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EX 8 Practice and Application

A simple random sample of size n=40 is obtained from a population with a mean of 20 and a standard deviation of 5. Is the sampling distribution normally distributed? Why?

- Yes, the sampling distribution is normally distributed because the sample size is greater than 30.
- Yes, the sampling distribution is normally distributed because the population is normally distributed.
- No, the sampling distribution is not normally distributed because the population is not normally distributed.
- No, the sampling distribution is not normally distributed because the population mean is less than 30.

A population has parameters μ=136μ=136 and σ=3σ=3. You intend to draw a random sample of size n=62n=62.

What is the mean of the distribution of sample means?

μx¯=

What is the standard deviation of the distribution of sample means?

(Report answer accurate to 2 decimal places.)

σx¯=

A population of values has a normal distribution with μ=41.8μ=41.8 and σ=94.7σ=94.7. You intend to draw a random sample of size n=24n=24.

Find the probability that a single randomly selected value is less than 53.4.

*P*(*X* < 53.4) =

Find the probability that a sample of size n=24n=24 is randomly selected with a mean less than 53.4.

*P*(*M* < 53.4) =

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact *z*-scores or *z*-scores rounded to 3 decimal places are accepted.

A manufacturer knows that their items have a lengths that are skewed right, with a mean of 7 inches, and standard deviation of 1 inches.

If 41 items are chosen at random, what is the probability that their mean length is greater than 7.2 inches?

(Round answer to four decimal places)

A particular fruit’s weights are normally distributed, with a mean of 502 grams and a standard deviation of 6 grams.

If you pick 14 fruits at random, then 18% of the time, their mean weight will be greater than how many grams?

Give your answer to the nearest gram.

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 77000 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 767788 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 40 1 cc specimens has a mean of at least 767788 pus cells?

e. Is this unusual? Use the rule of thumb that events with probability less than 5% are considered unusual.

- Yes
- No

f. Explain your results above and use them to make an argument that the assumed population mean is incorrect. (6 points) Structure your essay as follows:

- 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.

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