Final answer:
The probability that the mean pregnancy of 100 randomly selected women falls between 254 days and 266 days is approximately 0.1586.
Step-by-step explanation:
To find the probability that the mean pregnancy of 100 randomly selected women falls between 254 days and 266 days, we need to use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means will approach a normal distribution, regardless of the shape of the population distribution, as the sample size increases.
First, we need to find the z-scores for the lower and upper limits of the mean pregnancy range. The z-score is calculated using the formula:
z = (x - μ) / σ
where x is the sample mean, μ is the population mean, and σ is the population standard deviation. In this case, x is (254 + 266) / 2 = 260, μ is 246, and σ is 25. Plugging in these values, we can calculate the z-scores as z1 = (260 - 246) / 25 = 0.56 and z2 = (260 - 246) / 25 = 0.56, respectively.
Next, we need to find the probability associated with these z-scores using a standard normal distribution table or a calculator. The probability of having a z-score less than or equal to 0.56 is approximately 0.7123, and the probability of having a z-score less than or equal to 0.56 is also 0.7123.
To find the probability of the sample mean falling between 254 days and 266 days, we subtract the probability associated with the lower z-score from the probability associated with the upper z-score: P(254 ≤ X ≤ 266) = P(X ≤ 0.56) - P(X ≤ 0.56) = 0.7123 - 0.7123 = 0.1586.