Question

A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 4 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 15 cm with a standard deviation of 0.10 cm. For what lengths will a bolt be​ destroyed?Explain the meaning of the accompanying percentiles. ​(a) The 15th percentile of the head circumference of males 3 to 5 months of age in a certain city is 41.0 cm. ​(b) The 95th percentile of the waist circumference of females 2 years of age in a certain city is 52.7 cm. ​(c) Anthropometry involves the measurement of the human body. One goal of these measurements is to assess how body measurements may be changing over time. The following table represents the standing height of males aged 20 years or older for various age groups in a certain city in 2015. Based on the percentile measurements of the different age​ groups, what might you​ conclude?

Answer 1

For the bolts, any lengths that are more than 4 standard deviations away from the mean (15 cm) will be destroyed. This means that any bolts shorter than 14.6 cm or longer than 15.4 cm will be destroyed.

Percentiles represent the position of a particular measurement within a distribution. For example:
(a) The 15th percentile of head circumference for males 3 to 5 months of age means that 15% of males in that age range have a head circumference of 41.0 cm or smaller.
(b) The 95th percentile of waist circumference for females 2 years of age means that 95% of females in that age group have a waist circumference of 52.7 cm or smaller.

Based on the percentile measurements of standing height for males aged 20 years or older in different age groups, we can conclude that average height tends to decrease with increasing age.