Catfish weights are normally distributed with a mean of 3.2 pounds and a standard deviation of 0.8 pounds. What is the probability that a randomly selected catfish weighs between 3 and 4 pounds?
A “champion” catfish is one that weighs more than 98% of all catfish. How much must a catfish weigh in order to earn “champion” status?
A “champion” catfish is one that weighs more than 98% of all catfish. How much must a catfish weigh in order to earn “champion” status?
1 Answer
1) 0.44005 or 44% roughly
2) 4.842 pounds
Explanation:
Question 1
We know the weights are normally distributed according to a mean
To estimate the probability that a randomly selected catfish weighs between 3 and 4 points requires that we convert the weights to a standard normal distribution
First, I'll denote
(where
Thus, we find:
These values make sense in the context of the problem; a weight of 3 pounds is below the mean, thus we expect a negative z-score, while a weight of 4 pounds is exactly 1 standard deviation of weight above the mean, hence a positive z-score of 1.
Now, to find the probability we are seeking we relate this problem to the probability that a z-score
Doing this requires a little cleverness on our part and the use of a z-score table (or calculator). We can use tables to determine the left-tail probability
From tables, we see that
#P(-0.25 < Z < 1) = 0.84134 - 0.40129 = 0.44005.
There's about a 44% chance of a randomly selected catfish weight falling between 3 and 4 pounds.
Question 2
To find the weight a catfish should be so that it's weight falls in the top 2% of all catfish (and thus 98% of all catfish weights are below it) requires us to work backwards from a z-score table. In this case, we need to look up the z-score
Looking up a z-score table tells us that
Using the z-score formula, we can now calculate the weight this corresponds to under our original distribution