A factory produces a particular type of electronic component. The probability of a component being acceptable is 0.95. The components are packed in boxes of 24.
(i) Calculate the probability that a box, chosen at random, contains exactly 22 acceptable components.
All the boxes are inspected and a box is rejected if it contains fewer than 22 acceptable components.
(ii) Calculate the probability that a box, chosen at random is rejected. The factory produces 80 boxes per day over a long period of time.
(iii) Estimate the mean and standard deviation of the number of boxes rejected per day.
I want answers for this problem
1.5
I want the answer for this problem
To solve these problems, we will be using the binomial probability distribution. The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial, and each trial is independent of all others. In this case, the acceptable or rejected components are the two outcomes.
The probability mass function for a binomial distribution is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes (acceptable components)
n is the number of trials (components in a box)
k is the number of successes (acceptable components in a box)
p is the probability of success (probability of a component being acceptable)
(1 - p) is the probability of failure (probability of a component being rejected)
C(n, k) is the binomial coefficient, equal to n! / (k!(n - k)!)
(i) Calculate the probability that a box, chosen at random, contains exactly 22 acceptable components.
In this case, n = 24 (components in a box), k = 22 (acceptable components in a box), and p = 0.95 (probability of a component being acceptable).
Using the formula, we can calculate the probability as follows:
P(X = 22) = C(24, 22) * 0.95^22 * (1 - 0.95)^(24 - 22)
(ii) Calculate the probability that a box, chosen at random, is rejected.
A box is rejected if it contains fewer than 22 acceptable components. We can calculate the probability as the sum of the probabilities of getting 0 to 21 acceptable components:
P(rejected) = P(X < 22) = P(X = 0) + P(X = 1) + ... + P(X = 21)
This can be calculated using the binomial probability formula for each value of k and summing them up.
(iii) Estimate the mean and standard deviation of the number of boxes rejected per day.
To estimate the mean, we multiply the probability of rejection (calculated in part (ii)) by the number of boxes produced per day (80). The mean (expected value) will give us an estimate of the average number of boxes rejected per day.
To estimate the standard deviation, we can use the formula for the standard deviation of a binomial distribution:
SD(X) = sqrt(n * p * (1 - p))
Here, n = 24 (components in a box) and p = probability of a box being rejected (calculated in part (ii)). We then multiply the standard deviation by the square root of the number of boxes produced per day to estimate the daily standard deviation of boxes rejected.
With these steps explained, you can now proceed to calculate the probabilities and estimates as described.