Probability and Sampling Distributions

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Probability and Sampling Distributions
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Definitions

• Population parameters ?, ?, p
• Sample statistics x, s, p
• Statistics are often used to estimate parameters.
  Probability distributions of statistics allow us
  to attach measures of accuracy to the estimates.

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Definitions

• Random phenomenon individual outcomes are
  uncertain, but there is a regular distribution of
  outcomes in a large number of repetitions.
• Probability proportion of times the outcome
  would occur in a very large number of
  repetitions.
• Examples flipping a coin, rolling a die,
  tossing a tack, spinning a penny, shooting free
  throws?, etc.

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Examples

• Flipping a penny
• Binomial trials independent, 2 possible
  outcomes, constant probability of success
• What is streakiness?
• Spinning a pennywhat is the probability of
  heads?
• Shooting free throws (4.11 on p. 219)are free
  throws a random phenomenon or are there hot
  streaks and cold streaks?

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More definitions

• The sample space S of a random phenomenon is the
  set of all possible outcomes.
• An event is any outcome or a set of outcomes of a
  random phenomenon. (A subset of S)

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Probability Rules

• The probability of any event, A, satisfies 0 ?
  P(A) ? 1.
• P(S) 1
• P(A does not occur) 1-P(A)
• If two events A and B are disjoint (no outcomes
  in common), then P(A or B) P(A) P(B). (E.g.,
  probability of obtaining a 6 or an odd when a die
  is rolled 1/61/24/6. This does not work for
  prob. of a 5 or an odd.)

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Example

• Winning in Craps
• Probability of a 7 or 11 on first roll of dice?
• Probability of 2, 3, or 12 on first roll?
• The overall probability of winning gets
  complicated, but we know that it is (slightly)
  less than ____.

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More Definitions

• A random variable is a variable whose value is a
  numerical outcome of a random phenomenon.
  Examples X0 for tails, 1 for heads, Xarea of
  randomly selected rectangle, Xmean of a random
  sample of 5 rectangles,
• The probability distribution of a random variable
  tells us what values it can take and how to
  assign probabilities to those values.

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Sampling Distribution

• When the probability distribution is for a sample
  statistic (e.g. X) we refer to it as the sampling
  distribution. (See p. 241)
• Law of Large Numbers
• Central Limit Theorem

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Law of Large Numbers

• As the number of observations, n, in a random
  sample from any population increases, the sample
  mean, x, of the observed values is likely to be
  close to the true mean, ?, of the population.
• The sampling distribution of x has mean ?
  (unbiased) and standard deviation ?/?n. (Note
  the effect of n on the standard deviation.)

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Law of Large Numbers

• Apply to 4.43 for n1, n5, n50.

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Central Limit Theorem

• Minitab simulation.
• Draw an SRS of size n from any population
  (skewed, symmetric, discrete, etc.) with mean ?
  and finite standard deviation ?. When n is
  large, the sampling distribution of the sample
  mean x is approximately normal with mean ? and
  standard deviation ?/?n.
• How large n must be depends on the shape of the
  population, but often people use a guideline of
  25-30, which usually works well even for highly
  skewed data.

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Assignment

• Read Chapter 4
• Work 1-4, 6, 10, 12, 21, 24, 25, 33 (another
  casino game)
• Work 38, 39, 40 (use Minitab or Excel?), 41,
  42, 43, 44, 45
• More Ch. 4 problems to try 46, 47 (if you had
  trouble with 40), 51, 52, 53, 57, 61, 63, 65