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 114000 cells. The FDA then tasks you with checking to see if this is accurate. You collect a random sample of 55 specimens (1 cc each) which results in a sample mean of 771598 pus cells. Use this sample data to create a sampling distribution. Assume that the population mean is equal to the FDA's legal limit and see what the probability is for getting your random sample. a. Why is the sampling distribution approximately normal? Sample size is large enough, so that the distribution of the sampling distribution is approximately normal by central limit theorem. b. What is the mean of the sampling distribution? 750,000 c. What is the standard deviation of the sampling distribution? 15372 d. Assuming that the population mean is 750,000, what is the probability that a simple random sample of 55 1 ce specimens has a mean of at least 771598 pus cells? 0.0793 e. Is this unusual? Use the rule of thumb that events with probability less than 5% are considered unusual. No Yes

Answer 1

a. Why is the sampling distribution approximately normal?

Sample size is large enough, so that the distribution of the sampling distribution is approximately normal by central limit theorem.

(b) 750,00

(c ) 15371.756

(d)

we know that

I N - g/n

so

Z 771598 – 750000 114000/55 1.418

P(
T
$\geq$ 771598 ) = 1- area from left to (z=1.418)

= 1 - 0.921905

= 0.078095 Answer

(e) No

(f)

1.Population parameter = population mean =750000
2.sample size = 55 , statistics =
(the sample mean)
3.The distribution of the random variable
which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as the sample size(n), increases.
4.We know the standard deviation of the population. Selection of 55 specimens randomly is a sampling distribution.
5. If n > 30 ,sample is called large .Here sample size is large so we can use the cental limit theorem.
The sampling distribution of the mean approaches a normal distribution as the sample size increases.