Your best friend is 5 months pregnant and wants you to be there when the baby is born. Since the baby was conceived on Labor Day, the delivery might coincide with your annual trip to cabo. the doctor tells you that the gestation period for humans is normally distributed with a mean of 266 days and a standard deviation of 16 days.
a)the resort in cabo is booked for a week starting may 22nd (262 days after conception). what is the probability that the baby will be delivered during your 7 day trip?
b)suddenly you realize that if the baby is born during the 1st full week of may, you will have to miss your final. what is the probability that the baby is born after your statistics final on may 2nd?(242 days after conception)
c)starting to worry about the worse case scenario, you realize that there is a 1% chance that the pregnancy will be longer than how many days?
I bet you can do this !!!
266 - 262 = 4 days before mean
booked for 7 days so from 4 days before to two days after
like
262
263
264
265
266 mean
267
268
sigma = 16
so we are going from -4/16 to + 2/16 sigma from mean that is -.25 to +.125 from mean
integral of normal dist from -.25 to +.125 is about
.9 - .4 = .5 roughly
so you have about a fifty fifty chance
The rest of this is the same, wiggling around the normal distribution curve.
To answer these questions, we can use the concept of the standard normal distribution and Z-scores.
a) To find the probability that the baby will be delivered during your 7-day trip, we need to find the probability that the gestation period falls within the range of May 22nd to May 29th.
First, let's calculate the Z-score for May 22nd:
Z1 = (262 - 266) / 16 = -0.25
Next, let's calculate the Z-score for May 29th:
Z2 = (269 - 266) / 16 = 0.1875
To find the probability, we can use a Z-table or a calculator that can give us the probability associated with these Z-scores. The probability can be calculated as the area under the standard normal curve between Z1 and Z2.
Therefore, we can calculate:
P(Z1 < Z < Z2) = P(-0.25 < Z < 0.1875)
You can use a Z-table or a calculator to find the probability associated with Z1 and Z2. Subtracting the cumulative probabilities will give us the probability that the baby will be delivered during your trip.
b) To find the probability that the baby is born after your statistics final on May 2nd, we need to find the probability that the gestation period is longer than 242 days.
First, let's calculate the Z-score for May 2nd:
Z = (242 - 266) / 16 = -1.5
Using a Z-table or a calculator, find the probability associated with Z = -1.5. This probability will represent the likelihood that the baby is born after May 2nd.
c) To find the number of days that falls within the 1% chance of longer pregnancies, we need to find the Z-score that has an associated cumulative probability of 0.99 (since we want to find the 1% tail of the distribution).
First, find the Z-score associated with a cumulative probability of 0.99. You can use a Z-table or a calculator to find this value. Once you have the Z-score, use the formula:
Z = (X - 266) / 16
Rearrange the formula to solve for X, where X represents the number of days.
Therefore, X = (Z * 16) + 266
Substituting the Z-score associated with a cumulative probability of 0.99 will give you the number of days that represents the 1% chance of longer pregnancies.