Sampling Distributions

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Sampling Distributions Point Estimation
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Questions

• What is a sampling distribution?
• What is the standard error?
• What is the principle of maximum likelihood?
• What is bias (in the statistical sense)?
• What is a confidence interval?
• What is the central limit theorem?
• Why is the number 1.96 a big deal?

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Population

• Population Sample Space
• Population vs. sample
• Population parameter, sample statistic

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Parameter Estimation

• We use statistics to estimate parameters,
• e.g., effectiveness of pilot training,
  effectiveness of psychotherapy.

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Sampling Distribution (1)

• A sampling distribution is a distribution of a
  statistic over all possible samples.
• To get a sampling distribution,
• 1. Take a sample of size N (a given number like
  5, 10, or 1000) from a population
• 2. Compute the statistic (e.g., the mean) and
  record it.
• 3. Repeat 1 and 2 a lot (infinitely for large
  pops).
• 4. Plot the resulting sampling distribution, a
  distribution of a statistic over repeated
  samples.

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Suppose

• Population has 6 elements 1, 2, 3, 4, 5, 6 (like
  numbers on dice)
• We want to find the sampling distribution of the
  mean for N2
• If we sample with replacement, what can happen?

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1st 2nd M 1st 2nd M 1st 2nd M
1 1 1 3 1 2 5 1 3
1 2 1.5 3 2 2.5 5 2 3.5
1 3 2 3 3 3 5 3 4
1 4 2.5 3 4 3.5 5 4 4.5
1 5 3 3 5 4 5 5 5
1 6 3.5 3 6 4.5 5 6 5.5
2 1 1.5 4 1 2.5 6 1 3.5
2 2 2 4 2 3 6 2 4
2 3 2.5 4 3 3.5 6 3 4.5
2 4 3 4 4 4 6 4 5
2 5 3.5 4 5 4.5 6 5 5.5
2 6 4 4 6 5 6 6 6
Possible Outcomes
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Histogram
Sampling distribution for mean of 2 dice.
123456 21. 21/6 3.5
There is only 1 way to get a mean of 1, but 6
ways to get a mean of 3.5.
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Sampling Distribution (2)

• The sampling distribution shows the relation
  between the probability of a statistic and the
  statistics value for all possible samples of
  size N drawn from a population.

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Sampling Distribution Mean and SD

• The Mean of the sampling distribution is defined
  the same way as any other distribution (expected
  value).
• The SD of the sampling distribution is the
  Standard Error. Important and useful.
• Variance of sampling distribution is the expected
  value of the squared difference a mean square.
• Review

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Review

• What is a sampling distribution?
• What is the standard error of a statistic?

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Statistics as Estimators

• We use sample data compute statistics.
• The statistics estimate population values, e.g.,
• An estimator is a method for producing a best
  guess about a population value.
• An estimate is a specific value provided by an
  estimator.
• We want good estimates. What is a good
  estimator? What properties should it have?

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Maximum Likelihood (1)

• Likelihood is a conditional probability.
• L is the probability (say) that x has some value
  given that the parameter theta has some value.
  L1 is the probability of observing heights of 68
  inches and 70 data inches given adult
  malestheta. L2 is the probability of 68 and 70
  inches given adult females.
• Theta ( ) could be continuous or discrete.

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Maximum Likelihood (2)

• Suppose we know the function (e.g., binomial,
  normal) but not the value of theta.
• Maximum likelihood principle says take the
  estimate of theta that makes the likelihood of
  the data maximum.
• MLP says Choose the value of theta that makes
  this maximum

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Maximum Likelihood (3)

• Suppose we have 2 values hypothesized for
  proportions of male grad students at USF, 50 and
  40. We randomly sample 15 students and find that
  9 are male.
• Calculate likelihood for each using binomial
• The .50 estimate is better because the data are
  more likely.

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Likelihood Function
The binomial distribution computes probabilities
Likelihood
Theta (p value)
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Maximum Likelihood (4)

• In example, best (max like) estimate would be
  9/15 .60.
• There is a general class called maximum
  likelihood estimators that find values of theta
  that maximizes the likelihood of a sample result.
• ML is one principle of goodness of an estimator

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More Goodness (1)

• Bias. If E(statistic)parameter, the estimator
  is unbiased. If its unbiased, the mean of the
  sampling distribution equals the parameter. The
  sample mean has this property .
  Sample variance is biased.

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More Goodness (2)

• Efficiency size of the sampling variance.
• Relative Efficiency. Relative efficiency is the
  ratio of two sampling variances.
• More efficient statistics have smaller sampling
  variances, smaller standard error, and are
  preferred because if both are unbiased, one is
  closer than the other to the parameter on average.

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Goodness (3)

• Sometimes we trade off bias and efficiency. A
  biased estimator is sometime preferred if it is
  more efficient, especially if the magnitude of
  bias is known.
• Resistance. Indicates minimal influence of
  outliers. Median is more resistant than the mean.

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Sampling Distribution of the Mean

• Unbiased
• Variance of sampling distribution of means based
  on N obs
• Standard Error of the Mean
• Law of large numbers Large samples produce
  sample estimates very close to the parameter.

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Unbiased Estimate of Variance

• It can be shown that
• The sample variance is too small by a factor of
  (N-1)/N.
• We fix with
• Although the variance is unbiased, the SD is
  still biased, but most inferential work is based
  on the variance, not SD.

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Review

• What is the principle of maximum likelihood?
• Define
• Bias
• Efficiency
• Resistance
• Is the sample variance (SS divided by N) a biased
  estimator?

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Interval Estimation

• Use the standard error of the mean to create a
  bracket or confidence interval to show where good
  estimates of the mean are.
• The sampling distribution of the mean is nice
  when Ngt20. Therefore
• Suppose M100, SD14, N49. Then SDM14/72.
  Bracket 100-6 94 to 1006 106 is 94 to 106.
  P is probability of sample not mu.

Unimodal and symmetric
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Review

• What is a confidence interval?
• Suppose M 50, SD 10, and N 100. What is the
  confidence interval?

SEM 10/sqrt(100) 10/10 1 CI (lower)
M-3SEM 50-3 47 CI (upper) M3SEM 503
53 CI 47 to 53
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Central Limit Theorem

• 1. Sampling distribution of means becomes normal
  as N increases, regardless of shape of original
  distribution.
• 2. Binomial becomes normal as N increases.
• 3. Applies to other statistics as well (e.g.,
  variance)

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Properties of the Normal

• If a distribution is normal, the sampling
  distribution of the mean is normal regardless of
  N.
• If a distribution is normal, the sampling
  distributions of the mean and variance are
  independent.

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Confidence Intervals for the Mean

• Over samples of size N, the probability is .95
  for
• Similarly for sample values of the mean, the
  probability is .95 that
• The population mean is likely to be within 2
  standard errors of the sample mean.
• Can use the Normal to create any size confidence
  interval (85, 99, etc.)

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Size of the Confidence Interval

• The size of the confidence interval depends on
  desired certainty (e.g., 95 vs 99 pct) and the
  size of std error of mean ( ).
• Std err of mean is controlled by population SD
  and sample size. Can control sample size.
• SD 10. If N25 then SEM 2 and CI width is about
  8. If N100, then SEM 1 and CI width is about
  4. CI shrinks as N increases. As N gets large,
  decreasing change in CI because of square root.
  Less bang for buck as N gets big.

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Review

• What is the central limit theorem?
• Why is the number 1.96 a big deal?
• Assume that scores on a curiosity scale are
  normally distributed. If the sample mean is 50
  based on 100 people and the population SD is 10,
  find an approx 99 pct CI for the population mean.