Question 1161701: Please help me with this question
The lengths of pregnancies are normally distributed with a mean of 267 days and a standard deviation of 15 days.
a. Find the probability of a pregnancy lasting 308 days or longer.
b. If the length of pregnancy is in the lowest 3​%, then the baby is premature. Find the length that separates premature babies from those who are not premature.
a. The probability that a pregnancy will last 308 days or longer is
​(Round to four decimal places as​ needed.)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! mean is 267 days
standard deviation is 15 days.
probability of 308 days or longer is the the area to the right of the z-score of 308 days.
to find this area, do the following:
z = (x - m) / s
z is the z-score
x is the raw score
m is the raw mean
s is the standard deviation.
the formula becomes:
z = (308 - 267) / 15 = 2.73 rounded to two decimal laces.
look up in the z-score table for the area to the left of a z-score of 2.73.
you will find that area to be equal to .99683
take 1 minus that to get the area to the right of that z-score.
it will be equal to .00317
that's the probability that the pregnancy will last 308 days or longer.
to find the lowest 3%, look for an area in the z-score table that shows .03 to the left of the indicated z-score.
you will find that an area of .03005 is associated with a z-score of -1.88 and an area of .02938 is associated with a z-score of -1.89.
this indicates your answer is somewhere between those two z-scores
interpolation tells you that the z-score is approximately -1.880746269.
use of a calculator is more exact and tells you that the z-score is -1.88079361.
if you round to 3 decimal places, both calculators tell you that the z-score is equal to -1.881.
that amount of detail is usually sufficient for most problems.
to find the raw score, use the z-score formula and solve for x.
the formula of z = (x - m) / s becomes:
-1.881 = (x - 2.67) / 15
solve for x to get:
x = -1.881 * 15 + 2.67 = 238.785.
what this says is any pregnancy to term less than 238.785 days is considered premature.
you can probably round to the nearest number of days which would be 239.
the table i used can be found at https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf
the calculator i used to verify the results can be found at http://davidmlane.com/hyperstat/z_table.html
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