Answer to Question #272489 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

Let $X=$ sample mean: $X\sim N(\mu, \sigma^2/n).$

Given $\mu=15, \sigma=5, n=50.$

$$P(14<X<16.5)=P(X<16.5)-P(X\leq 14)$$

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

$$\approx P(Z<2.1213)-P(X\leq -1.4142)$$

$$\approx0.98305-0.07865\approx 0.9044$$

The probability that the average length of those fish (i.e. sample mean X ) is between 14 cm and 16.5 cm is 0.9044.