Statistics 270 - Lecture 20

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Statistics 270 - Lecture 20
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• Last Daycompleted 5.1
• Today Parts of Section 5.3 and 5.4

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Sampling

• In chapter 1, we concerned ourselves with
  numerical/graphical summeries of samples (x1, x2,
  , xn) from some population
• Can view each of the Xis as random variables
• We will be concerned with random samples
• The xis are independent
• The xis have the same probability distribution
• Often called

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Definitions

• A parameter is a numerical feature of a
  distribution or population
• Statistic is a function of sample data (e.g.,
  sample mean, sample median)
• We will be using statistics to estimate
  parameters (point estimates)

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• Suppose you draw a sample and compute the value
  of a statistic
• Suppose you draw another sample of the same size
  and compute the value of the statistic
• Would the 2 statistics be equal?

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• Use statistics to estimate parameters
• Will the statistics be exactly equal to the
  parameter?
• Observed value of the statistics depends on the
  sample
• There will be variability in the values of the
  statistic over repeated sampling
• The statistic has a distribution of its own

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• Probability distribution of a statistic is called
  the sampling distribution (or distribution of the
  statistic)
• Is the distribution of values for the statistic
  based on all possible samples of the same size
  from the population?
• Based on repeated random samples of the same size
  from the population

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Example

• Large population is described by the probability
  distribution
• If a random sample of size 2 is taken, what is
  the sampling distribution for the sample mean?

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Sampling Distribution of the Sample Mean

• Have a random sample of size n
• The sample mean is
• What is it estimating?

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Properties of the Sample Mean

• Expected value
• Variance
• Standard Deviation

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Properties of the Sample Mean

• Observations

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Sampling from a Normal Distribution

• Suppose have a sample of size n from a N(m,s2)
  distribution
• What is distribution of the sample mean?

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Example

• Distribution of moisture content per pound of a
  dehydrated protein concentrate is normally
  distributed with mean 3.5 and standard deviation
  of 0.6.
• Random sample of 36 specimens of this concentrate
  is taken
• Distribution of sample mean?
• What is probability that the sample mean is less
  than 3.5?

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Central Limit Theorem

• In a random sample (iid sample) from any
  population with mean, m, and standard
  deviation, s, when n is large, the distribution
  of the sample mean is approximately normal.
• That is,

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Central Limit Theorem

• For the sample total,

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Implications

• So, for random samples, if have enough data,
  sample mean is approximately normally
  distributed...even if data not normally
  distributed
• If have enough data, can use the normal
  distribution to make probability statements about
  

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Example

• A busy intersection has an average of 2.2
  accidents per week with a standard deviation of
  1.4 accidents
• Suppose you monitor this intersection of a given
  year, recording the number of accidents per week.
• Data takes on integers (0,1,2,...) thus
  distribution of number of accidents not normal.
• What is the distribution of the mean number of
  accidents per week based on a sample of 52 weeks
  of data

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Example

• What is the approximate probability that is
  less than 2
• What is the approximate probability that there
  are less than 100 accidents in a given year?