An economist with a major bank wants to learn, quantitatively, how much spending on luxury goods and services can be explained based on consumers’ perception about the current state of the economy and what do they expect in the near future (6 months ahead).  Consumers, of all income and wealth classes, were surveyed.  Every year, 1500 consumers were interviewed.  The bank having all of the data from the 1500 consumers interviewed every year, computed the average level of consumer confidence (an index ranging from 0 to 100, 100 being absolutely optimistic) and computed the average dollar amount spent on luxuries annually.  Below is the data shown for the last 24 years.

Date                 X                     Y (in thousands of dollars)

1994                79.1                 55.6

1995                79                    54.8

1996                80.2                 55.4

1997                80.5                 55.9

1998                81.2                 56.4

1999                80.8                 57.3

2000                81.2                 57

2001                80.7                 57.5

2002                80.3                 56.9

2003                79.4                 55.8

2004                78.6                 56.1

2005                78.3                 55.7

2006                78.3                 55.7

2007                77.8                 55

2008                77.7                 54.4

2009                77.6                 54

2010                77.6                 56

2011                78.5                 56.7

2012                78.3                 56.3

2013                78.5                 57.2

2014                78.9                 57.8

2015                79.8                 58.7

2016                80.4                 59.3

2017                80.7                 59.9

Questions:

1. Measure the strength of the linear association between consumers’ moods and the dollar amounts spent on luxury items.
2. Construct the linear regression model for the dollar amount spent on luxury goods and services.
3. Explain how you would interpret the slope and the intercept of the regression model.
4. How well does our model fit the data? Explain what it means.
5. Do you think that measuring the level of optimism is a good predictor for trying to forecast future spending on luxury items?  Explain why or why not.
6. How would you be able to improve on the model?  You must provide a minimum of two specific ways to go about improving the model.
7. If the economist expects that, by year’s end, the average level of consumer confidence will hit 81.5 points, how much will be expected by consumers to spend on luxury items?

5) Polychlorinated biphenyls (PCBs) are organic chlorine compounds that were widely used as dielectrics and coolants in electrical systems in the past. They were found to be a major environmental contaminant in the 1960s. In a study, the mean PCB content at each of thirteen sites was reported for the years 1982 and 1996 (from “The ratio of DDE to PCB concentrations in Great Lakes herring gull eggs and its use in interpreting contaminants data”, Journal of Great Lakes Research 24 (1): 12-31, 1998). The data are below.

a) Which test would be more appropriate in this case: a t-test for the difference between two population means, or a paired t-test Why
b) Do the data provide sufficient evidence to support the claim that the mean PCB level has decreased in the region Be sure to check all assumptions, write the null and alternative hypotheses, calculate the appropriate test statistic, calculate the p-value, and state your conclusion.
c) Construct a 95% confidence interval for the mean decrease in PCB level.

6) Salt sensitivities for ten patients before and after antihypertensive treatment are below.

a) Are the conditions met for running a paired t-test Explain, addressing each condition. If you need to check for normality, use R to perform the usual diagnostic checks.
b) Regardless of your answer to part (a), run a paired t-test with R. Be sure to state the null and alternative hypotheses, the test statistic, the p-value, and your conclusion.
c) What is a good estimate for the mean sensitivity change

Is the number of tornadoes increasing? In the last homework, data on the number of tornadoes in the United States between 1953 and 2014 were analyzed to see if there was a linear trend over time. Some argue that it’s not the number of tornadoes increasing over time, but rather the probability of sighting them because there are more people living in the United States. Let’s investigate this by including the U.S. census count (in thousands) as an additional explanatory variable (data in EX11-24TWISTER.csv).

Perform a multiple regression using both year and census count as explanatory variables. Write down the fitted model. Are year and census count respectively significant in the MLR model?

The below contains the actual data on COVID-19 cases in Ghana from 1st April, 2020 to 25th May, 2020 as presented by the Ghana Health Service. Use the information provided to answer the following questions:

1. Construct a graph to show the trends of the COVID-19 confirmed cases and fatalities in Ghana, and speculate on the factors that might have influenced these trend

From the table below, use a simple linear regression analysis to establish the relationship that may exist between a) number of confirmed cases and deaths; b) number of confirmed cases and number of tests performed; c) number of confirmed cases and number of recoveries; and d) number of deaths and number of recoveries. Briefly discuss these relations.

In the mid-1990s, Colgate-Palmolive developed a new toothpaste for the U.S. market, Colgate Total, with an antibacterial ingredient that was already being successfully sold overseas. At that time, the word antibacterial was not allowed for such products by the Food and Drug Administration (FDA). In response, the name “Total” was given to the product in the United States. The one word would convey that the toothpaste is the “total” package of various benefits. Young & Rubicam developed several commercials illustrating Total’s benefits and tested the commercials with focus groups. One commercial touting Total’s long-lasting benefits was particularly successful. The product was launched in the United States in January of 1998 using commercials that were designed from the more successful ideas of the focus group tests. Suppose 32% of all people in the United States saw the Total commercials. Of those who saw the commercials, 40% purchased Total at least once in the first 10 months of its introduction. According to U.S. Census Bureau data, approximately 20% of all Americans were in the 45-64 age category. Suppose 24% of the consumers who purchased Total for the first time during the initial 10-month period were in the 45-64 age category. Within three months of the Total launch, Colgate-Palmolive grabbed the number one market share for toothpaste. Ten months later, 21% of all U.S. households had purchased Total for the first time. The commercials and the new product were considered a success. During the first 10 months of its introduction, 43% of those who initially tried Total purchased it again.   

1. What percentage of U.S. households purchased Total at least twice in the first 10 months of its release?

2. Can you conclude the initial purchase of Total was independent of age? Use a quantitative argument to justify your answer.

3. Calculate the probability that a randomly selected U.S. consumer is either in the 45-64 age category or purchased Total during the initial 10-month period.

4. What is the probability that a randomly selected person purchased Total in the first 10 months given that the person is in the 45-64 age category?

e. What percentage of people who did not see the commercials purchased Total at least once in the first 10 months of its introduction?

Is the number of tornadoes increasing? In the last homework, data on the number of tornadoes in the United States between 1953 and 2014 were analyzed to see if there was a linear trend over time. Some argue that it’s not the number of tornadoes increasing over time, but rather the probability of sighting them because there are more people living in the United States. Let’s investigate this by including the U.S. census count (in thousands) as an additional explanatory variable (data in EX11-24TWISTER.csv).

Fit one SLR model with year as the predictor, another SLR model with census count as the predictor. Write down the two models. Are year and census count significant, respectively?

5. In the gizmo market, the supply curve is the typical upward-sloping straight line, and the demand curve is the
typical downward-sloping straight line. The equilibrium quantity in the gizmo market is 50,000 gizmos per month when there is no tax. Then a tax of $5 per gizmo is imposed. As a result, the government is able to raise $231,200 per month in tax revenue. We can conclude that the equilibrium quantity of gizmos has decreased by _______ gizmos per month.
A. more than 3,975
B. more than 3,750, but less than 3,975
C. more than 3,525, but less than 3,750
D. more than 3,300, but less than 3,525
E. more than 3,075, but less than 3,300

6. Which of the following variables decrease in response to a tax on a good?
A the effective price received by sellers of the good, the wedge between the effective price paid by buyers and the effective price received by sellers, and consumer surplus
B. the effective price paid by buyers of the good, the wedge between the effective price paid by buyers and the effective price received by sellers, producer surplus and consumer surplus
C. the equilibrium quantity in the market for the good, the effective price of the good paid by buyers, and consumer surplus
D. the equilibrium quantity in the market for the good, producer surplus, and the well-being of buyers of the good
E. None of the above is necessarily correct unless we know whether the tax is levied on buyers or on sellers.

7. In which of the following cases is it most likely that an increase in the size of a tax will increase tax revenue?
A. The price elasticity of demand and the price elasticity of supply are both large.
B. The price elasticity of demand and the price elasticity of supply are both small.
C. The price elasticity of demand is small, and the price elasticity of supply is large.
D. The price elasticity of demand is large, and the price elasticity of supply is small.
E. None of the above answers is plausible because an increase in the size of the tax will always increase tax revenue.