In a certain population of fish the lengths of the individua

In a certain population of fish, the lengths of the individual fish follow approximately a normal distribution with mean 54.0 mm and standard deviation 4.5 mm. We saw in Example 4.3.1 that in this situation, 65.68% of the fish are between 51 and 60 mm long. Suppose a random sample of four fish is chosen from the population. Find the probability that all four fish are between 51 and 60 mm long. the mean length of the four fish is between 51 and 60 mm.

Solution

Given X~Normal(mean=54, s=4.5)

b. P(51<xbar<60)

=P((51-54)/(4.5/2) <(xbar-mean)/(s/n) <(60-54)/(4.5/2))

=P(-1.33<Z<2.67)

=0.9044 (check standard normal table)