Y (in thousands of dollars) Date X 1998 79.1 55.6 1999 79 54.8 2000 80.2 55.4 2001 80.5 55.9 2002 81.2 56.4 2003 80.8 57.3 2004 81.2 57 2005 80.7 57.5 2006 80.3 56.9 2007 79.4 55.8 2008 78.6 56.1 2009 78.3 55.7 2010 78.3 55.7 2011 77.8 55 2012 77.7 54.4 2013 77.6 54 2014 77.6 56 2015 78.5 56.7 2016 78.3 56.3 2017 78.5 57.2 2018 78.9 57.8 2019 79.8 58.7 2020 80.4 59.3 2021 80.7 59.9 Part One: Analyzing Descriptive Statistics. Questions: (a) Estimate the overall level by measuring the mean level of optimism over the 24 year period. What can you infer from that? (b) Estimate the level of dispersion and what can you infer from that as well? (c) Many economists have indicated that, historically, the true average level of optimism consumers have exhibited is 80.65 points. Considering that the distribution of the sample data came from a normally distributed population: - - At the 5% level of significance, does the data support the claim many economists indicated that the true average level of optimism is 80.65 points? Determine its p-value. Construct a 95% confidence interval for the true average level of optimism. For all three methods you conducted, what can you infer from them? Please explain in the context of the assignment and not mathematically. Consider if we decide to adjust our level of significance from 5% to 1%, what can you infer from this? (d) At the same time, when economists have also claimed that the true average level of spending, annually, on luxury items is $55.2 (in thousands). Again, considering that it came from a normally distributed population: - At the 5% level of significance, does our sample data support the claim made by economists regarding that, historically, consumers spend, on average, $55.2 annually? Compute its p-value. Construct a 95% confidence interval for the true mean dollar amount spent on luxury items annually. Again, from all three methods, what can you infer from the sample data? If we are to adjust our level of significance to 1%, will our conclusions change as a result? Explain why or why not. Consider a study on consumer behavior. We want to learn more about the wealth effect in our economy. As we note, when a consumer's wealth rises, all things considered fixed, that consumer will be willing to spend more on luxury items and vice versa. We want to test the hypothesis how consumers' overall level of optimism affect their decisions to purchase more or fewer luxury items on an annual basis. To do so, we have collected data over the last 24 years and our dependent (endogenous) variable is the average dollar amount consumers, whom we interviewed (1500 annually), spent on luxury items. Our exogenous (explanatory) variable is the average level of optimism which was collected from the Conference Board. The level of consumer confidence is indexed from 0 to 100, 0 being absolutely pessimistic and 100 being perfectly optimistic.