Final answer:
To determine the probability that a shipment of batteries will be returned due to more than 1.5% being defective, we can use a normal approximation of the binomial distribution given a large sample size with a known true proportion of defects.
Step-by-step explanation:
The question concerns the calculation of the probability that a shipment of batteries will be returned due to an excessive proportion of defective units, given that the true proportion of defects is known. Specifically, we're asked to find the probability that the proportion of defective batteries in a sample of 280 will exceed the 1.5% threshold when the true proportion is 4%.
We can use the normal approximation to the binomial distribution to solve this problem because the sample size is large. The binomial distribution is used for a series of independent experiments (each battery being checked for defects), where each experiment has two possible outcomes (defective or not defective), and the probability of success (defective) is known (4%). When working with a large sample size like 280, the binomial distribution can be approximated with a normal distribution if np and n(1-p) are both greater than or equal to 10.
The mean of the binomial distribution is μ = np and the standard deviation is σ = √np(1-p). Hence, we can find a z-score that corresponds to a sample proportion of 1.5% and use the standard normal distribution to find the probability that the sample proportion will exceed this value.
After calculating the z-score and using the standard normal distribution table or a calculator, we'd obtain the probability which would correspond to one of the multiple-choice answers provided (A, B, C, or D).