Question

1. A pharmaceutical company producing vitamin capsules desires a proportion of calcium content between 40 and 55 ppm. A random sample of 20 capsules chosen from the output yields a sample mean calcium content of 44 ppm with a standard deviation of 3 ppm. Find the natural tolerance limits of the process. If the process is in control at the present values of its parameters, what proportion of the output will be nonconforming, assuming a normal distribution of the characteristic?

Answer 1

Answer:

this is my answer

Step-by-step explanation:

I hope it helps

Answer 2

If the process is in control at the present values of its parameters and the calcium content follows a normal distribution, approximately 0.06% of the output will be non-conforming.

Given:

A pharmaceutical company produces vitamin capsules.

They desire a proportion of calcium content between 40 and 55 ppm.

A random sample of 20 capsules chosen from the output yields a sample mean calcium content of 44 ppm with a standard deviation of 3 ppm.

To find:

Find the natural tolerance limits of the process. If the process is in control at the present values of its parameters, what proportion of the output will be nonconforming, assuming a normal distribution of the characteristic?  

Solution:

Formula used:

Upper natural tolerance limit = sample mean + 3 x SD  

Lower natural tolerance limit = sample mean - 3 x  SD  

Z - score z = (x - μ) / σ

From the data,

Sample mean, x = 44 ppm  

Standard deviation, SD = 3 ppm

Using the above formulas

The natural tolerance limits of the process can be calculated as follows:

Upper natural tolerance limit = 44 + (3 x 3) = 53

Lower natural tolerance limit = 44 - (3 x 3) = 35

∴ The natural tolerance limits of the process are 35 ppm and 53 ppm.

To find the proportion of the output calculate the z-scores corresponding to the natural tolerance limits  

Using the formula  z = (x - μ) / σ

For the upper natural tolerance limit:

z = (53 - 44) / (3 / √20) = 3.41

 

For the lower natural tolerance limit:

z = (35 - 44) / (3 / √20) = - 3.41

   

Using a standard normal distribution table or calculator, we can find the probabilities beyond these z-scores:

P(Z > 3.40) = 0.0003 (approximately)

P(Z < -3.40) = 0.0003 (approximately)

Therefore, the proportion of nonconforming output is:

P(nonconforming) = P(Z > 3.40) + P(Z < -3.40) = 0.0003 + 0.0003 = 0.0006

Therefore,

If the process is in control at the present values of its parameters and the calcium content follows a normal distribution, approximately 0.06% of the output will be non-conforming.

#SPJ3