Question

(a) According to the Ministry of Health, the height of Guinean travellers who were quarantined in Tamale for the novel coronavirus were normally distributed about a mean of 160cm and a standard deviation of 8cm. Find the probability that a traveller selected at random has (i) Height between 148cm and 175cm (ii) Height above 164cm
(iii) Height below 179cm

Answer 1

Let X = the random variable denoting the height of a Guinean traveller who was quarantined in Tamale for the novel corona virus

Then by the problem,

X ~ N($\mu$ = 160, $\sigma$² = 8²)

Therefore,

Z = $\frac{X-\mu}{\sigma}$ ~ N(0, 1), Z is the standard normal variate

(i) The probability that a traveler selected at random has height between 148 cm and 175 cm

= P(148 $\leq$ X $\leq$ 175)

= P($\frac{148-160}{8}\leq\frac{X-160}{8}\leq\frac{175-160}{8}$)

= P(-1.5 $\leq$ Z $\leq$ 1.88)

= $\Phi$(1.88) - $\Phi$(-1.5)

= 0.9699 - 0.0668

= 0.9031

Answer: The probability that a traveler selected at random has height between 148 cm and 175 cm is 0.9031.

(ii) The probability that a traveler selected at random has height above 164 cm

P(X $>$ 164)

= P($\frac{X-160}{8}>\frac{164-160}{8}$)

= P(Z $>$ 0.5)

= 1 - P(Z $\leq$ 0.5)

= 1 - $\Phi$(0.5)

= 1 - 0.6915

= 0.3085

Answer: The probability that a traveler selected at random has height above 164 cm 0.3085.

(iii) The probability that a traveler selected at random has height below 179 cm

P(X < 179)

= P($\frac{X-160}{8}<\frac{179-160}{8}$)

= P(Z < 2.38)

= $\Phi$(2.38)

= 0.9913

Answer: The probability that a traveler selected at random has height below 179 cm is 0.9913.