Step 1:

(a)

Let the random variable  be length of pregnancy.

The mean of random variable is 268 days.

The standard deviation of random variable is 15 days.

Find the probability of a pregnancy lasting 308 days or longer.

.

Now calculate the Z-score for the given probability.

,

Where is the length of pregnancy

is the mean and is the standard deviation.

For , Z-score is

From the Z-score table, Area of the region when Z = 2.67 is 0.9962.

The probability of a pregnancy lasting 308 days or longer is 0.0038.

Solution:

(a) The probability of a pregnancy lasting 308 days or longer is 0.0038.

Step 2:

(b)

Find the length of the pregnancy.

The length of the pregnancy is in the lowest 3% = 0.03.

Area of the region is 0.03

Z-score when area of the region is 0.03 is .

The length of the pregnancy is 240 days.

Solution:

(b) The length of the pregnancy is 240 days.