Walleye is a common game fish. Adult walleye have a length with a mean of 48 cm and a standard deviation of 3 cm and the distribution of lengths is approximately Normal. What fraction of walleye are shorter than 45 cm in length?
15.9% of the fish have lengths more than 1 standard deviation below the mean
45-48 divided by 3
-1 = z score
find z< -1 use a z-table or a calculator to find the p-value
which is 15.9%
To determine the fraction of walleye that are shorter than 45 cm in length, we need to calculate the cumulative probability up to 45 cm using the normal distribution.
The first step is to standardize the value of 45 cm using the z-score formula:
z = (x - μ) / σ
where:
x = 45 cm (the value we want to calculate the probability for)
μ = 48 cm (mean)
σ = 3 cm (standard deviation)
Substituting the values, we get:
z = (45 - 48) / 3
z = -1
Once we have the z-score, we can use a standard normal distribution table or a statistical calculator to find the cumulative probability associated with the z-score.
Using either method, we find that the cumulative probability for a z-score of -1 is approximately 0.1587.
Therefore, approximately 15.87% (or about 0.1587) of walleye are shorter than 45 cm in length.