The length of human pregnancies is approximately normal with mean $\mu=266$ days and standard deviation $\sigma=16$ days. Complete parts (a) through (f) (Iype integers or decimals rounded to tour decimal places as needed.) (c) What is the probability that a random sample of 20 pregnancies has a mean gestation period of 261 days or less? The probability that the mean of a random sample of 20 pregnancies is less than 261 days is approximately 0.0811 (Round to four decimal places as needed.) Interpret this probability Select the correct choice below and fill in the answer box within your choice (Round to the nearest integer as needed) A. If 100 independent random samples of size $n=20$ pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 2 B. If 100 independent random samples of size $n=20$ pregnancies were obtained from this poputation, we would expect. 8 sample(s) to have a sample mean of 26 C. If 100 independent random samples of size $n=20$ pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of $e$ (d) What is the probability that a random sample of 66 pregnancies has a mean gestation penod of 261 days or less? The probability that the mean of a random sample of 66 pregnancies is less than 261 days is approximately (Round to four decimal places as needed.)
Solution
Step 1 ：We are given that the length of human pregnancies is approximately normal with mean \(\mu=266\) days and standard deviation \(\sigma=16\) days.
Step 2 ：We are asked to find the probability that a random sample of 20 pregnancies has a mean gestation period of 261 days or less.
Step 3 ：This is a problem of normal distribution. We can use the formula for the z-score to find the probability.
Step 4 ：The z-score is calculated as \((X - \mu) / (\sigma / \sqrt{n})\), where X is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the standard deviation, and n is the sample size.
Step 5 ：Substituting the given values into the formula, we get \(z = (261 - 266) / (16 / \sqrt{20}) = -1.3975424859373686\).
Step 6 ：We can then use a z-table to find the probability corresponding to this z-score, which is approximately 0.0811.
Step 7 ：This means that if we take a random sample of 20 pregnancies, there is about an 8.11% chance that the mean gestation period will be 261 days or less.
Step 8 ：Final Answer: The probability that a random sample of 20 pregnancies has a mean gestation period of 261 days or less is approximately \(\boxed{0.0811}\).
