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 111000 cells. The FDA then tasks you with checking to see if this is accurate. You collect a random sample of 45 specimens (1 cc each) which results in a sample mean of 776297 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?
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 45 1 cc specimens has a mean of at least 776297 pus cells?
e. Is this unusual? Use the rule of thumb that events with probability less than 5% are considered unusual. Describe the population and parameter for this situation. Describe the sample and statistic for this situation. Give a brief explanation of what a sampling distribution is. Describe the sampling distribution for this situation. Explain why the Central Limit Theorem applies in this situation. Interpret the answer to part d. Use the answer to part e. 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. Explain what the FDA should do with this information.

Answer 1

a. The sampling distribution is approximately normal due to the Central Limit Theorem. b. The mean of the sampling distribution equals the population mean. c. The standard deviation of the sampling distribution is the population standard deviation divided by the square root of the sample size.

Step-by-step explanation:

a. The sampling distribution is approximately normal because of the Central Limit Theorem. According to the theorem, when the sample size is sufficiently large (typically around 30 or more), the distribution of sample means will be approximately normal, regardless of the shape of the population distribution.

b. The mean of the sampling distribution is equal to the population mean. In this case, the assumed population mean is 750,000.

c. The standard deviation of the sampling distribution, also known as the standard error of the mean, is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the standard deviation is 111,000 / sqrt(45) = 16,536.49.

d. To find the probability that a simple random sample of 45 specimens has a mean of at least 776,297 pus cells, we can use the z-score formula. The z-score is calculated as (sample mean - population mean) / standard deviation. In this case, the z-score is (776,297 - 750,000) / 16,536.49 = 1.59. Using a standard normal distribution table or a calculator, we can find the probability associated with a z-score of 1.59, which is approximately 0.0569 or 5.69%.

e. When an event has a probability less than 5%, it is considered unusual. In this case, the probability of getting a sample mean of at least 776,297 pus cells is 5.69%, which is greater than 5%. Therefore, it is not considered unusual.

The population in this situation is all the possible cow milk specimens, and the parameter is the population mean, which is assumed to be 750,000. The sample is the randomly collected 45 specimens, and the statistic is the sample mean of 776,297 pus cells.

A sampling distribution is a probability distribution that describes the possible values of a sample statistic. In this case, the sampling distribution represents the possible means of random samples of 45 cow milk specimens.

The Central Limit Theorem applies in this situation because the sample size is sufficiently large (45 is typically considered large enough) and the theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution.

The answer to part d indicates that the probability of getting a sample mean of at least 776,297 pus cells is approximately 5.69%. This means that if the assumed population mean of 750,000 is correct and the distribution of pus cells in cow milk follows the same pattern, the observed sample mean is not unusual. If the assumed population mean is incorrect, it could indicate that the actual population mean is different from 750,000, either greater or less depending on whether the observed sample mean is higher or lower than the assumed mean.

The FDA should consider investigating further and potentially adjust the allowed amount of somatic cells in cow milk if the assumed population mean is found to be incorrect.