Question

The heights of 2-year-old children are normally distributed with a mean of 32 inches and
a standard deviation of 1.5 inches. 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. [10 Marks]
a. Determine the heights of 2-year-old children that could be a problem. [4 Marks]
b. Find the probability of these events:
(i) A 2-year-old child is taller than 36 inches [2 Marks]
(ii) A 2-year-old child is shorter than 34 inches [2 Marks]
(iii) A 2-year-old child is between 30 and 33 inches tall [2 Marks]

Answer 1

a. $P(Z<z)=0.05\to z=-1.645\to -1.645=\frac{x-32}{1.5}\to x=29.5.$

$P(Z>z)=0.05\to z=1.645\to 1.645=\frac{x-32}{1.5}\to x=34.5$

The heights less than 29.5 in and greater than 34.5 in could be a problem.

b.

(i) $P(X>36)=P(Z>\frac{36-32}{1.5})=P(Z>2.67)=0.0038.$

(ii) $P(X<34)=P(Z<\frac{34-32}{1.5})=P(Z<1.33)=0.9082.$

(iii)

$P(30<X<33)=P(\frac{30-32}{1.5}<Z<\frac{33-32}{1.5})=\\=P(-1.33<Z<0.67)=0.6568.$