Using%20Probability%20to%20Make

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Chapter 6

• Using Probability to Make
• Decisions about Data

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Going Forward

• Your goals in the chapter are to learn
• What probability is
• How to compute the probability of raw scores and
  sample means using z-scores
• How random sampling should produce a
  representative sample
• How sampling error may produce an
  unrepresentative sample
• How to use a sampling distribution of means to
  decide whether a sample represents a particular
  population

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Random Sampling

• Selecting a sample so that all events or
  individuals in the population have an equal
  chance of being selected is known as random
  sampling.

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Probability

• The probability of an event is equal to the
  events relative frequency in the population of
  possible events that can occur
• The symbol for probability is p

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Probability Distributions
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Probability Distributions

• A probability distribution indicates the
  probability of all possible events in a
  population
• An empirical probability distribution is created
  by observing the relative frequency of every
  event in the population
• A theoretical probability distribution is based
  on how we assume nature distributes events in the
  population

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Obtaining Probability from the Standard Normal
Curve
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Probability of Individual Scores

• The proportion of the total area under the
  standard normal curve for particular scores
  equals the probability of those scores.

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The Probability of Sample Means

• One type of theoretical probability distribution
  known as the sampling distribution of means is
  used to determine the probability of randomly
  obtaining any particular sample means.

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Sampling Distribution of SATMeans When N 25
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Probability of Sample Means

• The probability of selecting a particular sample
  mean is the same as the probability of randomly
  selecting a sample of participants whose scores
  produce that sample mean
• The larger the absolute value of a sample means
  z-score, the less likely the mean is to occur
  when samples are drawn from the underlying raw
  score population

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Random Sampling and Sampling Error
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Representative Samples

• A representative sample is one in which the
  characteristics of the individuals and scores in
  the sample accurately reflect the characteristics
  of the individuals and scores in the population
• Sampling error occurs when random chance produces
  a sample statistic (e.g., s2) not equal to the
  population parameter it represents (e.g., s2)

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Which Is It?

• It is always possible to obtain a sample that is
  not representative
• Therefore, any sample might either poorly
  represent one population because of sampling
  error or accurately represent a different
  population

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Deciding Whether a Sample Represents a Population
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Likely vs. Unlikely

• Sample mean A is likely. Sample mean B is
  unlikely.

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Region of Rejection

• At some point, a sample mean is so far above or
  below the population mean it is unbelievable that
  chance produced such an unrepresentative sample
• The area beyond these points is called the region
  of rejection

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Region of Rejection

• The region of rejection is the part of a sampling
  distribution containing values so unlikely we
  reject the idea they represent the underlying
  raw score population.

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A Sampling Distribution of Means Showing the
Region of Rejection
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Criterion

• The criterion is the probability defining samples
  as unlikely to be representing the raw score
  population.

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Critical Value

• A critical value marks the inner edge of the
  region of rejection
• For a criterion of .05, the area in each tail
  equals .025
• 1.96 is the critical value of z for a criterion
  of .05 in a two-tailed test

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Rejection Rule

• When a samples z-score lies beyond the critical
  value, reject the idea the sample represents the
  underlying raw score population reflected by the
  sampling distribution
• When the z-score does not lie beyond the critical
  value, retain the idea the sample represents the
  underlying raw score population

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Summary

• Set up the sampling distribution
• Select the criterion (e.g., .05)
• Locate the region of rejection
• Determine the critical value (e.g., 1.96 in a
  two-tailed test with a criterion of .05)

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Summary

• Compute the sample mean and its z-score
• Compute the standard error of the mean ( )
• Compute z using and the m of the sampling
  distribution

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One-Tailed and Two-Tailed Tests

• Two-tailed testswe reject the idea the sample
  mean is representative if it falls in either the
  negative tail or the positive tail of the
  distribution
• One-tailed tests
• If we are interested in positive z-scores, reject
  the idea the sample mean is representative only
  if it falls in the positive tail
• If we are interested in negative z-scores, reject
  the idea the sample mean is representative only
  if it falls in the negative tail

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One-Tailed Tests
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Example

• A sample of 10 scores yields a sample mean of
  305. Does the sample represent the population
  where and ?

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Example

• With a criterion of 0.05 and a region of
  rejection in two tails, the critical value is
  ?1.96.
• Since the sample z of 2.53 is beyond 1.96, it
  is in the region of rejection. The sample does
  not represent the population.