Question

A car manufacturer makes a certain transmission gear with a mean circumference of 11 mm. This
is the size that allows the gear to fit precisely into the transmission to avoid unnecessary wear and
tear on the transmission’s other moving parts. The manufacturing process is not perfect, so there
is some variation in the circumference of the gear. The circumference measurements are
normally distributed, and the standard deviation of the gear’s circumference is 0.05mm.

a. Find the probability that a randomly selected gear produced by this company has a
circumference less than 10.9 mm.

b. Find the probability that 22 randomly selected gears produced by this company have a mean
circumference less than 10.9 mm. I got for the Z score (-9.09 but it says that the Zscore is supposed to be -9.38-- not sure where I messed up.. please clarify..Thank you)

Answer 1

a.

Given, = 11 mm, = 0.05 mm

Probability that a randomly selected gear produced by this company has a circumference less than 10.9 mm

= P[X < 10.9]

= P[Z < (10.9 - 11) / 0.05]

= P[Z < -2]

= 0.02275 (Using Standard normal Z table)

b.

Standard error of mean = / = 0.05 / = 0.01066

Probability that 22 randomly selected gears produced by this company have a mean circumference less than 10.9 mm =

P( < 10.9) = P[Z < (10.9 - 11) / 0.01066] {Z score = (10.9 - 11) / 0.0106 = -9.38}

= P[Z < -9.38]   

= 0.000