Question

In studying the dynamics of fish populations, knowing the length of a species at different ages is critical. Fisheries Science (2015) published a study of the length distributions of sardines inhabiting Japanese waters. At two years of age, fish have a length distribution that is approximately normally distributed, with a population mean, µ= 20.20 centimeters (cm) and a population standard deviation, σ = 0.65 cm. Find the probability that a two-year-old sardine inhabiting Japanese waters is between P (20 ≤ x ≤ 21) cm long.

Answer 1

Given,

= 20.20 , = 0.65

We convert this to standard normal as

P( X < x) = P( Z < x - / )

So,

P( 20 <= X <= 21) = P( X <= 21) - P( X <= 20)

= P( Z <= 21 - 20.20 / 0.65) - P( Z <= 20 - 20.20 / 0.65)

= P( Z <= 1.2308) - P( Z <= -0.3077)

= P( Z <= 1.2308) - ( 1 - P( Z <= 0.3077) )

= 0.8908 - ( 1 - 0.6208 ) (Probability calculated from Z table)

= 0.5116