The size of the left upper chamber of the heart is one measure of cardiovascular health. when the upper left chamber is enlarged, the risk of heart problems is increased. a paper described a study in which the left atrial size was measured for a large number of children age 5 to 15 years. based on this data, the authors concluded that for healthy children, left atrial diameter was approximately normally distributed with a mean of 26.2 mm and a standard deviation of 4.3 mm. (a) approximately what proportion of healthy children have left atrial diameters l

a)

b)

c)

d)

Step-by-step explanation:

1) 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".  

2) Part a : Approximately what proportion of healthy children have left atrial diameters less than 24 mm

Let X the random variable that represent the size of the left upper chamber of the heart 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 excel or a table, on this way:

3) Part b : Approximately what proportion of healthy children have left atrial diameters greater than 32 mm

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

We can use the complement rule and we can find this probability using excel or a table, on this way:

4) Part c : Approximately what proportion of healthy children have left atrial diameters between 25 and 30 mm

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

And we can find this probability on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

5) Part d: For healthy children, what is the value for which only about 20% have a larger left atrial diameter?

For this case we want a value that accumulates 0.2 of the area on th right tail of the distribution. And for this we can use the Z score in order fo find the X value.

First we find a z score that accumulates 0.2 of the area on the right tail and 0.8 of the area on the left, and this value is z=0.842

If we solve for X w got: