The amount of contaminants that are allowed in food products is determined by th

Question

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 arc told that the current amount of somatic cells (common name "pus") in 1 cc of cow milk is currently 750,000 You arc also told the standard deviation is 50000 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 762564 pus cells. Use this sample data to create a sampling distribution. a. Why is the sampling distribution approximately normal? b What is the mean of the sampling distribution? c. What is the standard deviation of the 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 762564 pus cells? e. Is this unusual? Use the rule of thumb that events with probability less than 5% are considered unusual. No Yes f. Explain your results above and use them to make an argument that the assumed population mean is incorrect. Structure your essay as follows: 1. describe the population and parameter for this situation. 2. Describe the sample and statistic for this situation. 3. Give a brief explanation of what a sampling distribution is. 4. Describe the sampling distribution for this situation. 5. Explain why the Central Limit Theorem applies in this situation. 6. Interpret the answer to part d. 7. Use the answer to part c. to argue that the assumed population mean is either correct or incorrect. If incorrect, indicate whether you think the actual population mean is greater or less than the assumed value. 8. Explain what the FDA should do with this information.

Explanation / Answer

a. Sampling distribution is approximately normal because sample size is more than 30 .

b Mean of sampling distribution xbar = 762564

c. Standard Deviation of sampling distribution s = population standard deviation = 50000

d. Standard error of sampling distribution se0= s/ n = 50000/ 40 = 7905.69

P - value = Pr(X >= 762564 ; 750000; 7905.69) = 1 -  Pr(X < 762564 ; 750000; 7905.69)

Z = (xbar - H)/ se0 = ( 762564 - 750000)/ 7905.69 = 1.59

so Pr(X >= 762564 ; 750000; 7905.69) = 1 - (1.59)

where is cumulative normal probability function.

P - value = 1 - 0.9441 = 0.0559

e. No, it is not unusual. events with probability less than 5% are considered normal.

f. Here FDA is checking the amount of somatic cells in 1cc of cow milk nad check that it the sample is under the actual permissible limit. A 40 sample specimen was selected and somatic cells in 1cc was measured by doing random sampling distribution. The sample is assumed as normal as sample size is above 30.

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