Summaries

Session 1.1

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.

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.

In our first case, we begin with a fair, six-sided die. We track predicted and observed face values in six random samples of 50 tosses of the die. We then compare our samples to what is expected under the fair model.

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 second case, we begin with a color bowl whose true color frequencies are known. We compute a population frequency model and then compute the expected structure for random samples from that bowl. We obtain six (6) random samples, each consisting of 50 draws with replacement (SRS/WR). We then compute sample color frequencies and compare them to 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

Process

We have a fair, six-sided die, with face values 1, 2, 3, 4, 5 and 6. Prior to each toss of the die, a member of the group predicts the face value that will be observed on that toss. Upon tossing the die, the group notes the observed face value, as well as the correctness (or lack thereof) of the prediction. Each group produces a sample of 50 tosses.

Sample Worksheet

Prediction and the Fair Die

Sample Grid n=50

Each cell corresponds to a single toss of the die.

X
X
X
X
X	X	X	X	X	X	X	X	X	X

Sample results are tabulated in the form below.

Face Value	Count	Prediction	Count
1		Hit(Correct)
2		Miss(Incorrrect)
3		Total
4
5
6
Total

Samples – Face Values and Predictions

Here are the results for our six samples. You should be able to begin with the counts in the table and work out the proportions and percentages.

6:30 SamplesIn 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

6:30

#1				#2				12
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	11	0.22	22	1	8	0.16	16	1	19	0.19	19
2	8	0.16	16	2	8	0.16	16	2	16	0.16	16
3	10	0.2	20	3	6	0.12	12	3	16	0.16	16
4	7	0.14	14	4	5	0.1	10	4	12	0.12	12
5	9	0.18	18	5	11	0.22	22	5	20	0.2	20
6	5	0.1	10	6	12	0.24	24	6	17	0.17	17
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	10	0.2	20	Hit	7	0.14	14	Hit	17	0.17	17
Miss	40	0.8	80	Miss	43	0.86	86	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	7	0.14	14	1	9	0.18	18	1	16	0.16	16
2	7	0.14	14	2	10	0.2	20	2	17	0.17	17
3	11	0.22	22	3	9	0.18	18	3	20	0.2	20
4	9	0.18	18	4	7	0.14	14	4	16	0.16	16
5	10	0.2	20	5	7	0.14	14	5	17	0.17	17
6	6	0.12	12	6	8	0.16	16	6	14	0.14	14
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	5	0.1	10	Hit	7	0.14	14	Hit	12	0.12	12
Miss	45	0.9	90	Miss	43	0.86	86	Miss	88	0.88	88
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	15	0.3	30	1	11	0.22	22	1	26	0.26	26
2	6	0.12	12	2	5	0.1	10	2	11	0.11	11
3	8	0.16	16	3	9	0.18	18	3	17	0.17	17
4	7	0.14	14	4	11	0.22	22	4	18	0.18	18
5	8	0.16	16	5	6	0.12	12	5	14	0.14	14
6	6	0.12	12	6	8	0.16	16	6	14	0.14	14
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
Pooled				Pooled				Pooled
135				246				123456
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	33	0.22	22	1	28	0.186666667	18.6667	1	61	0.203333333	20.3333
2	21	0.14	14	2	23	0.153333333	15.3333	2	44	0.146666667	14.6667
3	29	0.1933333	19.3333	3	24	0.16	16	3	53	0.176666667	17.6667
4	23	0.1533333	15.3333	4	23	0.153333333	15.3333	4	46	0.153333333	15.3333
5	27	0.18	18	5	24	0.16	16	5	51	0.17	17
6	17	0.1133333	11.3333	6	28	0.186666667	18.6667	6	45	0.15	15
Total	150	1	100	Total	150	1	100	Total	300	1	100
Prediction				Prediction				Prediction
Hit	21	0.14	14	Hit	21	0.14	14	Hit	42	0.14	14
Miss	129	0.86	86	Miss	129	0.86	86	Miss	258	0.86	86
Total	150	1	100	Total	150	1	100	Total	300	1	100

Face Value 1: 20.3% (Sample) versus 16.67% (Fair Model)

Face Value 2: 14.7% (Sample) versus 16.67% (Fair Model)

Face Value 3: 17.7% (Sample) versus 16.67% (Fair Model)

Face Value 4: 15.3% (Sample) versus 16.67% (Fair Model)

Face Value 5: 17% (Sample) versus 16.67% (Fair Model)

Face Value 6: 15% (Sample) versus 16.67% (Fair Model)

Prediction “Hit”: 14% (Sample) versus 16.67% (Fair Model)

Prediction “Miss”: 86% (Sample) versus 83.33% (Fair Model))

8:00 Samples

#1				#2				12
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	8	0.16	16	1	11	0.22	22	1	19	0.19	19
2	8	0.16	16	2	6	0.12	12	2	14	0.14	14
3	7	0.14	14	3	6	0.12	12	3	13	0.13	13
4	9	0.18	18	4	5	0.1	10	4	14	0.14	14
5	12	0.24	24	5	15	0.3	30	5	27	0.27	27
6	6	0.12	12	6	7	0.14	14	6	13	0.13	13
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	5	0.1	10	Hit	6	0.12	12	Hit	11	0.11	11
Miss	45	0.9	90	Miss	44	0.88	88	Miss	89	0.89	89
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	6	0.12	12	1	9	0.18	18	1	15	0.15	15
2	6	0.12	12	2	7	0.14	14	2	13	0.13	13
3	10	0.2	20	3	8	0.16	16	3	18	0.18	18
4	11	0.22	22	4	9	0.18	18	4	20	0.2	20
5	6	0.12	12	5	9	0.18	18	5	15	0.15	15
6	11	0.22	22	6	8	0.16	16	6	19	0.19	19
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	7	0.14	14	Hit	5	0.1	10	Hit	12	0.12	12
Miss	43	0.86	86	Miss	45	0.9	90	Miss	88	0.88	88
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	3	0.06	6	1	7	0.14	14	1	10	0.1	10
2	13	0.26	26	2	8	0.16	16	2	21	0.21	21
3	6	0.12	12	3	3	0.06	6	3	9	0.09	9
4	10	0.2	20	4	10	0.2	20	4	20	0.2	20
5	10	0.2	20	5	13	0.26	26	5	23	0.23	23
6	8	0.16	16	6	9	0.18	18	6	17	0.17	17
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	9	0.18	18	Hit	6	0.12	12	Hit	15	0.15	15
Miss	41	0.82	82	Miss	44	0.88	88	Miss	85	0.85	85
Total	50	1	100	Total	50	1	100	Total	100	1	100
Pooled				Pooled				Pooled
135				246				123456
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	17	0.113333	11.333	1	27	0.18	18	1	44	0.146667	14.667
2	27	0.18	18	2	21	0.14	14	2	48	0.16	16
3	23	0.153333	15.333	3	17	0.113333	11.333	3	40	0.133333	13.333
4	30	0.2	20	4	24	0.16	16	4	54	0.18	18
5	28	0.186667	18.667	5	37	0.246667	24.667	5	65	0.216667	21.667
6	25	0.166667	16.667	6	24	0.16	16	6	49	0.163333	16.333
Total	150	1	100	Total	150	1	100	Total	300	1	100
Prediction				Prediction				Prediction
Hit	21	0.14	14	Hit	17	0.113333	11.333	Hit	38	0.126667	12.667
Miss	129	0.86	86	Miss	133	0.886667	88.667	Miss	262	0.873333	87.333
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

8:00

Face Value 1: 14.67% (Sample) versus 16.67% (Fair Model)

Face Value 2: 16% (Sample) versus 16.67% (Fair Model)

Face Value 3: 13.3% (Sample) versus 16.67% (Fair Model)

Face Value 4: 18% (Sample) versus 16.67% (Fair Model)

Face Value 5: 21.7% (Sample) versus 16.67% (Fair Model)

Face Value 6: 16.3% (Sample) versus 16.67% (Fair Model)

Prediction “Hit”: 12.7% (Sample) versus 16.67% (Fair Model)

Prediction “Miss”: 87.3% (Sample) versus 83.33% (Fair Model))

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.

Process

We have a four color bowl, with blue, green, red and yellow marbles. Prior to each draw from the bowl, the bowl is thoroughly mixed, giving each resident marble an approximately equal chance of selection. After mixing, a blind (made without looking into the bowl) draw of a single marble is made. The group notes the color of the marble, and the marble is returned to the bowl – this is sampling with replacement. The mixing makes the sampling random. Each group produces a sample of 50 tosses.

Each cell corresponds to a single draw with replacement from the bowl.

Sample Grid (n=50)

0

0

0

0

0

0

0

0

0

0

Sample results are tabulated in the form below.

Table – Draws with Replacement

Color	Count
Blue
Green
Red
Yellow
Total

Samples from the Color Bowl

Here are the six samples from our groups. You should be able to begin with the counts in the table and work out the proportions and percentages.

6:30 Samples

#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	9	0.18	18	Blue	13	0.26	26	Blue	22	0.22	22	17.647
Green	9	0.18	18	Green	4	0.08	8	Green	13	0.13	13	11.765
Red	13	0.26	26	Red	18	0.36	36	Red	31	0.31	31	29.412
Yellow	19	0.38	38	Yellow	15	0.3	30	Yellow	34	0.34	34	41.176
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	8	0.16	16	Blue	7	0.14	14	Blue	15	0.15	15	17.647
Green	5	0.1	10	Green	6	0.12	12	Green	11	0.11	11	11.765
Red	17	0.34	34	Red	23	0.46	46	Red	40	0.4	40	29.412
Yellow	20	0.4	40	Yellow	14	0.28	28	Yellow	34	0.34	34	41.176
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	9	0.18	18	Blue	7	0.14	14	Blue	16	0.16	16	17.647
Green	1	0.02	2	Green	6	0.12	12	Green	7	0.07	7	11.765
Red	22	0.44	44	Red	14	0.28	28	Red	36	0.36	36	29.412
Yellow	18	0.36	36	Yellow	23	0.46	46	Yellow	41	0.41	41	41.176
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	26	0.173333	17.333	Blue	27	0.18	18	Blue	53	0.176667	17.667	17.647
Green	15	0.1	10	Green	16	0.106667	10.67	Green	31	0.103333	10.333	11.765
Red	52	0.346667	34.667	Red	55	0.366667	36.67	Red	107	0.356667	35.667	29.412
Yellow	57	0.38	38	Yellow	52	0.346667	34.67	Yellow	109	0.363333	36.333	41.176
Total	150	1	100	Total	150	1	100	Total	300	1	100	100
Bowl/Model
Color	Count	Proportion	Percent	E50	E100	E150	E200	E250	E300
Blue	3	0.176471	17.647	8.82	17.6	26.47059	35.29	44.11765	52.9
Green	2	0.117647	11.765	5.88	11.8	17.64706	23.53	29.41176	35.3
Red	5	0.294118	29.412	14.7	29.4	44.11765	58.82	73.52941	88.2
Yellow	7	0.411765	41.176	20.6	41.2	61.76471	82.35	102.9412	124
Total	17	1	100	50	100	150	200	250	300

Blue: 17.7% (Sample) versus17.6% (Model)

Green: 10.3% (Sample) versus 11.8% (Model)

Red: 35.7% (Sample) versus 29.4% (Model)

Yellow: 36.3% (Sample) versus 41.2% (Model)

8:00 Samples

8:00
Color Bowl I - Sampling with Replacement
#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	10	0.2	20	Blue	6	0.12	12	Blue	16	0.16	16	11.1111
Green	12	0.24	24	Green	8	0.16	16	Green	20	0.2	20	27.7778
Red	25	0.5	50	Red	31	0.62	62	Red	56	0.56	56	50
Yellow	3	0.06	6	Yellow	5	0.1	10	Yellow	8	0.08	8	11.1111
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	6	0.12	12	Blue	5	0.1	10	Blue	11	0.11	11	11.1111
Green	9	0.18	18	Green	14	0.28	28	Green	23	0.23	23	27.7778
Red	31	0.62	62	Red	26	0.52	52	Red	57	0.57	57	50
Yellow	4	0.08	8	Yellow	5	0.1	10	Yellow	9	0.09	9	11.1111
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	7	0.14	14	Blue	4	0.08	8	Blue	11	0.11	11	11.1111
Green	16	0.32	32	Green	13	0.26	26	Green	29	0.29	29	27.7778
Red	21	0.42	42	Red	29	0.58	58	Red	50	0.5	50	50
Yellow	6	0.12	12	Yellow	4	0.08	8	Yellow	10	0.1	10	11.1111
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	23	0.1533333	15.333	Blue	15	0.1	10	Blue	38	0.1266667	12.667	11.1111
Green	37	0.2466667	24.667	Green	35	0.2333333	23.333	Green	72	0.24	24	27.7778
Red	77	0.5133333	51.333	Red	86	0.5733333	57.333	Red	163	0.5433333	54.333	50
Yellow	13	0.0866667	8.6667	Yellow	14	0.0933333	9.3333	Yellow	27	0.09	9	11.1111
Total	150	1	100	Total	150	1	100	Total	300	1	100	100
Bowl/Model
Color	Count	Proportion	Percent	E50	E100	E150	E200	E250	E300
Blue	2	0.1111111	11.111	5.5555556	11.11	16.666667	22.222	27.777778	33.333
Green	5	0.2777778	27.778	13.888889	27.78	41.666667	55.556	69.444444	83.333
Red	9	0.5	50	25	50	75	100	125	150
Yellow	2	0.1111111	11.111	5.5555556	11.11	16.666667	22.222	27.777778	33.333
Total	18	1	100	50	100	150	200	250	300

Blue: 12.7% (Sample) versus 11.1% (Model)

Green: 24% (Sample) versus 27.8% (Model)

Red: 54.3% (Sample) versus 50% (Model)

Yellow: 9% (Sample) versus 11.1% (Model)

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,

nblue ~ number of blue draws in sample ntotal ~ total number of draws per sample pblue = nblue / ntotal pblue ~ proportion of sample draws showing blue pctblue = 100*pblue pctblue ~ percent of sample draws showing blue Nblue ~ number of blue marbles in bowl Ntotal ~ total number of marbles in bowl Pblue = Nblue / Nblue Pblue ~ proportion of marbles in bowl that are blue

ngreen ~ number of green draws in sample ntotal ~ total number of draws per sample pgreen = ngreen / ngreen pgreen ~ proportion of sample draws showing green pctgreen = 100*pgreen pctgreen ~ percent of sample draws showing green Ngreen ~ number of green marbles in bowl Ntotal ~ total number of marbles in bowl Pgreen = Ngreen / Ngreen Pgreen ~ proportion of marbles in bowl that are green

nred ~ number of red draws in sample ntotal ~ total number of draws per sample pred = nred / nred pred ~ proportion of sample draws showing red pctred = 100*pred pctred ~ percent of sample draws showing red Nred ~ number of red marbles in bowl Ntotal ~ total number of marbles in bowl Pred = Nred / Nred Pred ~ proportion of marbles in bowl that are red

nyellow ~ number of yellow draws in sample ntotal ~ total number of draws per sample pyellow = nyellow / nyellow pyellow ~ proportion of sample draws showing yellow pctyellow = 100*pyellow pctyellow ~ percent of sample draws showing yellow Nyellow ~ number of yellow marbles in bowl Ntotal ~ total number of marbles in bowl Pyellow = Nyellow / Nyellow Pyellow ~ proportion of marbles in bowl that are yellow

We didn’t get to these, so read up on Ellsberg I and Ellsberg II.

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.