How many of the 250 perch netted by a fisherman would you expect to have a mass more than 400g if the mass of the perch are normally distributed with a mean of 300g and a standard deviation of 80g.
400 is 100/80 = 1.25 standard deviations away from the mean. look up 1.25 in a cumulative normal distribution table (probably in your stats book). I get .8944. Meaning 89.44% of the time, each perch will be under 400g. Take it from here.
To find out how many of the 250 perch are expected to have a mass of more than 400g, we need to use the properties of the normal distribution.
Step 1: Find the z-score
The z-score measures how many standard deviations a value is from the mean. We can calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we are interested in (400g), μ is the mean (300g), and σ is the standard deviation (80g).
z = (400 - 300) / 80
z = 1.25
Step 2: Find the probability
The z-score can be used to find the probability of a value occurring in the normal distribution. We can use a standard normal distribution table or a calculator to find the probability associated with the z-score.
Using a standard normal table or calculator, we find that the probability associated with a z-score of 1.25 is approximately 0.8944.
Step 3: Convert probability to count
The probability calculated in the previous step corresponds to the area under the curve to the left of the z-score. To find the count of perch with a mass greater than 400g, we need to subtract this probability from 1 (since we are interested in the area to the right of the z-score).
count = (1 - probability) * total count
count = (1 - 0.8944) * 250
count = 0.1056 * 250
count ≈ 26.4
Therefore, we would expect around 26 perch out of the 250 to have a mass more than 400g.