The lengths of a particular animal\'s pregnancies are approximately normally dis

Question

The lengths of a particular animal's pregnancies are approximately normally distributed, with mean mu = 280 days and standard deviation sigma = 12 days. What proportion of pregnancies lasts more than 301 days? What proportion of pregnancies lasts between 259 and 286 days? What is the probability that a randomly selected pregnancy lasts no more than 277 days? A "very preterm" baby is one whose gestation period is less than 253 days. Are very preterm babies unusual? The proportion of pregnancies that last more than 301 days is 0.0401 The proportion of pregnancies that last between 259 and 286 days is 0.6514 The probability that a randomly selected pregnancy lasts no more than 277 days is 0.4013 A "very preterm" baby is one whose gestation period is less than 253 days Are very preterm babies unusual? The probability of this event is 0.0122, so it would be unusual because the probability is less than 0.05.

Explanation / Answer

We are given that

Mean=280

SD=12

1)

To find the proportion of pregnancies last more than 301 days, Because it is a normal distribution, we can use a z score.

z = (x - mean)/SD

z = (301 - 280)/12

z = 1.75

You can now determine the probability of getting a z-score this extreme or more extreme (the probability of having a pregnancy that lasts more than 301 days). You have to look this up on a z score table. Sometimes tables will give you the area to the left. The one I used gave the area to the left. Therefore, you would subtract that amount from 1.00 to get the area to the right since the whole area under the curve is equal to 1.

Thus, 1- 0.9599 = 0.0401.

So about 4.01% of pregnancies last more than 301 days.

2)

For finding the proportion of pregnancies last between 259 and 286 days.

For this question, we are going to find the two z scores, for 259 and 286. Then you are going to find the area in between these two points on the normal distribution curve. You can do this by finding the area to the left of the 286 point and then subtracting from that the area to the left of the 259 point. This will give you the area in between.

For x = 259 then z score is

z = (x - mean)/SD

z = (259 - 280)/12

z = -1.75

You have to look this up on a z score table.

0.0401

For x = 286 then z score is

z = (x - mean)/SD

z = (286 - 280)/12

z = 0.5

You have to look this up on a z score table.

0.6915

For finding the proportion of pregnancies last between 259 and 286 days. We can do this by finding the area to the left of the 286 point and then subtracting from that the area to the left of the 259 point. This will give you the area in between.

0.6915 – 0.0401 = 0.6514

So about 65.14% of pregnancies last between 259 and 286 days.

3)

To finding the probability that a randomly selected pregnancy lasts no more that 277 days

"No more than 277 days" suggests the area to the left of the point on the normal distribution (the proportion that it is below 277 days). So you would just take the probability to the left of that z score from the z table.

For x = 277 then z score is

z = (x - mean)/SD

z = (277 - 280)/12

z = -0.25

You have to look this up on a z score table.

0.4013

4)

A "very preterm" baby is one who gestation period is less that 253 days. Are very preterm babies unusual?

Find the z score for this point using the above formula

For x = 253 then z score is

z = (x - mean)/SD

z = (253 - 280)/12

z = -2.25

You have to look this up on a z score table.

0.0122

Therefore, the probability of this event is 0.0122, so it would be unusual because the probability is less than 0.05.

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