Statistical Hypothesis Testing for Discretionary Spending

by | Jul 30, 2024 | Marketing

The Federal Reserve stated in its latest report that the average amount of discretionary spending for a family of four last year was $3,457. You would like to test The Federal Reserves claim to see if this changed at the .05 level of significance, so you randomly select 50 families and get a sample mean of $3,635 with a standard deviation of $925. Chris OByrne 2024
A. What will be the alternative hypothesis? (just need to state 1-tail -Lower, 1-tail-Upper or 2-tail – just put one of those 3 answers in the body of the discussion board).
B. The value of the test statistic and which test statistic you used (example: z = 1.23 or t = 1.234) and make sure to round correctly depending on which test statistic you will be using.
C. What is the probability of the test statistic for this specific problem? (take into consideration if this is 1-tail or 2-tail test) Chris OByrne 2024
D. What conclusion will you come to? (just need to state Accept Ho or Reject Ho and Accept Ha just put one of those 2 answers in the body of the discussion board).
E. Using the critical value approach, at what critical value or values will you start rejecting the null hypothesis? (make sure you are very careful and detailed with your answer, pay attention to signs and values)

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.

Sample Answer

Sample Answer

 

Statistical Hypothesis Testing for Discretionary Spending

In this analysis, we will evaluate the Federal Reserve’s claim regarding the average discretionary spending of a family of four using a hypothesis testing approach. The key components of hypothesis testing will be addressed step-by-step.

A. Alternative Hypothesis

Since we want to see if there’s a change in the average amount of discretionary spending (either an increase or decrease), the appropriate alternative hypothesis is:

– 2-tail

B. Test Statistic Calculation

To calculate the test statistic, we will use the following formula for the t-test since the sample size is less than 30 and we have the sample standard deviation:

[
t = \frac{\bar{x} – \mu}{s / \sqrt{n}}
]

Where:

– (\bar{x}) = sample mean = $3,635
– (\mu) = population mean (Federal Reserve’s claim) = $3,457
– (s) = sample standard deviation = $925
– (n) = sample size = 50

Plugging in the values:

[
t = \frac{3635 – 3457}{925 / \sqrt{50}}
]
[
t = \frac{178}{130.43} \approx 1.36
]

Thus, the value of the test statistic is:

– t ≈ 1.36

C. Probability of the Test Statistic

To find the probability associated with this t-value, we need to refer to the t-distribution table (or use statistical software) with:

– Degrees of freedom (df) = n – 1 = 50 – 1 = 49

Using a t-table or calculator, we find:

– For t = 1.36 and df = 49, the corresponding two-tailed p-value is approximately 0.18.

D. Conclusion

Given our significance level ((\alpha)) of 0.05, and since our p-value (0.18) is greater than 0.05, we conclude:

– Accept (H_0)

E. Critical Value Approach

For a two-tailed test at the 0.05 significance level, we need to find the critical t-values for df = 49.

Using a t-table:

– The critical values for a two-tailed test at (\alpha = 0.05) are approximately ±2.009.

Thus, we will reject the null hypothesis if our test statistic is less than -2.009 or greater than +2.009.

In summary:

– Critical values: -2.009 and +2.009
– We will not reject (H_0) since our calculated t-value (1.36) falls within the range of -2.009 and +2.009.

This thorough analysis leads us to conclude that there is insufficient evidence to suggest that the average discretionary spending for a family of four has significantly changed from the Federal Reserve’s reported amount of $3,457.

 

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