In a certain population of fish the lengths of individual fi

In a certain population of fish, the lengths of individual fish follow a normal distribution with mean 54 mm and standard deviation 4.5 mm

a) What is the probability that a random chosen fish is between 51 and 60 mm long?

b) Suppose we sample 4 fish, what is the probability that the mean length of the four fish is between 51 and 60 mm long?

X be the length of the fish

X~N(54, 4.5^2)

The mean is 54 and the std. dev of X is 4.5 so, 4.5^2 is the variance of X

Let Y=(X-54) / 4.5 then Y~ N(0,1)

(a)

P(51<X<60)=P(-3<X-54<6)=P(-3/4.5<Y<6/4.5)=P(Y<6/4.5)-P(Y<-3/4.5)=P(Y<1.333333)-P(Y<-0.6666667)

=0.6562962

(b)

X₁,X₂,X₃,X₄ follow iid N(54,4.5^2)

The sample mean is Z= (X₁+X₂+X₃+X₄ )/4 ~ N(54, 4.5^2 /4)

i.e, the sample mean follows normal with mean 54 and std. dev 4.5 / 2 =2.25

Let W = (Z-54) / (4.5/2) = (Z-54) / 2.25 then, W~N(0,1)

P(51<Z<60)=P(-3/2.25<W<6/2.25)= 0.9049584