Question

A particular manufacturing design requires a shaft with a diameter of 17.000 ​mm, but shafts with diameters between 16.988 mm and 17.012 mm are acceptable. The manufacturing process yields shafts with diameters normally​ distributed, with a mean of 17.004 mm and a standard deviation of 0.004 mm.

Complete parts​ (a) through​ (d) below.

a. For this​ process, what is the proportion of shafts with a diameter between 16.988mm and 17.000 mm​?

The proportion of shafts with diameter between 16.988 mm and 17.000 mm is 0.1587.

​(Round to four decimal places as​ needed.)

b. For this​ process, what is the probability that a shaft is​ acceptable?

The probability that a shaft is acceptable is 0.9772.

​(Round to four decimal places as​ needed.)

c. For this​ process, what is the diameter that will be exceeded by only 2.5​% of the​ shafts?

The diameter that will be exceeded by only 2.5​% of the shafts is 17.0118 mm.

​(Round to four decimal places as​ needed.)

d. What would be your answers to parts​ (a) through​ (c) if the standard deviation of the shaft diameters were 0.003 ​mm? If the standard deviation is 0.003​mm, the proportion of shafts with diameter between 16.988 mm and 17.000 mm is

Answer 1

a) P(16.988 < X < 17)

= P((16.988 - )/ < (X - )/ < (17 - )/)

= P((16.988 - 17.004)/0.004 < Z < (17 - 17.004)/0.004)

= P(-4 < Z < -1)

= P(Z < -1) - P(Z < -4)

= 0.1587 - 0.000

= 0.1587

b) P(16.988 < X < 17.012)

= P((16.988 - )/ < (X - )/ < (17.012 - )/)

= P((16.988 - 17.004)/0.004 < Z < (17.012 - 17.004)/0.004)

= P(-4 < Z < 2)

= P(Z < 2) - P(Z < -4)

= 0.9772

c) P(X > x) = 0.025

or, P((X - )/ > (x - )/) = 0.025

or, P(Z > (x - 17.004)/0.004) = 0.025

or, P(Z < (x - 17.004)/0.004) = 0.975

or, (x - 17.004)/0.004 = 1.96

or, x = 1.96 * 0.004 + 17.004

or, x = 17.0118

d) i) P(16.988 < X < 17)

= P((16.988 - )/ < (X - )/ < (17 - )/)

= P((16.988 - 17.004)/0.003 < Z < (17 - 17.004)/0.003)

= P(-5.33 < Z < -1.33)

= P(Z < -1.33) - P(Z < -5.33)

= 0.0918 - 0.000

= 0.0918

b) P(16.988 < X < 17.012)

= P((16.988 - )/ < (X - )/ < (17.012 - )/)

= P((16.988 - 17.004)/0.003 < Z < (17.012 - 17.004)/0.003)

= P(-5.33 < Z < 2.67)

= P(Z < 2.67) - P(Z < -5.33)

= 0.9962 - 0.000 = 0.9962

c) P(X > x) = 0.025

or, P((X - )/ > (x - )/) = 0.025

or, P(Z > (x - 17.004)/0.003) = 0.025

or, P(Z < (x - 17.004)/0.003) = 0.975

or, (x - 17.004)/0.003 = 1.96

or, x = 1.96 * 0.003 + 17.004

or, x = 17.0099