Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact the fill volume is random with mean 12.01 oz and standard deviation 0.2 oz.

a. What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz?

b. If the population mean fill volume is increased to 12.03 oz what is the probability that the mean volume of a sample of size 144 will be less than 12oz?

Answer:

Step 1 of 2:

(a)

In this question, we are asked to find the mean volume of a random sample of bottles is less than oz.

Bottles filled by a certain machine are supposed to contain 12 oz of liquid.

Fill volume has mean 12.01 oz and standard deviation 0.2 oz.

Let denote the fill volume in the 144 bottles.

We need to find

Now the sample size is , which is a large sample.

Then according to Central Limit Theorem,

Let be a simple random sample from a population with mean

and variance .

Let

Hence can be approximately normal distributed.

CLT specifies that , variance and standard deviation

Hence our sample mean and standard deviation is,

= 0.01667

Therefore the score of 12 oz is

=

=

=

From the z table, the area to the left of is 0.2877.

Therefore = 0.2877, so only 28.77% of samples of size 144 will have fewer than 12 oz of fill volume.