BES Tutorial Sample Solutions, S1/13
WEEK 2 TUTORIAL EXERCISES (To be discussed in the week starting March 11) 1. What is meant by a variable in a statistical sense? Distinguish between qualitative and quantitative statistical variables, and between continuous and discrete variables. Give examples. A variable in a statistical sense is just some characteristic of an ‘object’. It may take different values. Data on a quantitative variable can be expressed numerically in a meaningful way (e.g. height of an individual, number of children in a family. Data on qualitative variables cannot be expressed numerically in a meaningful way; e.g. sex of an individual, hair colour). A discrete quantitative variable can assume only certain discrete numerical values on the number line (can be a finite or infinite number of these values). A continuous quantitative variable can assume any value in a specific range or interval; e.g. length of a pipe.

2. Distinguish between (a) a statistical population and a sample; (b) a parameter and a statistic. Give examples. A statistical population is the set of measurements or observations of a characteristic of interest for all elementary units in a frame; e.g the shoe sizes of all men in Australia. A statistical sample is a subset of a population; e.g. the shoe sizes of all the men in the class

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is a sample of the population represented by the shoe sizes of all men in Australia. A parameter is a numerical description of a population. For example, the average shoe size of all Australian men is a parameter (of the population of the shoe sizes of all Australian men). A statistic is a numerical description of a sample. For example, the average shoe size of all men in this class room is a statistic (calculated from the sample of the shoe sizes of all men in this class room). 3. In order to know the market better, the second-hand car dealership, Anzac Garage, wants to analyze the age of secondhand cars being sold. A sample of 20 advertisements for passenger cars is selected from the second-hand car advertising/listing website www.drive.com.au The ages of the vehicles at time of advertisement are listed below: 5, 5, 6, 14, 6, 2, 6, 4, 5, 9, 4, 10, 11, 2, 3, 7, 6, 6, 24, 11 (a) Calculate frequency, cumulative frequency and relative frequency distributions for the age data using the following bin classes: More than 0 to less than or equal to 8 years More than 8 to less than or equal to 16 years More than 16 to less than or equal to 24 years.
Bin 0 8 16 8 16 24 Relative Frequency 0.7 0.25 0.05 Frequency 14 5 1 Cumulative Frequency 14 19 20

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(b)Sketch a frequency histogram using the calculations in part (a). What can you say about the distribution of the age of these second-hand cars? Is there anything wrong with the frequency table and histogram? Specifically, is the choice of bin classes appropriate? What needs to be done?
Relative frequency histogram for Age
0.8 0.7 0.6 Frequency 0.5 0.4 0.3 0.2 0.1 0 8 16 Bin 24

From this graph (it was not necessary to use EXCEL although it is good practice), the Age distribution appears to be skewed to the right. 70% of observations have age between 0 and 8. However, this histogram only provides limited information about the Age distribution because there are too few bins and they are very wide.

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(c) Halve the width of the bins (0 to 4, 4 to 8, etc) and recalculate the frequency, cumulative frequency and relative frequency distributions. Using the new distributions and histogram, what can you now say about the distribution of the age of secondhand cars?
Bin
0 4

4 < Age ≤ 8 8 < Age ≤ 12 12 < Age ≤ 16 16 < Age ≤ 20 20 < Age ≤ 24

Cumulative Relative Frequency Frequency Frequency 0.25 5 5 0.45 9 14 0.2 4 18 0.05 1 19 0 0 19 0.05 1 20

Figure 3.1: Revised histogram for age of cars
10 9 8 7 6 5 4 3 2 1 0 2 6 10 14 Age 18 22

There still appears to be a skew to the right, but now we can also see that there is an outlier in the 21~24 Age category. 5~8 are the most frequently observed ages. A quite sizable proportion of the second-hand cars are relatively new (25% being less or equal to 4 years old).

Frequency

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4. SIA: Health expenditure A recent report by Access Economics provides a comparison of Australian expenditures on health with that of comparable OECD countries. Data from that report relating to 2005 have been used to reproduce their Figure 2.2 (below denoted as Figure 2.1). (a) What are the key features of these data?  A strong positive association – more per capita GDP implies more Health Expenditure per capita.  There are (at least) 2 outliers, the observation with the largest Health Expenditure (Luxembourg) and the observation with the highest GDP (USA). Without these 2 the relationship is approximately linear. With them, there is a suggestion of a non-linear relationship.  An indication of more variability in health expenditures when GDP is larger. (b)While this is a bivariate scatter plot, there are three variables involved: health expenditure, GDP and population. Why account for population by expressing health expenditure and GDP in per capita terms? This is recognition that there may be factors other than GDP associated with Health Expenditures and population size is one obvious factor. Expressing everything in per capita terms is one way to control for population variations and hence isolate the GDP Health Expenditure relationship.

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5. SIA: Australian private health insurance coverage Australia has a mixed public/private system for delivering hospital services. Medicare is publicly funded insurance that provides free treatment in public hospitals for all its citizens. In addition, Australians can purchase private health insurance to provide extra benefits such as choice of doctor, improved hospital accommodation and reduced waiting times. Figure 2.2 presents the changes over time in the proportion of the population who have private health insurance covering hospitalization. (a) What are the key features of these data?  Over the entire period there has been a trend down in percentage of Australians insured. In the early 70’s nearly 80% of Austalians were covered but now it’s less than 50%.  Overall trend has been subject to some “shocks” notably in the early 80’s then again in 2000.  Since the last shock coverage has been “relatively” constant compared to the variation beforehand. (b) What happened in 2000 that is associated with one of these key features?  A quick internet search will reveal that in response to the downward trend in coverage, the Australian Government introduced a new policy in 2000 called Lifetime Health Coverage (LHC) http://www.health.gov.au/internet/main/publishing.nsf/content/ private-1

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