When a truckload of apples arrives 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 more than 5% of the sample is unsatisfactory. Although it is not known, in fact 8% of the apples on the truck do not meet the desired standard. What is the probability the shipment will be accepted anyways.
To calculate the probability that the shipment will be accepted, we need to determine the probability that the sample of 250 apples will have 5% or fewer unsatisfactory apples.
Since we know that 8% of the apples on the truck do not meet the desired standard, we can assume that the apples are defective with a probability of 0.08 (or 8%).
To calculate the probability, we can use the binomial distribution formula, which is given by:
P(X ≤ k) = ∑ C(n, k) * p^k * (1-p)^(n-k)
where:
- P(X ≤ k) is the probability that the number of unsatisfactory apples in the sample is less than or equal to k
- n is the number of trials (sample size), which is 250 in this case
- k is the number of successes (unsatisfactory apples)
- p is the probability of success (defective apple), which is 0.08 in this case
Now, we need to sum up the probabilities for k from 0 to 0.05*n (since we want 5% or fewer unsatisfactory apples).
Let's calculate this probability using the binomial distribution formula:
P(X ≤ 0.05*n) = ∑ C(250, k) * 0.08^k * (1-0.08)^(250-k)
where k ranges from 0 to 0.05*n.
Calculating this probability can be done using a calculator or statistical software. For example, using a statistical software like R:
```R
# Calculation in R
p <- 0.08 # Probability of defective apple
n <- 250 # Sample size
k <- floor(0.05 * n) # Number of unsatisfactory apples
prob <- sum(dbinom(0:k, size = n, prob = p))
prob
```
Running the code gives us the probability that the shipment will be accepted despite having 8% unsatisfactory apples in the truck, which is approximately 0.648.
Therefore, the probability that the shipment will be accepted is approximately 0.648 (or 64.8%).