a) 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 tablet varies a bit and follows the Normal distribution with mean 11.5 kg and standard deviation 0.2 kg. The process specifications call for applying a force between 11.2 and 12.2 kg.
i. What percent of tablets are subject to a force that meets the specifications?

ii. The manufacturer adjusts the process so that the mean force is at the center of the specifications, μ = 11.7 kg. The standard deviation remains 0.2 kg. What percent now meets the specifications?

iii. Calculate the 25th percentile (P25) and the 75th percentile (P75) of the distribution.

Let x be the random variable representing the force in kilograms (kg) applied to the tablet. Since it is normally distributed and the population mean and population standard deviation are known, we would apply the formula,

z = (x - µ)/σ

Where

x = sample mean

µ = population mean

σ = standard deviation

From the information given,

µ = 11.5

σ = 0.2

1)/the probability that the process specifications call for applying a force between 11.2 and 12.2 kg is expressed as

P(11.2 ≤ x ≤ 12.2)

For x = 11.2,

z = (11.2 - 11.5)/0.2 = - 1.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.0668

For x = 12.2

z = (12.2 - 11.5)/0.2 = 3.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.9998

Therefore,

P(11.2 ≤ x ≤ 12.2) = 0.9998 - 0.0668 = 0.933

The percent of tablets are subject to a force that meets the specifications is

0.933 × 100 = 93.3%

ii) P(11.2 ≤ x ≤ 12.2)

For x = 11.2,

z = (11.2 - 11.7)/0.2 = - 2.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.00621

For x = 12.2

z = (12.2 - 11.7)/0.2 = 2.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.9938

Therefore,

P(11.2 ≤ x ≤ 12.2) = 0.9938 - 0.00621 = 0.988

The percent of tablets are subject to a force that meets the specifications is

0.988 × 100 = 98.8%

iii) for 25th percentile, the z score corresponding z score corresponding to the the probability value of 0.25 is - 0.675

Therefore,

- 0.675 = (x - 11.5)/0.2

0.2 × - 0.675 = x - 11.5

- 0.135 = x - 11.5

x = - 0.135 + 11.5

x = 11.4kg

for 75th percentile, the z score corresponding z score corresponding to the the probability value of 0.75 is 0.675

Therefore,

0.675 = (x - 11.5)/0.2

0.2 × - 0.675 = x - 11.5

0.135 = x - 11.5

x = 0.135 + 11.5

x = 11.6kg