Question

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32​ inches?

Answer 1

2.25 %

Step-by-step explanation:

65-95-99.7 is a rule to remember the precentages that lies around the mean.

at the range of mean (
`\mu`) plus or minus one standard deviation (
`\sigma`),
`P([\mu-\sigma \leq X \leq \mu+\sigma])\approx 68.3%`

at the range of mean plus or minus two standard deviation,
`P([\mu -2\sigma \leq X \leq \mu+2\sigma])\approx 95.5%`

at the range of mean plus or minus three standard deviation,
`P([\mu - 3\sigma\leq X \leq \mu+3\sigma])\approx 99.7%`

So, note that they are asking about the probability that it is greater than 0.32, that is the mean (0.3) plus two times the standard deviation (0.1) (
`P(X \leq \mu+2\sigma)`)

So we know that the 95.5% is between
`\mu - 2\sigma = 0.3 -2*0.1 = 0.28` and
`\mu + 2\sigma = 0.3 +2*0.1 = 0.32`, hence approximately the 4.5% (100%-95.5%) is greater than 0.32 or less than 0.28. But half (4.5%/2=2.25%) is greater than 0.32 and the other half is less than 0.28.

So
`P(X \leq \mu+2\sigma) \approx 2.25%`