Statistics Practice and Application

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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.

 

 

Sample Solution

Statistics Practice and Application

The distribution of sample means is defined as the set of means from all possible random samples of a specific size (n) selected from a specific population. The distribution has well-defined (and predictable) characteristics that are specified in the Central Limit Theorem. The mean of the distribution of sample means is called the Expected Value of M and is always equal to the population mean (mu). the concept of the distribution of sample means and its characteristics should be intuitively reasonable: you should realize that sample means are variable. If two or more samples are selected from the same population, the two samples probably will have different means. The sampling distribution of the sample mean: if repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population mean is (mu) and the population standard deviation is (sigma) then the mean of all sample means (x-bars) is population mean (mu).

There are principally three management objectives. First, it ensures organizational targets and goals are met with the minimum cost and least waste. The subsequent objective looks after welfare and health and safety for the staff. The third objective protects machinery and organization resources and includes human resources. Planning involves coming up with missions, objectives and actions to accomplish them. It needs decision making, which is choosing action for the future courses among alternatives. Plans entail the overall purposes and purpose to the detailed actions taken. There is no real plan which subsists unless a decision or commitment of human and material resources has been put in place. Thus, before the making of decision all which exists are the planning study, proposal and analysis.

People who work together in groups to gain some goals should have duties to take part in. Usually, the roles have been defined and also structured by someone who ensures people contribute is unique ways towards the group efforts. Organizing thus forms part of management which involves in the establishment of intentional structures of people roles to fill organization. Intentional in all the tasks are necessary to come up with goals which are assigned to people who can perform maximally. The purpose of any organizational structure is to assist in the creation of environment for human performance. However, designing of organizational structures in not a simple managerial task since most of problems are come across in making of structures which fit situations to include defining the types of jobs which must be done and getting of human power to do them.

Staffing entails filling and ensuring filled, the positions in the organizational structure. This is performed by identification of the work-force requirements, recruiting and inventorying of the people available, selecting, promoting, placing, appraising, compensating, planning the careers and training or developing current jobholders and both candidates to achieve their duties efficiently and effectively (Koontz

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