Question

1. 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. The paper “Left atrial size increases with body mass index in children” (International Journal of Cardiology [2009]: 1–7) 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 diameters were approximately normally distributed with a mean of 26.4 mm and a standard deviation of 4.2 mm.

a. About what proportion of healthy children have left atrial diameters greater than 32 mm?

b. About what proportion of healthy children have left atrial diameters between 25 mm and 30 mm?

c. Children in the top 10% of left atrial diameters are at risk of heart problems. What is the left atrial diameter value (the original x value) describing the upper 10% of extreme left atrial diameters? (Hint: What percentile is this, draw a figure).

Answer 1

a) Let X be the left atrial diameters

then ,

To find , P( X> 32)

= P( z> 1.33)

= 0.0918

proportion children having left atrial diameter greater than 32mm is 0.0918 or 9.18%

b)

To find , P( 25<X<30)

= P( -0.33 < z < 0.86)

= P( -0.33 < z <0) +P( 0<z<0.86)

= 0.1293 + 0.3051 (from z table)

= 0.4344

proportion children having left atrial diameter between 25mm and 30 mm is 0.4344 or 43.44 %

c) To find x such that

P( X> x) = 0.10

From z table ,

P( z> 1.29) = 0.10

Thus , we get