Final answer:
To determine proportions and probabilities of different gestation periods in a normal distribution with mean 274 days and standard deviation 16 days, one must calculate the z-scores and refer to a standard normal distribution table or use a calculator. For values such as 294, 254, 282, 266, and 250 days, we compute their respective z-scores and compare them to the distribution to determine the proportions and probabilities.
Step-by-step explanation:
The lengths of a particular animal's pregnancies are approximately normally distributed, with mean μ=274 days and standard deviation σ=16 days.
(a) To find the proportion of pregnancies that lasts more than 294 days, we need to compute the z-score associated with 294 days. The z-score is calculated by (X - μ) / σ. So, for 294 days: (294 - 274) / 16 = 20 / 16 = 1.25. We then check a standard normal distribution table, or use a calculator, to find the probability associated with a z-score of 1.25. This gives us the proportion of pregnancies less than 294 days, and we subtract from 1 to find the proportion greater than 294 days.
(b) The proportion of pregnancies lasting between 254 and 282 days can be found by computing the z-scores for both values and then finding the area between them in the standard normal distribution. So we calculate z-scores for 254 and 282 days and use a standard normal distribution table or calculator to find the probabilities.
(c) To determine the probability that a randomly selected pregnancy lasts no more than 266 days, we calculate the z-score for 266 days and use a standard normal distribution to find the corresponding probability.
(d) A "very preterm" baby is one whose gestation period is less than 250 days. To determine if this is unusual, we compute the z-score for 250 days and see where that lies on the standard normal distribution. Typically, if the z-score is below -2 or above +2, the outcome is considered unusual because it falls in the tails of the distribution, which comprise less than 5% of the area under the curve.