Problem
The length of human pregnancies is approximately normal with mean $\mu=266$ days and standard devation $\sigma=16$ days. Complete parts $(a)$ through $(f)$ (b) Suppose a random sample of 20 human pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies. The sampling distribution of $\bar{x}$ is normal ' with $\mu_{\bar{x}}=266$ and $\sigma_{\bar{x}}=3.577$ '? (Type integers or decimals rounded to four 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 261 days or more. B. 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 261 days or less. 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 exactly 261 days.
Solution
Step 1 ：The sampling distribution of the sample mean length of pregnancies is normal with mean \(\mu_{\bar{x}}=266\) and standard deviation \(\sigma_{\bar{x}}=3.5777\).
Step 2 ：The probability that a random sample of 20 pregnancies has a mean gestation period of 261 days or less is approximately 0.0811.
Step 3 ：If 100 independent random samples of size 20 pregnancies were obtained from this population, we would expect approximately 8 samples to have a sample mean of 261 days or less.
Step 4 ：Final Answer: If 100 independent random samples of size \(n=20\) pregnancies were obtained from this population, we would expect approximately \(\boxed{8}\) samples to have a sample mean of 261 days or less.
