The lengths of pregnancies are normally distributed with a mean of 268268 days and a standard deviation of 1515 days. a. Find the probability of a pregnancy lasting 307307 days or longer. b. If the length of pregnancy is in the lowest 33%, then the baby is premature. Find the length that separates premature babies from those who are not premature.

a)

And we can find this probability with the complement rule and with excel or a normal distribution table:

b)

And if we solve for a we got

So the value of length that separates the bottom 33% of data from the top 67% is 261.401.  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the lengths of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability with the complement rule and with excel or a normal distribution table:

Part b

For this part we want to find a value a, such that we satisfy this condition:

  (a)

  (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.33 of the area on the left and 0.67 of the area on the right it's z=-0.44. On this case P(Z<-0.44)=0.33 and P(z>-0.44)=0.67

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

So the value of height that separates the bottom 33% of data from the top 67% is 261.401.  

2.83941151 would be the answer you're looking for! : )

hello from mrbilldoesmath!

answer:   (7x + 6y) ( 49x^2 - 42xy + 36y^2)

discussion:

in general, the factorization of a^3 + b^3 is

a^3 + b^ 3 = (a + b) ( a^2 - ab + b^2)

in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:

343x^3 + 216y^3 = (7x)^3 + (6y)^3

if you think of a (above) as 7x and b as 6y the factorization becomes

343x^3 + 216y^3 =

((7x + 6y) (   (7x)^2 - 7x*6y + (6y)^2)   =

(7x + 6y) ( 49x^2 - 42xy + 36y^2)

you,

mrb

(a + b) (a^2 - a b + b^2)