Summaries

Session 1.2

20^th January 2010

Predicting Sample Behavior from a Model

We use a population model to predict the behavior of random samples. We check the predictions by direct inspection of samples. 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.

In general, if we are working with the correct model, then the predicted sample behavior reliably describes observed samples.

Session Overview

Estimation: Previously, we saw how random samples drawn from a population could be used to estimate the structure of a population as an empirical model.

Prediction: In this session, we continue our study of probability. We begin with a very basic example of a population with known structure, and use that structure to predict the behavior of random samples from that population.

We begin by constructing a probability model for our population, and define the concept of perfect sample. We then relate the perfect sample to random samples, both observed and unobserved. We then obtain real random samples, and check them against the perfect samples.

Exclude Case Study 1.2.1

Case Study 1.2.2

In this case study the idea of a perfect sample is introduced – a perfect sample matches perfectly the population from which it is sampled. On average, real samples corresponded nicely, though not perfectly, with their corresponding perfect samples.

We begin by building a color bowl. We then compute a probability model for draws with replacement(DWR) from the bowl, and then perfect samples of size 50, 100, 150, 200, 250 and 300. We then engage six groups to generate six samples each of n=50 DWR. We then compare sample frequencies and proportions to the model and to the perfect samples.

Bowl Counts and Perfect Sample Calculations

Expected Count Blue ( for Sample Size n) = n*P_Blue

Expected Count Green ( for Sample Size n) = n*P_Green

Expected Count Red ( for Sample Size n) = n*P_Red

Expected Count Yellow ( for Sample Size n) = n*P_Yellow

Revise for Spring 2010

The Bowl

Color	Count	Proportion	Percent
Blue	8	8/20 = 0.4	100*.40 = 40
Green	3	3/20 = 0.15	100*.15 = 15
Red	2	2/20 = 0.1	100*.10 = 10
Yellow	7	7/20 = 0.35	100*.35 = 35
Total	20	20/20 = 1	100*1 = 100

Probabilities with Long Run Interpretation

P_Blue = 8/20 = .40

In long runs of draws with replacement from the bowl, approximately 40% of draws show blue.

P_Green = 3/20 = .15

In long runs of draws with replacement from the bowl, approximately 15% of draws show green.

P_Red = 2/2= .10

In long runs of draws with replacement from the bowl, approximately 10% of draws show red.

P_Yellow = 7/20 = .35

In long runs of draws with replacement from the bowl, approximately 35% of draws show yellow.

Perfect Counts for the Bowl – n = 50, 100, 150, 200, 250 and 300 Draws with Replacement

Color	Count	Proportion	Percent
Blue	8	8/20 = 0.4	100*.40 = 40
Green	3	3/20 = 0.15	100*.15 = 15
Red	2	2/20 = 0.1	100*.10 = 10
Yellow	7	7/20 = 0.35	100*.35 = 35
Total	20	20/20 = 1	100*1 = 100

Color	E50	E100	E150	E200	E250	E300
Blue	50*.40 = 20	100*.40 = 40	150*.40 = 60	200*.40 = 80	250*.40 = 100	300*.40 =120
Green	50*.15 = 7.5	100*.15 = 15	150*.15 = 22.5	200*.15 = 30	250*.15 = 37.5	300*.15 = 45
Red	50*.10 = 5	100*.10 =10	150*.10 = 15	200*.10 = 20	250*.10 = 25	300*.10 = 30
Yellow	50*.35 =17.5	100*.35 = 35	100*.35 = 52.5	200*.35 = 70	250*.35 = 87.5	300*.35 = 105
Total	50	100	150	200	250	300

Perfect Samples 

n=50

E50_Blue  = n*P_Blue= 50*(8/20) = 20

E50_Green  = n*P_Green = 50*(3/20) = 7.5

E50_Red = n*P_Red = 50*(2/20) = 5

E50_Yellow = n*P_Yellow = 50*(7/20) = 17.5

In samples of 50 draws with replacement from the bowl, we expect approximately 20 blue draws, 7 or 8 green draws, 5 red draws, and 17 or 18 yellow draws.

n=100

E100_Blue  = n*P_Blue= 100*(8/20) = 40

E100_Green  = n*P_Green = 100*(3/20) = 15

E100_Red = n*P_Red = 100*(2/20) = 10

E100_Yellow = n*P_Yellow = 100*(7/20) = 35

In samples of 100 draws with replacement from the bowl, we expect approximately 40 blue draws, 15 green draws, 10 red draws, and 35 yellow draws.

n=150

E150_Blue  = n*P_Blue= 150*(8/20) = 60

E150_Green  = n*P_Green = 150*(3/20) = 22.5

E150_Red = n*P_Red = 150*(2/20) = 15

E150_Yellow = n*P_Yellow = 150*(7/20) = 42.5

In samples of 150 draws with replacement from the bowl, we expect approximately 60 blue draws, 22 or 23 green draws, 15 red draws, and 42 or 43 yellow draws.

n=200

E200_Blue  = n*P_Blue= 200*(8/20) = 80

E200_Green  = n*P_Green = 200*(3/20) = 30

E200_Red = n*P_Red = 200*(2/20) = 20

E200_Yellow = n*P_Yellow = 200*(7/20) = 70

In samples of 200 draws with replacement from the bowl, we expect approximately 80 blue draws, 30 green draws, 20 red draws, and 70 yellow draws.

n=250

E250_Blue  = n*P_Blue= 250*(8/20) = 100

E250_Green  = n*P_Green = 250*(3/20) = 37.5

E250_Red = n*P_Red = 250*(2/20) = 25

E250_Yellow = n*P_Yellow = 250*(7/20) = 87.5

In samples of 250 draws with replacement from the bowl, we expect approximately 100 blue draws, 37 or 38 green draws, 25 red draws, and 87 or 88 yellow draws.

n=300

E300_Blue  = n*P_Blue= 300*(8/20) = 120

E300_Green  = n*P_Green = 300*(3/20) = 45

E300_Red = n*P_Red = 300*(2/20) = 30

E300_Yellow = n*P_Yellow = 300*(7/20) = 105

In samples of 300 draws with replacement from the bowl, we expect approximately 120 blue draws, 45 green draws, 30 red draws, and 105 yellow draws.

Samples – 6.30

#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	22	0.44	44	Blue	24	0.48	48	Blue	46	0.46	46	40
Green	10	0.2	20	Green	9	0.18	18	Green	19	0.19	19	15
Red	3	0.06	6	Red	6	0.12	12	Red	9	0.09	9	10
Yellow	15	0.3	30	Yellow	11	0.22	22	Yellow	26	0.26	26	35
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#3				#4				Pooled 34
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	23	0.46	46	Blue	18	0.36	36	Blue	41	0.41	41	40
Green	7	0.14	14	Green	9	0.18	18	Green	16	0.16	16	15
Red	2	0.04	4	Red	3	0.06	6	Red	5	0.05	5	10
Yellow	18	0.36	36	Yellow	20	0.4	40	Yellow	38	0.38	38	35
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#5				#6				Pooled 56
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	22	0.44	44	Blue	18	0.36	36	Blue	40	0.4	40	40
Green	6	0.12	12	Green	9	0.18	18	Green	15	0.15	15	15
Red	5	0.1	10	Red	4	0.08	8	Red	9	0.09	9	10
Yellow	17	0.34	34	Yellow	19	0.38	38	Yellow	36	0.36	36	35
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	67	0.4466667	44.6667	Blue	60	0.4	40	Blue	127	0.4233333	42.3333	40
Green	23	0.1533333	15.3333	Green	27	0.18	18	Green	50	0.1666667	16.6667	15
Red	10	0.0666667	6.66667	Red	13	0.086667	8.6667	Red	23	0.0766667	7.66667	10
Yellow	50	0.3333333	33.3333	Yellow	50	0.333333	33.333	Yellow	100	0.3333333	33.3333	35
Total	150	1	100	Total	150	1	100	Total	300	1	100	100

Samples – 8.00

#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	22	0.44	44	Blue	26	0.52	52	Blue	48	0.48	48	40
Green	10	0.2	20	Green	3	0.06	6	Green	13	0.13	13	15
Red	6	0.12	12	Red	2	0.04	4	Red	8	0.08	8	10
Yellow	12	0.24	24	Yellow	19	0.38	38	Yellow	31	0.31	31	35
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#3				#4				Pooled 34
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	20	0.4	40	Blue	21	0.42	42	Blue	41	0.41	41	40
Green	8	0.16	16	Green	6	0.12	12	Green	14	0.14	14	15
Red	3	0.06	6	Red	3	0.06	6	Red	6	0.06	6	10
Yellow	19	0.38	38	Yellow	20	0.4	40	Yellow	39	0.39	39	35
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#5				#6				Pooled 56
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	15	0.3	30	Blue	23	0.46	46	Blue	38	0.38	38	40
Green	11	0.22	22	Green	12	0.24	24	Green	23	0.23	23	15
Red	6	0.12	12	Red	2	0.04	4	Red	8	0.08	8	10
Yellow	18	0.36	36	Yellow	13	0.26	26	Yellow	31	0.31	31	35
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	57	0.38	38	Blue	70	0.466667	46.667	Blue	127	0.4233333	42.333333	40
Green	29	0.193333	19.333	Green	21	0.14	14	Green	50	0.1666667	16.666667	15
Red	15	0.1	10	Red	7	0.046667	4.6667	Red	22	0.0733333	7.3333333	10
Yellow	49	0.326667	32.667	Yellow	52	0.346667	34.667	Yellow	101	0.3366667	33.666667	35
Total	150	1	100	Total	150	1	100	Total	300	1	100	100

Pooled across Sessions (6:30 + 8:00)

Super Pool
6:30
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	67	0.4467	44.667	Blue	60	0.4	40	Blue	127	0.42	42.33	40
Green	23	0.1533	15.333	Green	27	0.18	18	Green	50	0.17	16.67	15
Red	10	0.0667	6.6667	Red	13	0.087	8.66667	Red	23	0.08	7.667	10
Yellow	50	0.3333	33.333	Yellow	50	0.333	33.3333	Yellow	100	0.33	33.33	35
Total	150	1	100	Total	150	1	100	Total	300	1	100	100
8:00
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	57	0.38	38	Blue	70	0.467	46.6667	Blue	127	0.42	42.33	40
Green	29	0.1933	19.333	Green	21	0.14	14	Green	50	0.17	16.67	15
Red	15	0.1	10	Red	7	0.047	4.66667	Red	22	0.07	7.333	10
Yellow	49	0.3267	32.667	Yellow	52	0.347	34.6667	Yellow	101	0.34	33.67	35
Total	150	1	100	Total	150	1	100	Total	300	1	100	100
	Pooled 300		Pooled 300		Pooled 600
	n	p	n	p	n	p	Truth
Blue	124	0.4133	130	0.433	254	0.423	0.4
Green	52	0.1733	48	0.16	100	0.167	0.15
Red	25	0.0833	20	0.067	45	0.075	0.1
Yellow	99	0.33	102	0.34	201	0.335	0.35
Total	300	1	300	1	600	1	1

The structure of the bowl, expressed as color proportions, determines the basic structure of samples drawn from the bowl. Probability models allow the prediction of sample behavior, but said predictions are only as reliable as the validity of the original model and of the sampling procedures.

The foundation of statistical applications is the careful preparation of a study population and the random sampling procedures to go with it. Proper execution of this sampling procedure ensures a potable sample.

You are now ready to learn the Long Run Argument and Perfect Sample case types in 1^st Hourly Stuff.

Case Data (Excel)