Question

The heights of children two years old are normally distributed with a mean of 80cm and a standard deviation of 3.6cm. Pediatricians regularly measure the heights of toddlers to determine whether there is a problem. There may be a problem when a child is in the top or bottom 5% of heights.

a) Determine the heights of two-year old children that could be a problem.

b) Fund the probability of these events

i. A two-year old child is taller than 90 cm

ii. A two-year old child is shorter than 85 cm

iii. A two-year old child is between 75 and 85 cm

Answer 1

a) Determine the heights of two-year old children that could be a problem.

b) Fund the probability of these events

i. A two-year old child is taller than 90 cm

ii. A two-year old child is shorter than 85 cm

iii. A two-year old child is between 75 and 85 cm

a) P(Z<z) = 0.05

$z = -1.645 \\ \frac{x-80}{3.6} = -1.645 \\ x = 74.08 \;cm \\ P(Z>z) = 0.05 \\ z = 1.645 \\ \frac{x-80}{3.6} = 1.645 \\ x = 85.92 \;cm$

The heights of two-year old children that could be a problem: less than 74.08 cm and greater than 85.92 cm.

b)

i. $P(X>90) = P(Z> \frac{90-80}{3.6})$

= P(Z>2.77)

= 1 – P(Z<2.77)

= 1 – 0.8437

= 0.1563

ii. $P(X<85) = P(Z<\frac{85-80}{3.6})$

= P(Z<1.38)

= 0.9162

iii. $P(75<X<85) = P(\frac{75-80}{3.6}<Z<\frac{85-80}{3.6})$

= P(-1.38<Z<1.38)

= P(Z<1.38) – P(Z<-1.38)

= 0.9162 – 0.0838

= 0.8324