Question

Applying the Central Limit Theorem:

The amount of contaminants that are allowed in food products is determined by the FDA (Food and Drug Administration). Common contaminants in cow milk include feces, blood, hormones, and antibiotics. Suppose you work for the FDA and are told that the current amount of somatic cells (common name "pus") in 1 cc of cow milk is currently 750,000 (note: this is the actual allowed amount in the US!). You are also told the standard deviation is 127000 cells. The FDA then tasks you with checking to see if this is accurate.

You collect a random sample of 40 specimens (1 cc each) which results in a sample mean of 793576 pus cells. Use this sample data to create a sampling distribution.

d. Assuming that the population mean is 750,000, what is the probability that a simple random sample of 40 1 cc specimens has a mean of at least 793576 pus cells?

Answer 1

Solution:

Given:

Mean =

Standard deviation =

Sample size = n = 40

Sample mean =

Use this sample data to create a sampling distribution.

Applying the Central Limit Theorem:

Since sample size n = 40 ( n > 30 ) is large , we can use Central limit theorem which states that for large sample size n ,
sampling distribution of sample mean is approximately normal with mean of sample means:

and standard deviation of sample means is:

Part d) Assuming that the population mean is 750,000, what is the probability that a simple random sample of 40 1 cc specimens has a mean of at least 793576 pus cells?

that is find:

Thus find z score for

thus we get:

Look in z table for z = 2.1 and 0.07 and find corresponding area.

P( Z < 2.17) = 0.9850

thus