A survey of 50 lobster fishermen on Funafuti (an island in Tuvalu), found that they catch an average of 32 pounds of lobster per day with a standard deviation of four pounds. If a fisherman is selected randomly, what is the probability that his catch is less than 20 pounds?
0.29
To find the probability that a fisherman's catch is less than 20 pounds, we need to use the concept of the normal distribution. Since we have the average and standard deviation of the sample, we can assume that the distribution of catches among the fishermen follows a normal distribution.
To calculate the probability, we need to convert the given information to a standardized z-score. The formula for calculating the z-score is:
z = (x - μ) / σ
where:
- z is the standardized z-score
- x is the value we want to find the probability for (in this case, 20 pounds)
- μ is the mean of the distribution (32 pounds)
- σ is the standard deviation of the distribution (4 pounds)
Plugging in the values into the formula, we get:
z = (20 - 32) / 4
z = -12 / 4
z = -3
Now that we have the z-score, we need to find the corresponding probability using a z-table or a statistical calculator. The z-table provides the area under the standard normal curve for different z-scores. In this case, we are interested in finding the area to the left of the z-score -3.
Looking up -3 in the z-table, we find that the area to the left of z = -3 is approximately 0.0013. This indicates that the probability of a fisherman's catch being less than 20 pounds is approximately 0.0013 or 0.13%.
Therefore, the probability that a randomly selected fisherman's catch is less than 20 pounds is around 0.0013 or 0.13%.