(a) To find the probability of randomly selecting a woman and her pregnancy lasting less than 277 days, we need to use the normal distribution.
First, we calculate the z-score using the formula:
z = (x - mean) / standard deviation
Where x is the value we want to find the probability for, in this case, 277 days.
z = (277 - 266) / 16 = 11 / 16 = 0.6875
Next, we look up the z-score in the z-table (or use a calculator) to find the corresponding probability. The z-table tells us that the probability for a z-score of 0.6875 is approximately 0.7520.
Therefore, the probability of randomly selecting a woman and her pregnancy lasting less than 277 days is approximately 0.7520, or 75.20%.
(b) To find the number of days that the longest 25% of all pregnancies should last, we need to find the z-score that corresponds to the upper 25% of the normal distribution.
Using the z-table, we find that the z-score for the upper 25% is approximately 0.6745.
Now, we can use the formula:
x = mean + (z * standard deviation)
x = 266 + (0.6745 * 16) = 266 + 10.792 = 276.792
Rounding to the nearest day, the longest 25% of all pregnancies should last at least 277 days.
(c) To find the probability that 10 pregnant women will have a combined pregnancy length greater than 2,730 days, we need to use the Central Limit Theorem and the normal distribution.
First, we calculate the mean and standard deviation of the combined pregnancy length for the 10 pregnant women. Since the mean and standard deviation are given for a single pregnancy, we can use the properties of the normal distribution.
The mean of the combined pregnancy length is 10 * 266 = 2660 days.
The standard deviation of the combined pregnancy length is sqrt(10) * 16 = 50.91 days.
Next, we calculate the z-score using the formula:
z = (x - mean) / standard deviation
Where x is the value we want to find the probability for, in this case, 2730 days.
z = (2730 - 2660) / 50.91 = 70 / 50.91 = 1.375
Using the z-table (or a calculator), we find that the probability for a z-score of 1.375 is approximately 0.9157.
Therefore, the probability that these 10 pregnant women will have a combined pregnancy length greater than 2,730 days is approximately 0.9157, or 91.57%.