The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week. Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of .82 million tons.
a. What is the probability that the port handles less than 5 million tons of cargo per week (to 4 decimals)? b. What is the probability that the port handles 3 or more million tons of cargo per week( to 4 decimals)? c. What is the probability that the port handles between 3 million and 4 million tons of cargo per week ( to 4 decimals)?
a) To find the probability that the port handles less than 5 million tons of cargo per week, we need to standardize the value of 5 million using the given mean and standard deviation. We can then use the standardized value to find the corresponding probability using a standard normal distribution table.
First, we calculate the z-score:
z = (x - μ) / σ
where x is the value (5 million tons), μ is the mean (4.5 million tons), and σ is the standard deviation (0.82 million tons).
z = (5 - 4.5) / 0.82
z = 0.6097561
Next, we use the standard normal distribution table or a calculator to find the probability corresponding to the z-score of 0.6098. The probability is approximately 0.7272.
Therefore, the probability that the port handles less than 5 million tons of cargo per week is approximately 0.7272.
b) To find the probability that the port handles 3 or more million tons of cargo per week, we need to calculate the probability of the port handling less than 3 million tons and subtract it from 1.
First, we calculate the z-score for 3 million tons:
z = (3 - 4.5) / 0.82
z = -1.829268
Next, we find the probability corresponding to the z-score of -1.8293 using the standard normal distribution table or a calculator. The probability is approximately 0.033.
Finally, we subtract the probability from 1 to get the probability of handling 3 or more million tons:
1 - 0.033 = 0.967
Therefore, the probability that the port handles 3 or more million tons of cargo per week is approximately 0.967.
c) To find the probability that the port handles between 3 million and 4 million tons of cargo per week, we need to find the probability of handling less than 4 million tons and subtract the probability of handling less than 3 million tons.
First, we calculate the z-score for 4 million tons:
z = (4 - 4.5) / 0.82
z = -0.609756
Next, we find the probability corresponding to the z-score of -0.6098 using the standard normal distribution table or a calculator. The probability is approximately 0.2728.
Then, we calculate the probability of handling less than 3 million tons as explained in part b, which is approximately 0.033.
Finally, we subtract the probability of handling less than 3 million tons from the probability of handling less than 4 million tons:
0.2728 - 0.033 = 0.2398
Therefore, the probability that the port handles between 3 million and 4 million tons of cargo per week is approximately 0.2398.