In the last part of the 20th century, scientists developed the theory that the planet was warming and that the primary cause was the increasing amounts of atmospheric carbon dioxide (CO₂), which are the product of burning oil, natural gas, and coal (fossil fuels). Although many climatologists believe in the so-called greenhouse effect, many others do not subscribe to this theory. Further, Earth's temperature has increased and decreased many times in its long history. We have had higher temperatures and we have had lower temperatures, including various ice ages. In fact, a period called the 'little ice age' ended around the middle to the end of the nineteenth century. Then the temperature rose until about 1940, at which point it decreased until 1975. In fact, a Newsweek article published 28 April 1975, discussed the possibility of global cooling, which seemed to be the consensus among scientists at the time. There are three critical questions that need to be answered in order to resolve the issue.

1. Is Earth actually warming?
2. If the planet is warming, is there a human cause or is it natural fluctuation?
3. If the planet is warming, is CO₂ the cause?

In terms of data, the generally accepted procedure is to record monthly temperature anomalies. To do so, we calculate the average for each month over many years. We then calculate any deviations between the latest month's temperature reading and the monthly average calculated above. A positive anomaly would represent a month's temperature that is above the average. A negative anomaly indicates a month where the temperature is less than the average.

Is Earth actually warming?

File GLOBALA.xlsx contains the monthly temperature anomalies (C) from 1880 to 2020. Using the data answer the following:

1. Considering the monthly temperature anomalies data over the data period 1880-2020, identify the data type and use an appropriate graphical technique to display the data. Discuss the trend in data and comment whether there is global warming. [Hint: Insert a trend line.]
2. Considering the population mean of the temperature anomalies (m), test to show that, on average, there is global warming at the 5 percent level of significance. [Hint: Test whether the population mean of monthly temperature anomalies is positive (m > 0)].

If the planet is warming, is there a human cause or is it natural fluctuation?

We need to consider the temperature anomalies at various periods to consider this belief. File GLOBALB1.xlsxstores the monthly temperature anomalies (C) for the time period 1880 to 1940, GLOBALB2.xlsxstores the data from 1941 to 1975, GLOBALB3.xlsxstores the data from 1976 to 1997 and GLOBALB4.xlsx stores the data from 1998 to 2020. Using these data answer the following:

c. For each of the four time-period data, estimate the least squares line and the coefficient of determination. Report and interpret your findings. Has there been global warming in each of the four periods?

If the planet is warming, is CO₂ the cause?

Data for CO₂ levels (ppm) in the atmosphere together with the temperature anomalies for March 1958 to August 2020 are stored in file GLOBALC.xlsx.

d. Use a graphical technique to determine whether there is a linear relationship between temperature anomalies and CO₂ levels. On the plot, insert the trend line and the coefficient of determination (R²). Comment on the fitness of the model.

1. Using the estimated trend line equation, predict the temperature anomaly if the CO₂ level reaches 400ppm?

A university in Victoria is investigating expanding its evening programs. It wants to target people aged between 25 and 35 years, who have completed high school but not a university degree. To help determine the extent and type of offerings, the university needs to know the size of its target market. A survey of 320 adults aged between 25 and 35 years was drawn and each person was asked to identify his or her highest educational attainment. The responses are:

1 Did not complete high school

2 Completed high school only

3 Completed high school and some vocational study only

4 A university graduate

The responses are recorded and stored in file EDUCATION.xlsx.

1. Estimate with 95% confidence the proportion of adults in Victoria aged between 25 and 35, who belong to the market segment targeted by the university.

There are about 1,147,315 people between the ages of 25 and 35 in Victoria.

1. Estimate with 95% confidence the number of adults in Victoria aged between 25 and 35, who belong to the market segment targeted by the university.

The university administration claims that more than 45% of the adults aged between 25 and 35 years are in the market segment the university wishes to target.

1. Test the University's claim at the 5% level of significance.

[Hint: Follow the 6-step process for testing hypotheses.]

QUESTION3: Life Insurance and Longevity

Life insurance companies are keenly interested in predicting how long their customers will live, because their premiums and profitability depend on such numbers. An actuary for one insurance company suspects that the longevity (age at death) of a male customer is linearly related to the age at death of his father. To verify this, he gathered data from 100 recently deceased male customers. He recorded the age at death of the customer and the age at death of his father. These data are recorded in columns 1 to 2, respectively, in file LONGEVITY.xlsx.

a. Estimate a linear relationship between the longevity of a male customer and age at death of his father and interpret your results.

b. Provide three measures to verify the fitness of the model. Do you think that this model is good enough to be used to estimate and predict the longevity of a male customer? Briefly explain the reasons for your answer.

c. Using the estimated regression model,

i predict the longevity of a male customer whose father lived to the age of 70.

ii estimate with 95% confidence the mean longevity of a male customer whose father lived to the age of 70.

iii predict with 95% confidence the longevity of a male customer whose father lived to the age of 70.

Looking at the estimation results using the simple linear regression model above, the actuary wants to further investigate whether in addition to the father's age at death, other factors such as the age at death his mother and his grandparents also have some effect on his longevity. For the same 100 recently deceased male customers, he recorded the age at death of the customer plus the ages at death of his father and mother, the mean ages at death of his grandfathers and the mean ages at death of his grandmothers. These data are recorded in columns 1 to 5, respectively, in file LONGEVITY.xlsx.

d. Develop a multiple regression model and discuss the possible signs of the coefficients.

e. Estimate the model using the recorded data. Write the estimated regression model with standard errors and p-values in the standard format.

f. Interpret the coefficient estimates of the independent variables.

g. Test to determine whether each of the independent variables is linearly related to longevity of the male customer. (a = 0.05)

h. Test the overall utility of the model. Is the model likely to be useful in predicting men's longevity? (a = 0.05)

i. Discuss the required conditions for the estimation of the multiple regression model.

j. Predict the longevity of a man whose parents lived to the age of 70, whose grandfathers averaged 75 years and whose grandmothers averaged 80.

QUESTION4: Predicting Quarterly Tourist Arrivals[1+1+1+2.5+1+1+2+1+1.5+1+1+1=15 marks]

The tourism industry in Australia is to some extent subject to enormous seasonal variation. The Australian Bureau of Statistics (ABS) publishes various information on tourism-related variables. The quarterly short-term inbound tourist arrival numbers to Australia for the years 2014(1)-2019(4) are recorded in file ARRIVALS.xlsx.

a. Plot the time series. Does there appear to be any seasonal pattern?

Measuring the seasonal effect using the moving averages:

b. Calculate the four-quarter centred moving averages.

c. On the same graph, plot the series and the four-quarter centred moving averages.

d. Calculate the quarterly seasonal indexes.

e. Calculate the seasonally adjusted series.

Measuring the seasonal effect using the trend method:

f. Estimate the linear trend line (from regression analysis) for the data and present the estimated trend line.

g. Calculate the quarterly seasonal indexes, using this trend line.

h. Calculate the seasonally adjusted series and plot the unadjusted and adjusted series.

i. Forecast the number of inbound tourist arrivals in Australia during the four quarters in 2020-2023.

Updating data and comparing with the forecast - Effect of the Covid-19 pandemic in 2020 and beyond:

j. Collect the actual quarterly tourist arrivals data for 2020-2023 and present it in a table (source should also be provided).

k. Compare the actual and predicted quarterly tourist arrivals for 2020 (Covid-19 pandemic year).

l. Considering the quarterly tourist arrivals for 2021-2023, comment whether the increases are on track to reach the pre-pandemic 2019 tourists arrival levels in Australia.