This paper aims to test the module ILOs using a practical real-life case-study. In this case study, you are going to play the role of an analyst for one of the corporates, let us call it Company-Z. Therefore, let us first introduce the case-study random variables: 

Y1 is the Company-Z daily sales along the period Pre-COVID-19 (November and December 2019); 

Y2 is the same Company-Z daily sales but during the period Post-COVID-19 (April and May 2020); 

X1 is the daily inventory holding cost during the period Pre-COVID-19 (November and December 2019); 

while X2 is the same daily inventory holding cost but during the period Post-COVID-19 (April and May 2020). 

Accordingly, in your analysis, you will depend on two main random variables over two time series. The first random variable is the Company-Z daily sales, and the second random variable is the Company-Z daily inventory holding cost. Therefore, the two random variables will be chosen over two periods of time: the first is (Pre-COVID-19: November and December 2019), and the second is (Post-COVID-19: April and May 2020). Your task is to prepare a comprehensive report to the company, fulfilling specific requirements outlined below in point (4).

1. a). What are the appropriate descriptive statistics to summarize the Company-Z daily sales in Pre- and Post- COVID-19 Y1 & Y2? Can you visualize both random variables separately using the graphing technique? Explain why you used these descriptive statistics and this graphing technique?      
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3. b). Calculate the average daily inventory holding cost during Pre- and Post- COVID-19 X1& X2? Explain whether this average is considered as population parameters or sample statistics? Justify your answer?    
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5. c). What is the 95 percent confidence intervals for the average daily inventory holding cost Pre- and Post- COVID-19 X1& X2? And what do you conclude by comparing these intervals? Also what is the 99 percent confidence interval for the average daily inventory holding cost Post- COVID-19 X2? And what do you conclude by comparing the 95 and 99 percent confidence intervals for the average daily inventory holding cost Post- COVID-19 X2?                                                                
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7. d). Suppose there is a hypothesis arguing that the population mean of the daily inventory holding cost is 1.5 times the value of average daily inventory holding cost during the selected period (November and December 2019) Pre-COVID-19 X1. List the full analytical steps to test this hypothesis? Comment on the result and write your conclusion regarding the hypothesis?             
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9. e). To what extent is there a change in the pattern or the power of the relationship Pre- and Post- COVID-19 between the daily inventory holding cost X and the daily sales Y? Support your answer with the appropriate statistical evidences.                                                                              
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11. f). Discuss the appropriate statistical technique that can explain the variation in the daily sales because of the daily inventory holding cost Post-COVID-19 post COVID 19. Justify your answer.