Question

The length,x centimeters,of eels in a river may be assumed to be normally distributed with mean 48 and standard deviation 8.An angler an eel from the river.Determine the probability that be length of the eel is : a) exactly 60 cm b) less than 60 cm c) within 5% of the mean length

Answer 1

$X\sim N(\mu,\sigma^2)$

$$Z={X-\mu \over \sigma}\sim N(0,1)$$

Given $\mu=48,\sigma=8$

a)

$$P(X=60)=0$$

b)

$$P(X<60)=P(Z<{60-48 \over 8})=P(Z<1.5)\approx0.9332$$

c)

$$P(\mu-0.025\mu<X<\mu+0.025\mu)=$$

$$=P(X<0.975(48))-P(X<1.025(48))=$$

$$=P(X<49.2)-P(X<46.8)=$$

$$=P(Z<{49.2-48 \over 8})-P(Z<{46.8-48 \over 8})=$$

$$=P(Z<0.15)-P(Z<-0.15)\approx$$

$$\approx0.559618-0.440382\approx0.1192$$