In a recent study, the center for disease control and preventation reported that the diastolic blood pressures of adult women in the united states are apporcimately normally distrubuted with mean 80.5 and standard deviation 9.9.

a. Find the probability that a randomly selected adult woman will have a diastolic blood pressure greater than 85?
b. Find the 40th percentile of the blood pressures.
c. A random sample of size 81will be drawn from the population. What is the probability that the sample mean will be greater than 78.5?

a)

And we can find this probability using the complement rule and the normal standard table and we got:

b)

And if we solve for a we got

So the value of height that separates the bottom 40% of data from the top 60% is 77.995.  

c)

And using the complement rule a calculator, excel ir the normal standard table we have that:

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 diastolic pressure 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 using the complement rule and the normal standard table and we got:

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 0.4 of the area on the left and 0.6 of the area on the right it's z=-0.253. On this case P(Z<-0.253)=0.4 and P(z>-0.253)=0.6

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 40% of data from the top 60% is 77.995.  

Part c

Since the distribution for X is normal then the distribution for the sample mean is given by:

We can find the individual probability like this:

And using the complement rule a calculator, excel ir the normal standard table we have that: