In: Statistics and Probability
A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The force in kilograms (kg) applied to the tablets varies a bit, with the N(11.4, 0.3) distribution. The process specifications call for applying a force between 11.3 and 12.3 kg. (a) What percent of tablets are subject to a force that meets the specifications? % (b) The manufacturer adjusts the process so that the mean force is at the center of the specifications, μ = 11.8 kg. The standard deviation remains 0.3 kg. What percent now meet the specifications? %
SOLUTION:
From given data,
A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The force in kilograms (kg) applied to the tablets varies a bit, with the N(11.4, 0.3) distribution. The process specifications call for applying a force between 11.3 and 12.3 kg.
N(11.4, 0.3) distribution
Mean = = 11.4
Standard deviation = = 0.3
applying a force between 11.3 and 12.3 kg.
z = ( X - ) /
(a) What percent of tablets are subject to a force that meets the specifications
at X = 11.3
z = ( 11.3 - 11.4 ) / 0.3
= -0.1 /0.3
= -0.33
at X = 12.3
z = ( 12.3 - 11.4 ) / 0.3
= 0.9 /0.3
= 3
P( 11.3 < X < 12.3 ) = P( -0.33 < z < 3)
= P( Z < 3) - P( Z < -0.33 )
= 0.9987 - 0.3707
= 0.628
= 62.8%
(b) The manufacturer adjusts the process so that the mean force is at the center of the specifications, μ = 11.8 kg. The standard deviation remains 0.3 kg. What percent now meet the specifications
Mean = = 11.8
Standard deviation = = 0.3
at X = 11.3
z = ( 11.3 - 11.8 ) / 0.3
= -0.5 /0.3
= -1.66
at X = 12.3
z = ( 12.3 - 11.8 ) / 0.3
= 0.5 /0.3
= 1.66
P( 11.3 < X < 12.3 ) = P( -1.66 < z < 1.66)
= P( Z < 1.66) - P( Z < -1.66 )
= 0.9515 - 0.0485
= 0.903
= 90.3%