Lecture 15

Sampling

DATA 8

Summer 2017

Slides created by John DeNero (denero@berkeley.edu), Ani Adhikari (adhikari@berkeley.edu), and Sam Lau (samlau95@berkeley.edu)

Announcements

Monty Hall Problem

Probability

Probability

• Lowest value: 0
• Chance of event that is impossible
• Highest value: 1 (or 100%)
• Chance of event that is certain

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• If an event has chance 70%, then the chance that it doesn’t happen is
• 100% - 70% = 30%
• 1 - 0.7 = 0.3

Equally Likely Outcomes

Assuming all outcomes are equally likely, the chance of an event A is:

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number of outcomes that make A happen

P(A) = ---------------------------------------------------------------

total number of outcomes

(Demo)

Fraction of a Fraction

(Demo)

Multiplication Rule

Chance that two events A and B both happen

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= P(A happens) x P(B happens given that A has happened)

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• The answer is less than or equal to each of the two chances being multiplied
• The more conditions you have to satisfy, the less likely you are to satisfy them all

(Demo)

Addition Rule

If event A can happen in exactly one of two ways, then

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P(A) = P(first way) + P(second way)

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• The answer is greater than or equal to the chance of each individual way

Example: At Least One Head

• In 3 tosses:
• Any outcome except TTT
• P(TTT) = (½) x (½) x (½) = ⅛
• P(at least one head) = 1 - P(TTT) = ⅞ = 87.5%

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• In 10 tosses:
• 1 - (½)**10
• 99.9%

(Demo)

Attendance

bit.ly/at-d8

Sampling

Sampling

• Deterministic sample:
• Sampling scheme doesn’t involve chance

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• Probability sample:
• Before the sample is drawn, you have to know the selection probability of every group of people in the population
• Not all individuals have to have equal chance of being selected

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(Demo)

Sample of Convenience

• Example: sample consists of whoever walks by
• Just because you think you’re sampling “at random”, doesn’t mean you are.
• If you can’t figure out ahead of time
• what’s the population
• what’s the chance of selection, for each group in the population

then you don’t have a random sample

Distributions

Probability Distribution

• Random quantity with various possible values

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• “Probability distribution”:
• All the possible values of the quantity
• The probability of each of those values

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• In some cases, the probability distribution can be worked out mathematically without ever generating �(or simulating) the random quantity

(Demo)

Empirical Distribution

• Based on observations

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• Observations can be from repetitions of an experiment

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• “Empirical Distribution”
• All observed values
• The proportion of counts of each value

(Demo)

Large Random Samples

Law of Averages

If a chance experiment is repeated many times,

independently and under the same conditions,

then the proportion of times that an event occurs

gets closer to the theoretical probability of the event

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As you increase the number of rolls of a die, the proportion of times you see the face with five spots gets closer to 1/6

(Demo)

Large Random Samples

If the sample size is large,

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then the empirical distribution of a uniform random sample

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resembles the distribution of the population,

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with high probability