Distributions, Probability, and the Normal Curve

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Distributions, Probability, and the Normal Curve

• Session 9

2
The Z-Table

• The body is the larger part of the distribution,
  the tail is the smaller section whether it is the
  right or left side
• Because the normal distribution is symmetrical,
  the proportions on the right hand side are
  exactly the same as the corresponding proportions
  on the left hand side.
• Although the z-score values will change signs
  from one side to the other, the proportions will
  always be positive.

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Probabilities, Proportions, and Z-Scores

• Finding proportions/probabilities for specific
  z-score values.
• Example z .25
• Example z 1.00
• Example z 1.50
• Example z -.50
• Example Life Expectancy
• Example What z score separates the top 10 of
  the distribution

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Probabilities and Proportions for Scores from a
Normal Distribution

• Transform the X values into z-scores.
• Use the z-table to look up the proportions
  corresponding to the z-scores values.
• Example p (X lt 130) where the mean is 100 and
  the standard deviation is 15

5

• Example p (55 lt X lt 65) where the mean is 58 and
  the standard deviation is 10
• Example What z score separates the top 15 of
  SAT scores?

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Looking Ahead to Inferential Statistics
Scores with a mean of 400, where plus 1.96
standard deviations is 440 and minus 1.96 is 360,
the probability is less than .05 or 5 that you
will find a score of greater than 440 from the
original population, so you would say that it is
likely that the treatment had in influence.
Original Population
Treatment
Sample
Treated Sample
7
The Logic of Statistical Inference
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Samples and Sampling Error

• Sampling Error is the discrepancy, or amount of
  error, between a sample statistic and its
  corresponding population parameter
• Samples include sampling error. They are also
  variable, in other words, samples will differ
  from one another even when drawn from the same
  population.

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Distribution of Sample Means

• The distribution of sample means is the
  collection of sample means for all the possible
  random samples of a particular size (n) that can
  be obtained from a population
• A sampling distribution is a distribution of
  statistics obtained by selecting all the possible
  samples of a specific size from a population

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Characteristics of the Distribution of Sample
Means

• The sample means tend to pile up around the
  population mean
• The distribution of sample means is approximately
  normal in shape
• We can use the distribution of sample means to
  answer probability questions about sample means
• If we have a distribution of 16 sample means, and
  only 1 is greater than 7, the probability of
  finding a mean greater than 7 is 1/16.

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Central Limit Theorem

• Even with four scores and a sample size of 2,
  there are 16 possible samples. So we rely on
  what we know about the central limit theorem to
  avoid having to actually find all possible
  samples.
• Central limit theorem For any population with a
  given mean and standard deviation, the
  distribution of sample means will approach a
  normal distribution as n approaches infinity.
• The mean of the distribution of sample means is
  equal to the population mean and is called the
  expected value of M

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• The standard deviation of the distribution of
  sample means is called the standard error of M.
  The standard error measures the standard amount
  of difference between the sample mean and the
  population mean that is reasonable to expect
  simply by chance.

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More about the Standard Error

• The size of the standard error is determined by
• The sample size large samples make the standard
  error smaller
• The population standard deviation large
  population standard deviations make the standard
  error bigger
• standard error is the population variance divided
  by the square root of the sample size

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Probability and the Distribution of Sample Means

• The primary use of the distribution of sample
  means is to find the probability associated with
  any specific sample statistic
• Example SAT scores
• The distribution of sample means will have the
  following characteristics
• The distribution is normal because the population
  of SAT scores is normal
• The distribution has a mean of 500 because the
  population mean is µ 500
• The distribution has a standard error of sM

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• A z-score for sample means
• Note when computing z for a single score, use
  the standard deviation, s. When computing z for
  a sample mean, you must use the standard error,
  sM
• Example What kind of sample mean would we
  probably obtain if the population mean is 500?

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More about the Standard Error

• Sampling Error The general concept of sampling
  error is that a sample typically will not provide
  a perfectly accurate representation of its
  population. There is typically a discrepancy.
• Standard Error The standard error tells us
  exactly how much error, on average, should exist
  between the sample mean and the unknown
  population mean.