The length of human pregnancies is approximately normal with mean $\mu=266$ days and standard deviation $\sigma=16$ days. Complete parts (a) through (f). Click here to viow the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2). C. If 100 pregnant individuals were selected independently from this population, we would expect pregnancies to last more than 258 days. (b) Suppose a random sample of 14 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 $\sigma_{\bar{x}}=4.2762$. (Round to four decimal places as needed.) (c) What is the probability that a random sample of 14 pregnancies has a mean gestation period of 258 days or less? The probability that the mean of a random sample of 14 pregnancies is less than 258 days is (Round to four decimal places as needed) iew an example Get more help - Clear all Check answer
Solution
Step 1 ：The length of human pregnancies is approximately normal with mean \(\mu=266\) days and standard deviation \(\sigma=16\) days.
Step 2 ：A random sample of 14 human pregnancies is obtained. The sampling distribution of the sample mean length of pregnancies is normal with \(\mu_{\bar{x}}=266\) and \(\sigma_{\bar{x}}=4.2762\).
Step 3 ：We are asked to find the probability that a random sample of 14 pregnancies has a mean gestation period of 258 days or less.
Step 4 ：This is a question about the sampling distribution of the mean, which is normally distributed with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.
Step 5 ：We can use the z-score formula to standardize the sample mean and then use the standard normal distribution to find the probability.
Step 6 ：Given that \(\mu = 266\), \(\sigma = 16\), \(n = 14\), and \(\bar{x} = 258\), we find that \(\sigma_{\bar{x}} = 4.27617987059879\) and \(z = -1.8708286933869707\).
Step 7 ：Using these values, we find that the probability \(p = 0.030684414569701078\).
Step 8 ：Final Answer: The probability that the mean of a random sample of 14 pregnancies is less than 258 days is approximately \(\boxed{0.0307}\).
