Final answer:
To find probabilities related to the lengths of pregnancies, we use the standard normal distribution. We calculate the z-scores for each question and then use them to find the areas on the standard normal distribution. (a) The proportion of pregnancies that last more than 280 days is 6.68%. (b) The proportion of pregnancies that last between 253 and 271 days is 48.34%. (c) The probability that a randomly selected pregnancy lasts no more than 241 days is 4.09%. (d) Very preterm babies are unusual, with only 0.62% of pregnancies having a gestation period of less than 232 days.
Step-by-step explanation:
To find the probabilities in this problem, we will use the standard normal distribution. First, we need to calculate the z-scores for each question. The formula for calculating z-score is z = (x - u) / o, where x is the value we are interested in, u is the mean, and o is the standard deviation.
(a) To find the proportion of pregnancies that last more than 280 days, we need to find the area to the right of 280 on the standard normal distribution. The z-score for 280 days is z = (280 - 262) / 12 = 1.5. Using a standard normal table or calculator, we find that the area to the right of 1.5 is 0.0668, or 6.68%.
(b) To find the proportion of pregnancies that last between 253 and 271 days, we need to find the area between these two values on the standard normal distribution. The z-scores for 253 and 271 are z1 = (253 - 262) / 12 = -0.75 and z2 = (271 - 262) / 12 = 0.75. Using a standard normal table or calculator, we find that the area between -0.75 and 0.75 is 0.4834, or 48.34%.
(c) To find the probability that a randomly selected pregnancy lasts no more than 241 days, we need to find the area to the left of 241 on the standard normal distribution. The z-score for 241 days is z = (241 - 262) / 12 = -1.75. Using a standard normal table or calculator, we find that the area to the left of -1.75 is 0.0409, or 4.09%.
(d) To determine if very preterm babies are unusual, we need to find the proportion of pregnancies that have a gestation period of less than 232 days. The z-score for 232 days is z = (232 - 262) / 12 = -2.5. Using a standard normal table or calculator, we find that the area to the left of -2.5 is 0.0062, or 0.62%. Since this is a small proportion, we can conclude that very preterm babies are unusual.