Statistics Practice and Application

by | Nov 23, 2021 | Statistics

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

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.


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God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

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