A quality control audit has been devised to check on the sampling procedure when a truckload of potatoes arrive at a packing plant. A random sample of 250 is selected and examined for bruises and other defects. The whole truckload will be rejected if 5% of the sample is unsatisfactory. Determine the probability that the shipment will be accepted anyway if the load were to have 8% of the potatoes on the truck not meeting the desired standard.State assumptions and conditions and if they are met.
To determine the probability that the shipment will be accepted, we need to use the binomial distribution since we are interested in the probability of a certain number of successes (unsatisfactory potatoes) out of a fixed number of trials (sample size).
Assumptions and conditions for using the binomial distribution:
1. The potato samples are independent of each other - This assumption is reasonable if the sampling procedure is truly random, and each potato's quality does not depend on the quality of other potatoes.
2. The probability of success (a potato being unsatisfactory) remains constant - This assumption assumes that the proportion of unsatisfactory potatoes in the truckload is consistent throughout the entire shipment.
3. The sample size is relatively small compared to the population size (truckload) - This assumption allows us to treat the sampling process as independent from one trial to the next. Since the population (truckload) is likely much larger than 250 potatoes, this condition is satisfied.
Now, let's calculate the probability:
The probability that a single potato is unsatisfactory is given as 8% or 0.08. Therefore, the probability of a potato being satisfactory is 1 minus the unsatisfactory probability, which is 0.92 (1 - 0.08).
Since the samples are selected randomly, the number of unsatisfactory potatoes in the sample follows a binomial distribution with parameters n = 250 (sample size) and p = 0.08 (probability of unsatisfactory potato).
To determine the probability that the shipment will be accepted, we need to find the probability of having less than or equal to 5% of the sample being unsatisfactory. We calculate this by summing the probabilities of having 0, 1, 2, 3, 4, or 5 unsatisfactory potatoes.
P(X ≤ 0) + P(X ≤ 1) + P(X ≤ 2) + P(X ≤ 3) + P(X ≤ 4) + P(X ≤ 5)
Using a binomial probability calculator or a statistical software, we find that the probability is approximately 0.99999.