Suppose that Stephen is the quality control supervisor for a food distribution company. A shipment containing many thousands of apples has just arrived. Unknown to Stephen, 11% of the apples are damaged due to bruising, worms, or other defects. If Stephen samples 10 apples from the shipment, use the binomial distribution to estimate the probability that his sample will contain at least one damaged apple.Give your answer as a decimal precise to at least four decimal places.P(X≥1)= Select the true statement.Stephen should sample with replacement so that the probability is exactly binomialStephen can use a sample of size 10 to reliably determine if the truck load contains damaged apples.Stephen could use a sample of size less than 10 to reliably determine if the truck load contains damaged apples.A sample of size 10 is too small to reliably determine if the truck load contains damaged apples.Stephen could have used a normal approximation to determine this probability.

The probability that at least 1 apple is damaged in the sample of 10 is 0.6882. Therefore, the sample size is too low to determine anything with accuracy.

Step-by-step explanation:

Let X be the total of damaged apples in the sample of 10. X has binomial distribution with parameters p =0.11 and n = 10. The probability that at least 1 apple is damaged is P(X≥1). This probability can be computed using the probability complementary event, P(X≥1) = 1 - P(X=0); and

[tex]P(X=0) = {10 \choose 0} * 0.11^0*0.89^{10} = 0.89^{10} = 0.3118[/tex]

Thus, the probability that at least 1 apple is damaged is 1-0.3118 = 0,6882. The sample size is too small to determine anything, since Steven will be mistaken almost half of the time (it should be at least 30 to obtain significant results). Replacement of the apples isnt necessary because the shipment has thousands of apples and he is just picking a few, that wouldnt change the probability that an apple is damaged in a significant way. Last, but not least, a Normal approximation should not give accurate results, because the value of n is too low (n should be at the very least 20, but better if it is at least 30).