Answer to Question #272487 in Statistics and Probability for Cheppie

Question

The population of fish in a particular pond is known to have a mean length µ = 15 cm with standard deviation σ = 5. You catch fifty fish from the pond and measure their lengths. What is the probability that the average length of those fish (i.e. sample mean X ) is between 14 cm and 16.5 cm?

Answer 1

Mean (µ) = 15 cm

Standard deviation (σ) = 5

If fifty fish catch from the pond then the probability that the average length of those fish is between 14 cm and 16.5 cm would be:

$Z =\frac{\bar{X}-\mu}{\sigma /\sqrt{n}}$

$P(14<X<16.5)=P(\frac{14-15}{5 /\sqrt{50}} <Z<\frac{16.5-15}{5 /\sqrt{50}})$

$P(14<X<16.5)=P(-1.41 <Z<2.12)$

$P(14<X<16.5)=P(Z<2.12)-P(Z<-1.41)$

From the Z table;

$P(14<X<16.5)=0.9830-0.0793$

$P(14<X<16.5)=0.9037$