A machine produces vitamin C tablets, the diameters of which are normally distributed
with mean 10mm and standard deviation 0.1mm. A tablet is acceptable if its diameter
lies between 9.81mm and 10.19mm. To pass a quality test, at least 90% of the tablets in
a pack should be acceptable.
(a) Determine the probability that a tablet is acceptable.
(b) What is the probability that a pack of 20 tablets will pass the quality test? Use
binomial probability distribution.
(c) What is the probability that a pack of 200 tablets will pass the quality test?
(d) If you have a choice of submitting a pack of 20 tablets or a pack of 200 tablets for
the quality test, which one would you prefer and why?
To answer these questions, we can use the properties of the normal distribution and the binomial distribution.
(a) To determine the probability that a tablet is acceptable, we can use the properties of the normal distribution. We know that the mean diameter is 10mm and the standard deviation is 0.1mm. We need to find the probability that the diameter lies between 9.81mm and 10.19mm.
First, we need to standardize the values using the z-score formula:
z = (x - mu) / sigma
where x is the given value, mu is the mean, and sigma is the standard deviation.
For 9.81mm:
z1 = (9.81 - 10) / 0.1 = -1.9
For 10.19mm:
z2 = (10.19 - 10) / 0.1 = 1.9
Now, we can use the z-table or a calculator to find the corresponding probabilities for these z-scores.
Using the z-table, we find that the probability associated with a z-score of -1.9 is approximately 0.0287, and the probability associated with a z-score of 1.9 is also approximately 0.9713.
To find the probability that a tablet is acceptable, we subtract the probability of the lower value from the probability of the upper value:
P(9.81mm <= diameter <= 10.19mm) = 0.9713 - 0.0287 = 0.9426
Therefore, the probability that a tablet is acceptable is 0.9426, or 94.26%.
(b) To find the probability that a pack of 20 tablets will pass the quality test, we can use the binomial probability distribution. In this case, each tablet has a probability of being acceptable equal to 0.9426.
Using the formula for the binomial probability:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials, k is the number of successes, p is the probability of success, and (n choose k) is the binomial coefficient.
In this case, n = 20, k is at least 90% of 20 (which is 18), and p = 0.9426.
P(X >= 18) = P(X = 18) + P(X = 19) + P(X = 20)
Using the binomial probability formula, we can calculate these probabilities:
P(X = 18) = (20 choose 18) * (0.9426^18) * (0.0574^2)
P(X = 19) = (20 choose 19) * (0.9426^19) * (0.0574^1)
P(X = 20) = (20 choose 20) * (0.9426^20) * (0.0574^0)
Calculating these values and summing them up will give the probability that a pack of 20 tablets will pass the quality test.
(c) To find the probability that a pack of 200 tablets will pass the quality test, we can use the same approach as in part (b), but now n = 200.
Using the binomial probability formula, we can calculate the probability of k successes out of 200 trials where each tablet has a probability of success equal to 0.9426.
P(X >= 180) = P(X = 180) + P(X = 181) + ... + P(X = 200)
We can calculate each term individually and sum them up to find the final probability.
(d) To determine whether to submit a pack of 20 tablets or a pack of 200 tablets for the quality test, we can compare the probabilities found in parts (b) and (c).
By comparing the probabilities, we can see which one has a higher chance of passing the quality test. If the probability of the pack of 200 tablets passing the test is greater than the probability of the pack of 20 tablets passing the test, then it would be preferable to submit the pack of 200 tablets. Conversely, if the probability of the pack of 20 tablets passing the test is greater, then it would be preferable to submit the pack of 20 tablets.
By comparing the probabilities found in parts (b) and (c), we can make an informed decision on which pack to submit for the quality test.