Question

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 ages 5 to 15 years. Based on this data, the authors conclude that for healthy children, left atrial diameter was approximately normally distributed with a mean of 26.5 mm and a standard deviation of 4.8 mm.

Required:
a. Approximately what proportion of healthy children has left atrial diameters less than 24 mm?
b. Approximately what proportion of healthy children has left atrial diameters greater than 32 mm?
c. Approximately what proportion of healthy children has left atrial diameters between 25 and 30 mm?
d. For healthy children, what is the value for which only about 20% have a larger left atrial diameter?

Answer 1

a) P [ X < 24 mm ] = 0,3015 or P [ X < 24 mm ] = 30,15 %

b) P [ X > 32 mm ] = 0,1251 or P [ X > 32 mm ] = 12,51 %

c) P [ 25 < X < 30 ] = 0,4964 or P [ 25 < X < 30 ] = 49,64 %

d) z(s) = 0,84

Explanation:

Normal Distribution N ( μ₀ ; σ ) is N ( 26,5 ; 4,8 )

a) P [ X < 24 mm ] = ( X - μ₀ ) / σ

P [ X < 24 mm ] = (24 - 26,5)/ 4,8 = - 0,5208 ≈ - 0,52

P [ X < 24 mm ] = - 0,52

And from z-table we find area for z score

P [ X < 24 mm ] = 0,3015 or P [ X < 24 mm ] = 30,15 %

b)P [ X > 32 mm ] = 1 - P [ X < 32 mm ]

P [ X < 32 mm ] = ( 32 - 26,5 ) / 4,8

P [ X < 32 mm ] = 5,5/4,8 = 1,1458 ≈ 1,15

P [ X < 32 mm ] = 1,15

And from z-table we get

P [ X < 32 mm ] = 0,8749

Then:

P [ X > 32 mm ] = 1 - 0,8749

P [ X > 32 mm ] = 0,1251 or P [ X > 32 mm ] = 12,51 %

c) P [ 25 < X < 30 ] = P [ X < 30 ] - P [ X < 25 ]

P [ X < 30 ] = 30 - 26,5 / 4,8 = 0,73

From z-table P [ X < 30 ] = 0,7673

P [ X < 25 ] = 25 - 26,5 / 4,8 = - 0,3125 ≈ - 0,31

From z-table P [ X < 25 ] = 0,2709

Then

P [ 25 < X < 30 ] = 0,7673 - 0,2709

P [ 25 < X < 30 ] = 0,4964 or P [ 25 < X < 30 ] = 49,64 %

d) If 20 %

z- score for 20% is from z-table

z(s) = 0,84