QUESTION 1            (25 points and each item 5 points)                                                                                  

Suppose that Australian government conducts a survey and collects 400 observations of works in Australia. We are informed that 36 works are unemployed (UN) and 150 works have 16 years of education or higher (BA). We also know 220 works have 15 years of education less and are employed. Let EN denote an event of being employed and let HS denote an event of having 15 years of education less. To answer questions in 1.2-1.5 below, use 2 decimal places.

1. Obtain a frequency table (contingent table)

2. Obtain a joint probability table

3. It the table in Question 1.2, interpret the number in a cell of (Total, UN). [less than 20 words.]
4. Obtain the following quantities

• P(UN|HS) =
• P(UN|BA) =

Explain these quantities (less than 30 words).

5. Obtain the following quantities

• P(UN|HS) P(HS)=
• P(UN) P(HS)=

Compare two values and explain the result (less than 30 words).

QUESTION 2 (10 points)

Implement the following procedure:

• Using an Excel function, RAND(), generate 100 uniform random variables.
• Transform those random variables to binary random variables taking 1 if a uniform random variable is strictly greater than 0.5 and 0 otherwise. (note: StatTools can create “dummy” variables, see StatTools->Data Utilities->Dummy).

Report an average of 100 binary variables up to 2 decimal places and compare it with the population mean (less than 30 words).

QUESTION 3 (20 points and 5 points for each)

To answer this question, use an Excel file, jq17_provisional_pi_time_series.xlsx, which is provided by Australian government.  We are interested in quarterly data (not yearly) the number of bankruptcies of VIC state, which is reported in the sheet of “Bankrupties” and covers the periods of 1986:Q3-2017:Q2, where Q3 means the third quarter and Q2 means the second one. Use 3 decimal places.

1. User the data from 1986:Q3 to 2017:Q2 and obtain its mean, variance, max and min.
2. Let X be that the number of bankruptcies and assume that X follows Poisson distribution. Using Excel, fill out the following table (i.e., obtain probabilities and cumulative distribution function):

3. Keep the assumption that X follows Poisson distribution and obtain

• P(1000 <X1050)=
• P(500x<1000) =

4. Comparing historical data and the predictions under the Poisson assumption and discuss whether the assumption is suitable for this data (less than 50 words).

QUESTION 4 (45 points and 9 points for each)

Let X be follow normal distribution with mean 21 and standard deviation 3.5, i.e., X ~ N(21, 3.5). To answer questions, use Table for normal distribution in Moodle with some approximation. Use 4 decimal places for your answer (not necessarily for the process of obtaining your answer).

1. Obtain P(X < 18).
2. Obtain P(29 < X).
3. Obtain P(10 < X < 20).
4. Obtain x such that P(X < x) = 0.9394.
5. Obtain x such that P(X < x) = 0.0606.