Hypothesis Testing

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Hypothesis Testing

• CJ 526

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Probability

• Review
• P number of times an even can occur/
• Total number of possible event
• Bounding rule of probability
• Minimum value is 0
• Maximum value is 1

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Probability

• Probability of an event NOT occurring is the
  complement of an event
• Probability of an illness .2
• Probability that illness will not occur
• 1probability of event or 1 - .2 .8
• Odds of an event is the ratio
• Odds of illness .2/.8 or 1 to 4 or odds of not
  getting ill are 4 to 1

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Addition rule of p

• What is the probability of either one event OR
  another occurring?
• If the events are mutually exclusive, simply add
  the probabilities (Venn diagram)
• What is the p of having a boy or a girl?
• P 5. .5 1

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Addition rule

• If events are not mutually exclusive, the
  probability of A or B must be p (A) p (B) minus
  p (A and B both occurring at the same time)

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Multiplication rule

• What is the probability of A and B occurring?
• If the events are independent of one another,
  they can be multiplied
• What is the p of having both schizophrenia and
  epilepsy?

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Multiplication rule

• If two events are not independent then the p of
  two events occurring simultaneously equals the
  unconditional probability of A and the
  conditional probability of B given A

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Probability distributions

• A probability distribution is theoreticalwe
  expect it based on the laws of probability
• That is different from an empirical
  distributionone which we actually observe

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Normal probability distribution

• Probability distribution for continuous events
• Probability of an event occurring is higher in
  the center of the curve
• Declines for events at each of the two ends
  (tails) of the distribution
• Neither of the tails touches the x axis (infinity)

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Normal distribution

• Theoretical probability distribution
• Unimodal, symmetrical, bell-shaped curve
• Symmetrical draw a line down the center, left
  and right halves would be mirror images
• Can be expressed as a mathematical formula (p.
  220)

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Normal distribution

• Family of normal distributions
• Dependent on mean and SD
• (Illustrate)
• More spread out larger SD
• Narrower smaller SD

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Variations

• Skewness
• Skewed to the right or the left, as opposed to
  symmetry
• Kurtosis degree of peakedness or flatness

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Area under the normal curve

• Remember that for any continuous distribution
  there is a mean and SD
• Example Mean 10 and SD 2
• If the distribution is not skewed, the majority
  of scores will be from 8 to 12
• 8 and 12 are each one SD from the mean
• See p. 225

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Area under the normal curve

• If a distribution is normal, we can express
  standard deviation in terms of z scores
• A z score (a score the mean)/SD
• If we convert all our raw scores to z scores,
  then we get what is call the standard normal
  distribution
• It STANDARDIZES our scores

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Standard normal distribution

• Then distributions of different measures can be
  compared against one another
• The standard normal distribution has a mean of 0
  and an SD of one
• If you use the formula for z scores, all the
  scores can be converted
• If a distribution has a mean of 10, the z score
  for 10 will be (10-10)/SD 0

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Standard normal distribution

• If a distribution has a mean of 10 and an SD of
  2, the z score for 12 would be z (12-10)/2 1
• The z score for 8 would be z (8-10)/2 -1
• The negative and positive sign have meaning a
  sign means a score is above the mean

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Standard normal distribution

• A minus sign means the score is less than the
  mean
• The z score also tell about magnitudethe larger
  the z score, the further from the mean, and the
  smaller the z score, the closer to the mean

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Standard normal distribution

• We can also make statements about where an
  individual score is in relation to the rest of
  the distribution
• .3413 (or 34.13) of scores will fall between the
  mean and 1 SD
• .3413 (or 34.13) of scores will fall between the
  mean and 1 SD

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Standard normal distribution

• .6826 (0r 68.26) of scores will be between -1 and
  1 SD on a normal distribution
• Thus, when we see a mean and SD, if it is
  normally distributed, about 2/3 of the scores
  will fall between the mean the SD and the mean
  the SD

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Standard normal distribution

• 50 of the scores will be above the mean
• 50 of the scores will be below the mean
• .1359 (13.59) will fall between -1 and -2 SD and
  between 1 and 2 SD
• .0215 (2.15) will fall between -2 and -3 SD and
  2 and 3 SD
• See p. 223, illustrate

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Standardized normal distribution

• Tells us about any distribution
• Example of IQ scores, mean 100, SD 15
• 2/3 between 85 and 115
• Less (13.5) between 115 and 130, and 70 and 85
• About 2 between 130 and 145, and 55 and 70

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Standardized normal

• SAT scores, mean 500, SD 100
• Illustrate
• Use of z table, p. 724
• Reading the table

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Utility of the normal distribution

• Use of the normal distribution underlies many
  statistical tests
• Many variables not normally distributed
• However, the normal distribution useful anyway
  because of the apparently validity of the Central
  Limit Theorem

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Sampling distributions

• To understand the Central Limit Theorem, need to
  understand sampling distributions
• Say we draw many samples, and calculate a
  statistic for each sample, such as a mean
• When we draw the samples, the mean will not be
  the same each timethere will be variation

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Sampling distributions

• If you were to obtain some measure on several
  samples of patients with the same disorder, there
  would be variation in the mean of the measure for
  each sample.
• There is an actual mean for the entire population
  of patients that have the disorder, but that is
  not know, because we dont have measures for the
  whole population

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Sampling distributions

• However, we could obtain means based on a large
  number of samples
• Central limit theorem if an infinite number of
  random samples of size n are drawn from a
  population, the sampling distribution of the
  sample means will itself approach being normally
  distributed (even if the measure is not itself
  normally distributed)

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Number of subjects

• With sample sizes greater than 100, the Central
  Limit Theorem can be used
• If the measure is not terribly skewed, then
  samples could be around 50
• With sample sizes of less than 50, the central
  limit theorem probably should not be used.
• Application of the central limit theorem (ex)