Assignment 1

4

1. [Theory] The 1921 census of Canada is now public. Below is a list of 50 observations of yearly income in downtown Victoria (District 24, Sub-District 1) taken from the census forms. I’ve included only positive incomes – no zeroes. If you’re curious, I’ve used items e002877457 - e002877461. You can access these images directly by typing in

http://central.bac-lac.gc.ca/.item/?app=Census1921&op=img&id=e002877461

and replacing the last two numbers with 57,58,59,60 or (as shown) 61.

Index	Yearly Income	Index	Yearly Income
1	$100	26	$1,900
2	$50	27	$1,000
3	$500	28	$1,700
4	$1,900	29	$310
5	$4,600	30	$1,500
6	$2,432	31	$240
7	$1,450	32	$1,440
8	$400	33	$2,250
9	$20	34	$75
10	$950	35	$2,300
11	$50	36	$960
12	$1,500	37	$970
13	$1,000	38	$1,760
14	$1,580	39	$75
15	$1,680	40	$450
16	$800	41	$2,000
17	$1,350	42	$1,300
18	$900	43	$1,280
19	$1,600	44	$1,050
20	$400	45	$560
21	$3,073	46	$1,440
22	$1,200	47	$1,440
23	$2,160	48	$790
24	$2,200	49	$360
25	$1,750	50	$1,610

For this question, you will run very simple Monte Carlo and bootstrap analyses to calculate a cost-acceptability curve and mean value for downtown Victoria. To allow you to do this by hand, we will only be running 10 trials.

a. (1 mark) [Monte Carlo] I’ve calculated the minimum, maximum and average values for the incomes reported. Let’s assume that income follows a normal distribution centred at that average, and that our minimum and maximum values cover 6 standard deviations. Calculate an estimate of the standard deviation by taking the difference between the maximum and minimum and dividing by 6 [(MAX – MIN)/6]:

MINIMUM	$20
MAXIMUM	$4,600
AVERAGE	$1,248

Estimated Standard Deviation = _______________________

b. (2 marks) Go to random.org’s Gaussian Random Number generator and generate 10 random numbers. Use the average income as the mean, and your answer for part a. as the standard deviation. List the random numbers below.

Gaussian Random Number Generator: https://www.random.org/gaussian-distributions/

To make things more readable, try using 2 significant digits.

Note: 1.2e+3 means 1.2 x 1000 = 1,200, 1.2e+2 = 1.2 x 100 = 120, etc.

Mean used: ________________

Standard deviation used: _________________

Random Numbers Generated:

1. ________________

2. ________________

3. ________________

4. ________________

5. ________________

6. ________________

7. ________________

8. ________________

9. ________________

10. ________________

c. (5 marks) Use the numbers you generated and the techniques in class to plot a cost-acceptability curve. (Sort the numbers from smallest to largest, plot them with income on the horizontal axis and % on the vertical axis. Your % will go up by 10% at a time, since you have 10 observations.) Be sure to label your axes.

d. (2 marks) [Bootstrap] Go to random.org’s integer generator and generate 10 random whole numbers between 1 and 50: https://www.random.org/integers/ . Look those numbers up in the table below, and write both the number and the income that matches it below. For example, if your random number was 27, you would write (27, $1,000).

Random numbers generated, and matching incomes:

1. ________________

2. ________________

3. ________________

4. ________________

5. ________________

6. ________________

7. ________________

8. ________________

9. ________________

10. ________________

e. (5 marks) Use the numbers you generated and the techniques in class to plot a cost-acceptability curve. (Sort the numbers from smallest to largest, plot them with income on the horizontal axis and % on the vertical axis. Your % will go up by 10% at a time, since you have 10 observations.) Be sure to label your axes.