Step 1 :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 number of samples we would expect to have a sample mean of 261 days or more if 100 independent random samples of size 20 pregnancies were obtained from this population.
Step 3 :To solve this, we need to calculate the z-score for a sample mean of 261 days, and then use the standard normal distribution to find the probability of obtaining a z-score greater than or equal to this value.
Step 4 :The z-score is calculated as \((\text{sample mean} - \text{population mean}) / (\text{standard deviation} / \sqrt{\text{sample size}})\).
Step 5 :Substituting the given values, we get \(z = -1.3975424859373686\).
Step 6 :The probability of obtaining a z-score greater than or equal to a given value is found by looking up the value in a standard normal distribution table or using a function that calculates this probability.
Step 7 :Using the calculated z-score, we find that the probability is \(p = 0.9188747500763841\).
Step 8 :Once we have the probability, we can multiply it by the number of samples (100) to find the expected number of samples with a mean of 261 days or more.
Step 9 :Doing this, we get \(\text{expected samples} = 91.88747500763841\).
Step 10 :The expected number of samples with a mean of 261 days or more is approximately 92. However, the question asks for the number of samples we would expect to have a sample mean of 261 days or more, so we should round this number to the nearest whole number.
Step 11 :Final Answer: \(\boxed{92}\)