Answer:
Step-by-step explanation:
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
