Summaries

Session 0.1/1.1

1^st June 2009

Sampling a Simple Population

We use random sampling to estimate an empirical model of a population. We check the empirical model by direct inspection of the population. We repeat sampling with replacement, obtaining multiple random samples from the same population, obtained in the same process. We combine (pool) compatible samples to form larger samples. Pooling samples of size 50, we obtain samples of size 100, 150 and 300. In general, as sample size increases, samples become more precise and reliable, provided that the sampling process is reliable.

Random sampling is the basis for obtaining information in statistical activities. Sampling is necessary, tedious, time consuming and expensive. Random sampling incorporates reliability, precision and uncertainty.

Session Overview

In this session, we begin the study of probability. We begin with a very basic example of a population, and explore the process of sampling a population.

We examine two modes of sampling a population: census (total enumeration), in which every member of the population is examined; and random sampling with replacement (SRS/WR), in which single members are repeatedly selected from the population. One practical reason why we would want a sampling process is that we wish to estimate some property of the population. Total enumeration allows a definitive settling of the question, and random sampling allows an approximate answer. In most practical settings, the populations of interest are too difficult to totally enumerate – the population is too large, or too complex, or cannot be accessed in total. In practical applications, it is sufficient (and usually necessary) to use a suitable random sample in lieu of the total population.

In our first case, we begin with a color bowl whose true color frequencies are not known. We obtain six (6) random samples, each consisting of 50 draws with replacement (SRS/WR). We then compute sample color frequencies in order to estimate the population color frequencies, and then we check the estimates against the true structure of the bowl.

We then explore a bit of decision theory by playing with Ellsberg’s Urns.

Prediction and Probabilistic Randomness: Predicting the Behavior of a Six-sided Die

Samples – Face Values and Predictions

You should be able to begin with the counts in the table and work out the proportions and percentages.

Prediction and the Fair Die
Samples				Samples				Pooled
#1				#2				12
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	7	0.14	14	1	15	0.3	30	1	22	0.22	22
2	8	0.16	16	2	7	0.14	14	2	15	0.15	15
3	5	0.1	10	3	7	0.14	14	3	12	0.12	12
4	8	0.16	16	4	9	0.18	18	4	17	0.17	17
5	14	0.28	28	5	8	0.16	16	5	22	0.22	22
6	8	0.16	16	6	4	0.08	8	6	12	0.12	12
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	12	0.24	24	Hit	5	0.1	10	Hit	17	0.17	17
Miss	38	0.76	76	Miss	45	0.9	90	Miss	83	0.83	83
Total	50	1	100	Total	50	1	100	Total	100	1	100
Samples				Samples				Pooled
#3				#4				34
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	10	0.2	20	1	7	0.14	14	1	17	0.17	17
2	9	0.18	18	2	3	0.06	6	2	12	0.12	12
3	7	0.14	14	3	15	0.3	30	3	22	0.22	22
4	10	0.2	20	4	9	0.18	18	4	19	0.19	19
5	10	0.2	20	5	10	0.2	20	5	20	0.2	20
6	4	0.08	8	6	6	0.12	12	6	10	0.1	10
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	6	0.12	12	Hit	7	0.14	14	Hit	13	0.13	13
Miss	44	0.88	88	Miss	43	0.86	86	Miss	87	0.87	87
Total	50	1	100	Total	50	1	100	Total	100	1	100
Samples				Samples				Pooled
#5				#6				56
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	7	0.14	14	1	7	0.14	14	1	14	0.14	14
2	12	0.24	24	2	8	0.16	16	2	20	0.2	20
3	5	0.1	10	3	9	0.18	18	3	14	0.14	14
4	11	0.22	22	4	6	0.12	12	4	17	0.17	17
5	9	0.18	18	5	7	0.14	14	5	16	0.16	16
6	6	0.12	12	6	13	0.26	26	6	19	0.19	19
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	10	0.2	20	Hit	4	0.08	8	Hit	14	0.14	14
Miss	40	0.8	80	Miss	46	0.92	92	Miss	86	0.86	86
Total	50	1	100	Total	50	1	100	Total	100	1	100
Pooled				Pooled				Pooled
135				246				All
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	24	0.16	16	1	29	0.19	19.33	1	53	0.18	17.7
2	29	0.193	19.33	2	18	0.12	12	2	47	0.16	15.7
3	17	0.113	11.33	3	31	0.21	20.67	3	48	0.16	16
4	29	0.193	19.33	4	24	0.16	16	4	53	0.18	17.7
5	33	0.22	22	5	25	0.17	16.67	5	58	0.19	19.3
6	18	0.12	12	6	23	0.15	15.33	6	41	0.14	13.7
Total	150	1	100	Total	150	1	100	Total	300	1	100
Prediction				Prediction				Prediction
Hit	28	0.187	18.67	Hit	16	0.11	10.67	Hit	44	0.15	14.7
Miss	122	0.813	81.33	Miss	134	0.89	89.33	Miss	256	0.85	85.3
Total	150	1	100	Total	150	1	100	Total	300	1	100

In the fair die model for this case, in long runs of tosses of the die: approximately 16⅔% of tosses show “1”, approximately 16⅔% of tosses show “2”, approximately 16⅔% of tosses show “3”, approximately 16⅔% of tosses show “4”, approximately 16⅔% of tosses show “5”, and approximately 16⅔% of tosses show “6.” The sample data are generally compatible with a fair die assumption (equally-likely face values) and with a baseline expected prediction success rate of (1/6), or 16⅔%. Sample performance seems to improve with increasing sample size – but the samples do not exactly fit the fair assumption.

Sample versus Fair Model

Face Value 1: 17.7% versus 16.7%

Face Value 2: 15.7% versus 16.7%

Face Value 3: 16% versus 16.7%

Face Value 4: 17.7% versus 16.7%

Face Value 5: 19.3% versus 16.7%

Face Value 6: 13.7% versus 16.7%

Prediction “Hit”: 14.7% versus 16.7%

Prediction “Miss”: 85.3% versus 83.3%

Case Study 1.1: A Color Bowl

In random sampling, we might get a complete list of colors - we'd need a total sample (census) for that kind of listing. The sample proportions of each listed color approximate the corresponding model proportion in the bowl itself. In census sampling, every object in the bowl is counted. The listing is complete, and the model proportions may be calculated directly.

The basic idea in case study 1.1 is that random samples give imperfect pictures of what is being sampled. However, with sufficiently large samples, these samples can reliably yield good pictures of the processes or populations being sampled. And the essence of many statistical applications is the study of selected processes or populations. For a sense of the efficiency of the samples, compare sample and true percentages.

Some Formulas – Proportions, Percentages, Counts

The class represents some property or attribute, for example, blue, or red. Each member, or unit, of a sample can be classified – the result of the classification of the unit is the unit’s class.

Sample Proportion (p)

n_class ~ number of units of sample in class

n_total ~ total number of units in sample

p_class = n_class / n_total

p_class ~ proportion of sample in class

Sample Percent (pct)

n_class ~ number of units of sample in class

n_total ~ total number of units in sample

p_class = n_class / n_total

pct_class = 100*(n_class / n_total)

pct_class = 100* p_class

pct_class ~ percent of sample in class

Population Proportion (P)

N_class ~ number of units of population in class

N_total ~ total number of units in population

P_class = N_class / N_total

P_class ~ proportion of population in class

Population Percent (PCT)

N_class ~ number of units of population in class

N_total ~ total number of units in population

P_class = N_class / N_total

PCT_class = 100*(N_class / N_total)

PCT_class = 100* P_class

PCT_class ~ percent of population in class

In this setting,

n_blue ~ number of blue draws in sample

n_total ~ total number of draws per sample

p_blue = n_blue / n_total

p_blue ~ proportion of sample draws showing blue

pct_blue = 100*p_blue

pct_blue ~ percent of sample draws showing blue

N_blue ~ number of blue marbles in bowl

N_total ~ total number of marbles in bowl

P_blue = N_blue / N_blue

P_blue ~ proportion of marbles in bowl that are blue

n_green ~ number of green draws in sample

n_total ~ total number of draws per sample

p_green = n_green / n_green

p_green ~ proportion of sample draws showing green

pct_green = 100*p_green

pct_green ~ percent of sample draws showing green

N_green ~ number of green marbles in bowl

N_total ~ total number of marbles in bowl

P_green = N_green / N_green

P_green ~ proportion of marbles in bowl that are green

n_red ~ number of red draws in sample

n_total ~ total number of draws per sample

p_red = n_red / n_red

p_red ~ proportion of sample draws showing red

pct_red = 100*p_red

pct_red ~ percent of sample draws showing red

N_red ~ number of red marbles in bowl

N_total ~ total number of marbles in bowl

P_red = N_red / N_red

P_red ~ proportion of marbles in bowl that are red

n_yellow ~ number of yellow draws in sample

n_total ~ total number of draws per sample

p_yellow = n_yellow / n_yellow

p_yellow ~ proportion of sample draws showing yellow

pct_yellow = 100*p_yellow

pct_yellow ~ percent of sample draws showing yellow

N_yellow ~ number of yellow marbles in bowl

N_total ~ total number of marbles in bowl

P_yellow = N_yellow / N_yellow

P_yellow ~ proportion of marbles in bowl that are yellow

Samples – Bowl

You should be able to begin with the counts in the table and work out the proportions and percentages.

#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent
Blue	23	0.46	46	Blue	19	0.38	38	Blue	42	0.42	42
Green	8	0.16	16	Green	13	0.26	26	Green	21	0.21	21
Red	19	0.38	38	Red	13	0.26	26	Red	32	0.32	32
Yellow	0	0	0	Yellow	5	0.1	10	Yellow	5	0.05	5
Total	50	1	100	Total	50	1	100	Total	100	1	100
#3				#4				Pooled 34
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent
Blue	22	0.44	44	Blue	17	0.34	34	Blue	39	0.39	39
Green	6	0.12	12	Green	15	0.3	30	Green	21	0.21	21
Red	19	0.38	38	Red	13	0.26	26	Red	32	0.32	32
Yellow	3	0.06	6	Yellow	5	0.1	10	Yellow	8	0.08	8
Total	50	1	100	Total	50	1	100	Total	100	1	100
#5				#6				Pooled 56
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent
Blue	13	0.26	26	Blue	23	0.46	46	Blue	36	0.36	36
Green	19	0.38	38	Green	10	0.2	20	Green	29	0.29	29
Red	10	0.2	20	Red	15	0.3	30	Red	25	0.25	25
Yellow	8	0.16	16	Yellow	2	0.04	4	Yellow	10	0.1	10
Total	50	1	100	Total	50	1	100	Total	100	1	100
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent
Blue	58	0.386667	38.67	Blue	59	0.393333	39.333	Blue	117	0.39	39
Green	33	0.22	22	Green	38	0.253333	25.333	Green	71	0.236667	23.667
Red	48	0.32	32	Red	41	0.273333	27.333	Red	89	0.296667	29.667
Yellow	11	0.073333	7.333	Yellow	12	0.08	8	Yellow	23	0.076667	7.6667
Total	150	1	100	Total	150	1	100	Total	300	1	100

The True State of the Bowl

Color	Count	Proportion	Percent	E50	E100	E150	E200	E250	E300
Blue	8	0.333333	33.33	16.667	33.3	50	66.7	83.3	100
Green	6	0.25	25	12.5	25	37.5	50	62.5	75
Red	8	0.333333	33.33	16.667	33.3	50	66.7	83.3	100
Yellow	2	0.083333	8.333	4.1667	8.33	12.5	16.7	20.8	25
Total	24	1	100	50	100	150	200	250	300

Sample versus Population

39% versus 33.3%

23.7% versus 25%

29.7% versus 33.3%

7.7% versus 8.3%

The true proportions are probabilities:

In long runs of draws with replacement from the bowl, approximately 33.3 percent of draws show blue.

In long runs of draws with replacement from the bowl, approximately 25 percent of draws show green.

In long runs of draws with replacement from the bowl, approximately 33.3 percent of draws show red.

In long runs of draws with replacement from the bowl, approximately 8.3 percent of draws show yellow.

We see reasonable, but not exact matches between the sample proportions (p) and the probabilities (P).

The probabilities imply perfect or expected counts: E=n*P:

In samples of 50 draws with replacement from the bowl, approximately 16 or 17 draws show blue.

In samples of 50 draws with replacement from the bowl, approximately 12 or 13 draws show green.

In samples of 50 draws with replacement from the bowl, approximately 16 or 17 draws show red.

In samples of 50 draws with replacement from the bowl, approximately 4 or 5 draws show yellow.

In samples of 100 draws with replacement from the bowl, approximately 33 or 34 draws show blue.

In samples of 100 draws with replacement from the bowl, approximately 25 draws show green.

In samples of 100 draws with replacement from the bowl, approximately 33 or 34 draws show red.

In samples of 100 draws with replacement from the bowl, approximately 8 or 9 draws show yellow.

In samples of 150 draws with replacement from the bowl, approximately 50 draws show blue.

In samples of 150 draws with replacement from the bowl, approximately 37 or 38 draws show green.

In samples of 150 draws with replacement from the bowl, approximately 50 draws show red.

In samples of 150 draws with replacement from the bowl, approximately 12 or 13 draws show yellow.

In samples of 200 draws with replacement from the bowl, approximately 66 or 67 draws show blue.

In samples of 200 draws with replacement from the bowl, approximately 50 draws show green.

In samples of 200 draws with replacement from the bowl, approximately 66 or 67 draws show red.

In samples of 200 draws with replacement from the bowl, approximately 16 or 17 draws show yellow.

In samples of 250 draws with replacement from the bowl, approximately 83 or 84 draws show blue.

In samples of 250 draws with replacement from the bowl, approximately 62 or 63 draws show green.

In samples of 250 draws with replacement from the bowl, approximately 83 or 84 draws show red.

In samples of 250 draws with replacement from the bowl, approximately 20 or 21 draws show yellow.

In samples of 300 draws with replacement from the bowl, approximately 100 draws show blue.

In samples of 300 draws with replacement from the bowl, approximately 75 draws show green.

In samples of 300 draws with replacement from the bowl, approximately 100 draws show red.

In samples of 300 draws with replacement from the bowl, approximately 25 draws show yellow.

We see reasonable, but not exact matches between the sample proportions and the probabilities.

We didn’t get to these, so read up.

Regarding Ellsberg I

The 1^st Game: The first bowl is 50%/50% split between blue and green. The best we can do is break even, regardless of strategy. The simplest strategy involves picking one of the colors and always betting on that color.

The 2^nd Game: The second bowl is an unknown composite of red and yellow. We might be able to win this game if 1) there is a dominant color and 2) we can determine that dominant color. A simple strategy here is to pick one color and ride it for awhile. Then stop betting and check the number of winning bets. If the color being betted is losing on a regular basis, switch colors.

The 3^rd Game: This game only makes sense if the second bowl is dominant in red, bet on red – if red consistently shows, stay on the second bowl. Otherwise, either stop playing, or stick with the first bowl.

Regarding Ellsberg II

The 1^st Game: The first bowl is 20% red / 40% black / 40% white. The simplest strategy involves picking one of the colors and always betting on that color. Regardless of betting choice, there is a 40% chance of losing for the single bet, and 20% for getting kicked off the game.

The 2nd Game: The second bowl is 20% red / 80% black or white. The simplest strategy involves picking one of the colors and always betting on that color. If either white or black is sufficiently dominant, this game might be worth playing. The problem is that regardless of the possible advantage in the white/black part of the bowl, there is still a 20% chance of getting killed (permanently losing). But to detect this advantage, one is forced to pick a betting color (white or black) and spend some money.

The idea underlying the Ellsberg games is to illustrate the concept of making decisions about selected processes or populations by making decisions using random samples.