1.     Probability: (4)

a.    Mozart wrote 27 piano concertos and 41 symphonies. How many ways are there to listen first to a Mozart piano concerto and then to a symphony?

b.    You have a compact disc player with 20 different songs on it. Your CD player can be programmed to play songs in any order. If you decide to listen to just 6 of the songs, the number of possible ways to do this is?

c.     You have identified nine courses that you are seriously considering taking next term. How many ways are there to select five from among the eight without regard to order?

d.    The Government of B.C. has decided to develop its own lottery called 4/20. To win, one must correctly pick 4 out of the 20 possible numbers. If you purchased one ticket, what is the probability of winning?

2. As a consultant to the United Nations Development Programme (UNDP), you have been hired to conduct a study that will update the literacy data for rural villages in Thailand. In a 1995 study, the mean adult literacy was 63.1% with a standard deviation of 9.5%. It is now 2002. Given the various development initiatives undertaken by the government and international aid agencies, things may have changed. However, before you hop on a plane to Bangkok, you need to provide the following information to UNDP.

1. What type of sampling technique will be used for this research? (provide a rationale). Describe how the sampling will be done. Has it been necessary to make any assumptions about the villages or the people in them?  (5)
    
2. The budget for this research is $200,000, but you must allocate $50,000 for living and travel expenses for the study team. You want to sample as many villages as feasible in order to achieve an accurate estimate of the regional literacy rate. The average cost of sampling one village is $2,500. What is the highest level of accuracy (lowest error) possible given your research budget?  (8)

    

   • Determine the maximum sample size possible within your budget.

    

   • Calculate the sample size recommended for tolerable errors of 1, 2, 3 and 4 for mean adult literacy (use whole numbers instead of decimals, i.e. 1 instead of 1% or 0.01).

    

    

   
   

   • Graph the relationship between tolerable error (X axis) and sample size (Y axis). Sketch a line to connect the points.

    

    

    

    

    

   • Use the Error vs Sample Size graph to determine lowest error possible for your sample size.

    

    

   3. The file Human_dev.sav contains data from the Human Development Research Office for 174 countries.

   Status: level of human development (HD - high, MD - medium, LD - low)
   Name: country name
   Life: life expectancy at birth (years)
   Lit: adult literacy rate (per 100)
   GDP: Gross Domestic Product per capita ($)
   Rank: overall ranking based on combination of Life, Lit and GDP variables

   Using this dataset as our population, we will draw several different samples from it and examine the relationship between sample statistics and population parameters. Although several variables are available, we will focus on life expectancy.

   a)  Calculate and for life expectancy. Note: As SPSS is designed for the analysis of sample data, the program uses n-1 in the denominator to calculate standard deviation. You must use the calculator functions available in SPSS (Transform -> Compute) to obtain yourself. (2)

b)  Describe the shape of the population distribution. Speculate on factors ( social, cultural, economic etc) that may explain the shape of the life expectancy distribution. Remember that the data represents the average life expectancy  of 174 individual countries. (3)
 

 

 

 

c)  How many random samples of 5 countries can be drawn from this population? (1)
 

 

 

d)  Select random samples of 5, 20 and 100 countries from the population. For each sample, describe the shape of the distribution and calculate and s. Summarize the population and sample calculations in a table. Are the samples representative of the population? Does the change in sample size affect the accuracy of the sample statistics? Explain why or why not. (10)

 

 

 

 

 

 

 

 

e)  Using the simple random sampling technique, there is a chance that a sample (n=5) might contain only highly developed (HD) countries. Calculate the probability that a sample of 5 would contain only HD countries. What implication would this have on subsequent analyses? (4)
 

 

 

 

 

f)  For some additional analyses, you need a sample of 45 countries. Recommend a sampling design and explain how you could apply this method to ensure that you have a representative sample from the population. (2)