Chapter 5 Section 1

1
Chapter 5 Section 1

• Sampling Distributions for Counts and Proportions

2
The Distribution of a Statistic

• A statistic from a random sample or randomized
  experiment is a random variable. The probability
  distribution of the statistic is its sampling
  distribution.
• Example 5.1

3
Population Distribution

• The population distribution of a variable is the
  distribution of its values for all members of the
  population. The population distribution is also
  the probability distribution of the variable when
  we choose one individual from the population at
  random. Ex N(64.5, 2.5)

4
The binomial distributions for sample counts

• The random variable X is a count of the
  occurrences of some outcome in a fixed number of
  observations.
• A sample proportion is p X / n, where n is the
  number of observations
• The goal of this section is to find the sampling
  distributions.

5
The Binomial Setting

• There are a fixed number n of observations.
• The n observations are all independent.
• Each observation falls into one of just two
  categories, which for convenience we call
  success and failure
• The probability of a success, call it p, is the
  same for each observation.

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Binomial Distributions

• The distribution of the count X of successes in
  the binomial setting is called the binomial
  distribution with parameters n and p. The
  parameter n is the number of observations, and p
  is the probability of a success on any one
  observation. The possible values of X are the
  whole numbers from 0 to n. As an abbreviation, we
  say that X is B(n,p).
• Example 5.2

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Binomial distributions in statistical sampling

• Example 5.3
• Sampling Distribution of a Count
• When the population is much larger than the
  sample, the count X of successes in an SRS of
  size n has approximately the B(n,p) distribution
  if the population proportion of success is p. (we
  will use this when the n is 10x the sample)

8
Finding binomial probabilities tables

• Table C, the calculator, and Example 5.4
• Example 5.5

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Binomial mean and standard deviation

• If a count X has the binomial distribution
  B(n,p), then
• Mu of X np
• Sigma of X sqrt(np(1-p))
• Example 5.6

10
Sample proportions

• Count is whole numbers between 0 and n.
• Proportions are always between 0 and 1.
• Example 5.7 (lengthythere must be a faster way)

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Mean and Standard Deviation of a Sample Proportion

• Let p-hat be the sample proportion of successes
  in an SRS of size n drawn from a large population
  having population proportion p of successes. The
  mean and standard deviation of p-hat are
• Mu of p-hat p
• Sigma of p-hat sqrt((p(1-p))/n)
• Example 5.8

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Normal Approximation for Counts and Proportions

• Draw an SRS of size n from a large population
  having population proportion p of success. Let X
  be the count of successes in the sample and p-hat
  X/n the sample proportion of successes. When n
  is large, the sampling distributions of these
  statistics are approximately normal
• X is approximately N(np, sqrt(np(1-p)))
• P-hat is approximately N(p, sqrt((p(1-p))/n))
• (we would prefer np gt 10 and n(1-p) gt 10)

13

• Example 5.9
• Example 5.10

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Binomial formulas

• Example 5.11
• Binomial Coefficient
• The number of ways of arranging k successes among
  n observations is given by the binomial
  coefficient
• n choose k n!/(k!(n-k)!)
• For k 0,1,2,,n

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Binomial Probability

• If X has the binomial distribution B(n,p) with n
  observations and probability p of success on each
  observation, the possible values of X are
  0,1,2,,n. If k is any one of these values, the
  binomial probability is
• P(X k) (n choose k)(pk)(1-p)(n-k)
• Example 5.12

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Daily Work

• Pg 390 -397
• 1, 5, 8, 10,
• 13, 16, 19, 21