Drums labeled 30 L are filled with a solution from a large vat. The amount of solution put into each drum is random with mean 30.01 L and standard deviation 0.1 L.

a. What is the probability that the total amount of solution contained in 50 drums is more than 1500 L?

b. If the total amount of solution in the vat is 2401 L, what is the probability that 80 drums can be filled without running out?

c. How much solution should the vat contain so that the probability is 0.9 that 80 drums can be filled without running out?

Answer:

Step 1 of 3:

(a)

In this question, we are asked to find the probability that the total amount of solution contained in 50 drums is more than 1500.

Drums labeled are filled with a solution from a large vat.

The amount of solution has mean and standard deviation .

We need to find

Let is the amount of solution in drums.

Let = .

Hence can be approximately normal distributed.

CLT specifies that and for sum of the sample items.

Hence we achieve,

Mean

Standard deviation

Now we will calculate the score because our distribution is approximately normal.

thus the score of 1500 is

=

=

= -0.7071

From the z table, the area to the left of is 0.2206.

Therefore

Hence the probability that the total amount of solution contained in 50 drums is more than 1500is 0.7794.

Step 2 of 3:

(b)

In this question, we are asked to find the probability that 80 drums can be filled without running out.

If the total amount of solution in the vat is 2401.

We need to find

Let is the amount of solution in drums.

Let = .

Mean

Standard deviation

.Now we will calculate the score because our distribution is approximately normal.

thus the score of 2401 is

=

=

= 0.2236

From the z table, the area to the left of is 0.5948.

Therefore

Hence the probability that the total amount of solution contained in 80 drums is 2401is 0.5948.