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 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 26.7 mm and a standard deviation of 4.4 mm.

(a)

Approximately what proportion of healthy children have left atrial diameters less than 24 mm? (Round your answer to four decimal places.)

(b)

Approximately what proportion of healthy children have left atrial diameters greater than 32 mm? (Round your answer to four decimal places.)

(c)

Approximately what proportion of healthy children have left atrial diameters between 25 and 30 mm? (Round your answer to four decimal places.)

(d)

For healthy children, what is the value for which only about 20% have a larger left atrial diameter? (Round your answer to two decimal places.)

mm

Answer 1

Solution :

Given that mean μ = 26.7 and a standard deviation σ = 4.4

(a) P(x < 24) = P((x - μ)/σ < (24 - 26.7)/4.4)

= P(Z < -0.6136)

= 0.2697

=> The proportion of healthy children have left atrial diameters less than 24 mm is 0.2697

(b) P(x > 32) = P((x - μ)/σ > (32 - 26.7)/4.4)

= P(Z > 1.2045)

= 0.1142

=> The proportion of healthy children have left atrial diameters greater than 32 mm is 0.1142

(c) P(25 < x < 30) = P((25 - 26.7)/4.4 < (x - μ)/σ < (30 - 26.7)/4.4)

= P(-0.3864 < Z < 0.75)

= 0.4238

=> The proportion of healthy children have left atrial diameters between 25 and 30 mm is 0.4238

(d) for larger 20% , Z = 0.8416

=> X = μ + Z*σ

= 26.7 + 0.8416*4.4

= 30.40 (rounded)

=> the value 30.40 is for only about 20% have a larger left atrial diameter