Final answer:
The probability that the sample mean of nine women's pregnancy lengths is less than 270 days can be calculated using the Central Limit Theorem and normal distribution; it's approximately 77.34%.
Step-by-step explanation:
The question asks us to find the probability that the sample mean for a random sample of nine women's pregnancies is less than 270 days. The length of human pregnancies is normally distributed with a mean of 266 days and a standard deviation of 16 days.
To find the probability, we will use the Central Limit Theorem which states that the distribution of the sample means will be normally distributed with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n).
First, we calculate the standard deviation of the sample mean, which is σ/√n = 16/√9 = 16/3. Next, we find the z-score for 270 days using the formula Z = (X - μ) / (σ/√n). Plugging in our values, we get Z = (270 - 266) / (16/3) = 4 / (16/3) = 0.75.
Now, using a z-table or a normal distribution calculator, we can find the probability that Z is less than 0.75. This will give us the probability that the sample mean is less than 270 days.
Assuming the Z-score corresponds to a probability of approximately 0.7734, there is a 77.34% chance that the sample mean pregnancy is less than 270 days.