Final answer:
The probability that the mean pregnancy length for 64 women is between 271 days and 273 days is found by converting the range to z-scores and looking up the cumulative probabilities from a standard normal distribution table. The calculated probability closely matches option C) 0.2912.
Step-by-step explanation:
To find the probability that the mean pregnancy length for 64 randomly selected women is between 271 days and 273 days, we use the normal distribution. Since the problem states that the lengths of pregnancies are normally distributed with a mean (μ) of 271 days and a standard deviation (σ) of 20 days, and we are considering a sample size (n) of 64, we first find the standard error of the mean, which is σ/√n = 20/√64 = 20/8 = 2.5 days.
Next, we convert the range of mean pregnancies to z-scores. For the lower limit (271 days), Z = (271 - 271) / 2.5 = 0. For the upper limit (273 days), Z = (273 - 271) / 2.5 = 2/2.5 = 0.8.
To find the probability of a z-score being between 0 and 0.8, we look up the values in the standard normal distribution table or use a calculator with normal distribution functions. The cumulative probability for Z = 0 is 0.5, and for Z = 0.8, it is approximately 0.7881. So, the probability that the mean is between 271 days and 273 days is the difference between these two values, which is 0.7881 - 0.5 = 0.2881. The closest answer choice to this calculated probability is C) 0.2912.