If 15,000 machine parts are manufactured, how many of these will be rejected?

Answer 1

The aanswerr is 750

Explanation:

Answer 2

Let
`X` be the random variable for the part's diameter, and
`Z` the same variable transformed into one that follows the standard normal distribution. You're looking for the proportion of the parts that lie 2 standard deviations away from the mean, which translates to


`\mathbb P(|X|>15.000+2*0.030)=\mathbb P(|X|>15.060)=\mathbb P(|Z|>2)`

Recall the empirical rule, which says that approximately 95% of a normal distribution falls within 2 standard deviations of the mean, or
`\mathbb P(|Z|<2)\approx0.95`. This means


`\mathbb P(|Z|>2)=1-\mathbb P(|Z|<2)\approx0.05=5\%`

So out of 15000 parts, you should expect about 5% of them, or about 750 parts to be rejected.