Probability and Hypothesis Testing Review

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Probability and Hypothesis Testing Review

• Exam 2

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Basic Probability Terms Ideas

• Independent Events
• Events are independent when the occurrence of one
  has no effect on the probability of occurrence of
  the other
• Mutually Exclusive Events
• Two events are mutually exclusive when the
  occurrence of one precludes the occurrence of the
  other
• Exhaustive
• A set of events that represents all possible
  outcomes (uses the entire sample space)
• Probability range from 0.0 to 1.0
• The closer you get to 0.0 or 1.0 the more
  unlikely the event
• Probabilities CANNOT be negative or greater than
  1.0

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Sampling with Replacement

• Sampling in which the item drawn on trial N is
  replaced before the drawing on trial N1
• The total number of outcomes the number of
  outcomes in the event does not change
• Ex
• If I reach into the bag of MMs what is the
  probability of getting a green?
• I end up drawing a green MM and I put it back.
  Now what is the probability of drawing a green
  MM?

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Sampling without Replacement

• Sampling in which the item drawn on trial N is
  NOT replaced before the drawing on trial N1
• The total number of outcomes always changes the
  number of outcomes in the event could also change
• Ex 1
• If I reach into the bag of MMs what is the
  probability of getting a green?
• I end up drawing a green MM and I eat it. Now
  what is the probability of drawing a green MM?

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Additive Rule

• Given a set of mutually exclusive events, the
  probability of the occurrence of one event or
  another is equal to the sum of their separate
  probabilities
• p(A or B) p(A) p(B) if mutually exclusive
• Look for the keyword or or a description of one
  or another event happening
• Ex 1
• What is the probability of grabbing a brown or a
  red MM?

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Multiplicative Rule

• The probability of the joint occurrence of 2 or
  more independent events is the product of their
  individual probabilities
• p(A and B) p(A) p(B) for independent events
• Look for the keyword and or a description of
  events happening together
• Ex 1
• I reach into the MM again. The is the
  probability that I will grab both a red and green
  MM?
• Notice that the probability decreases as we add
  events

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Types Of Probability

• Marginal Probability
• The probability of one event ignoring the
  occurrence or non-occurrence of some other
  (Unconditional P)
• Joint Probability
• The probability of the co-occurrence of 2 or more
  events denoted p(A,B) or p(A and B)
• Ex The p of grabbing a red and green MM
• Conditional Probability
• The probability of 1 event given the occurrence
  of some other event denoted p(AB)

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Hypothesis Testing Ideas Definitions

• We use sample statistics to estimate population
  parameters
• The exact value of the sample statistic can vary
  from sample to sample due to who is in the sample
• Sampling Error
• Variability of a statistic from sample to sample
  due to chance

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Sampling Distributions

• The distribution of a statistic over repeated
  sampling from a specified population
• This assumes a normal shape and imagines an
  infinite number of samples drawn
• It tells us specifically what degree of
  sample-to-sample variability we can expect by
  chance as a function of our sampling error
• Standard Error
• The standard deviation of a sampling distribution
  (the variability associated with the values)

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Sampling Distributions (cont.)

• We can have a sampling distribution of a
  statistic such as, the mean, mode, median, etc.
• Sampling distribution of the mean
• The distribution of sample means over repeated
  sampling from 1 population
• Standard error of the mean (?x)
• ?x ? / sqrt(n)
• This tells us on average how far the sample means
  are from the population mean

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Central Limit Theorem

• As n increases, sampling distribution statistics
  become better estimates of the population
  parameters
• For the sampling distribution of means, as n
  increases or for large n,
• ?x ?
• ?x ?/ Sqrt(n)
• As n increase the sampling distribution becomes
  normal regardless of the parent population

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Hypotheses

• Alternative or Research hypothesis (H1)
• The question the experiment was designed to
  examine
• This is the hypothesis we want to support
• ?A gt ?B or ?A lt ?B or ?A ? ?B
• Null Hypothesis (H0)
• The hypothesis of no difference or no
  relationship
• This is the hypothesis we test
• ?A lt ?B or ?A gt ?B or ?A ?B
• We always start off by assuming that the null
  hypothesis is true create a sampling
  distribution of the statistic based on this
  assumption
• If we find a difference, we can say the null
  hypothesis is false (i.e. reject the null)

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The Process of Hypothesis Testing

• Step 1 We form our hypotheses both H1 and H0
• Step 2 We determine our decision criteria
• Alpha, 1 or 2 tailed, the critical values
• Step 3 We compute our test statistic
• Step 4 We evaluate the statistical result
  against our decision criteria

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Evaluating the Hypothesis

• Significance Level (Rejection level)
• The probability with which we are willing to
  reject the Null (H0)
• Usually set the probability at .05 or .01
• So if the probability is less than .05 or .01, we
  reject the null otherwise we retain the null
• The area that is more extreme than the
  significance level is called the Rejection Region
  (or critical region)
• It is the range of values that are not probable
  under H0
• Basically this is an area under the curve. If a
  score falls within this area we reject the null.
• The size of the critical region is alpha

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Evaluating the Hypothesis (cont.)

• The rejection region is marked by the Critical
  value
• The value of a test statistic at or beyond which
  we will reject H0
• If obtained value gt critical value, we reject the
  null
• REMEMBER for a two-tailed test, this value can
  be larger in the negative direction, as well

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One Tailed Tests

• One-tailed test (or directional test)
• A test that rejects extreme outcomes in only one
  specified tail of the distribution gt or lt
• The advantage of a one-tailed test is that the
  critical region is large because it is
  concentrated in only one tail, so we have a
  better chance of rejecting the null
• With the directional test, the sign of your test
  statistic matters
• To determine if you should use a 1-tailed test,
  look for words describing one thing as being
  larger or smaller than something else

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Two Tailed Tests

• Two-tailed test (or non-directional test)
• A test that rejects extreme outcomes in either
  tail of the distribution ?
• The advantage of two-tailed is that it protects
  us if we find a relationship in the wrong
  direction
• We divide ? /2 this tells what to put in each
  tail
• Ex For ?.05, we divide .05 by 2. This equals
  .025. We put .025 in each tail
• This makes the critical region smaller and thus
  more difficult to reject the null
• To determine if you should use a 2-tailed test,
  look for words describing one thing as being only
  different than something else

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Type I Error

• The error of rejecting H0 when it is true
• We reject the null when we shouldnt
• The probability of a Type I Error is called
• Alpha (?)
• It is the size of our rejection region (.05)
• The probability of a correct decision is 1- ?
• Usually this is .95
• We have a 95 chance of making a correct decision
  when we reject the null
• The smaller the value of ?, the less likely we
  are to make a Type I error

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Type II Error

• The error of not rejecting H0 when it is false
• We retain the null when should reject it
• The probability of a Type II error is called Beta
  (?)
• Power
• The probability of correctly rejecting a false H0
  (1-?)
• The smaller the value of ?, the less likely we
  are to make a Type II error

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• True State of the World
• Decision H0 True H0 False
• Reject H0 Type I Error Correct
  Decision
  p ? p 1
  - ? Power
• Fail to Correct Decision Type II
  Error
• Reject H0 p 1 - ? p
  ?

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?, ?, Type I, and Type II

• Alpha, Beta, Type I and Type II errors are
  approximately inversely related
• So if we decrease our alpha level from .05 to
  .01, we increase our ability to make a correct
  decision about the null when it is true
• This means we decrease the probability of Type I
  errors
• However, this increases beta, which decreases our
  ability to make a correct decision when the null
  is false (i.e. weve decrease our power)
• This means we increase the probability of a Type
  II error

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Factors affects decisions about H0

• The actual obtained difference (X - ?), (X
  - ?), or (D-0)
• larger the difference the larger the test
  statistic
• The magnitude of the sample variance (s2)
• As s2 decreases the test statistic increases
• The sample size (N)
• as n increases the test statistic increases

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Factors affects decisions about H0

• The significance level (?)
• A higher alpha (e.g. .09) will have a lower
  critical value, so the test statistic does not
  have to be as large to reject the null
• Whether the test is a 1 or 2 tailed test
• 2 tailed has a higher critical value, so a larger
  test statistic is needed to reject the null

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Remember

• The sign of the critical values matter
• if 2 tailed use /- in front of the CV
• if 1 tailed use - or depending on the direction
  of the hypothesis
• To reject the null, we need obtained values that
  are more extreme than the critical value

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Z-test for a single score

• To test a hypothesis about whether a score
  belongs to a particular distribution, we use the
  z-score formula
• z (X - ?) / ?
• To determine the critical value for the z-test
  for single scores (use the Standard Normal
  Table)
• Determine alpha if the test if 1 or 2 tailed
• If one tailed, look up alpha in the smaller
  portion column see what z this corresponds to
• If 2 tailed, divide alpha in half, look up this
  value in the smaller portion column, see what z
  this corresponds to

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Z-test for a one-sample means

• We use the z-test for one-sample means when ?
  (population standard deviation) is known
• To test if a mean comes from population, we use
  the z-test
• z (X - ?) / ?x
• ?x ? / sqrt(n)
• To determine the critical value follow the same
  procedure as the z-test for single scores

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t-test for a one-sample means

• When ? is unknown, use the t-test for one-sample
  means
• t (X - ?) / sx
• Where sx s / sqrt(n)
• Degrees of Freedom, df n 1
• To determine the critical value for the
  one-sample means t-test
• We need to compute our df, determine alpha,
  determine 1 or 2 tailed test
• Use this information to look up the critical
  value in the t-distribution tables

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Appendix E.6

• Note that the t-table does not give you
  probabilities but critical values associated with
  your df, the type of test (1 or 2 tailed), and
  your alpha
• To use the table, you first look up your df. Next
  find the column that you alpha level and type of
  test
• The number in this location is the critical value
  

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Related measures t-test

• t (D - 0) / sD
• Where sD sD / sqrt(n)
• Where D is the mean of the difference scores
• D x1 - x2
• For the mean (D) ?D/ n
• For the variance (sD2) ?D2 (?D)2
• (same as Ch.5 n
• but with D scores) n-1
• For the SD (sD) Sqrt (sD2 )
• To determine the critical value, we follow the
  same steps as the one-sample t-test

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Confidence Limits on the Mean

• Two types of estimates
• Point estimate
• The specific value taken as the estimate of a
  parameter (one estimate) (e.g. X )
• Interval estimate
• A range of values estimated to include the
  parameter
• Confidence limits or confidence intervals
• An interval, with limits at either end, with a
  specified probability of including the parameter
  being estimated

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Confidence Limits on the Mean

• How much different would the population mean have
  to be to change our results
• CI.95 X /- (t.025 Sx) when ? .05
• CI.99 X /- (t.005 Sx) when ? .01
• Step 1
• We should already have the sample mean and
  standard error, so we just plug those in
• Step 2
• We need to look up the t critical value using our
  df
• Remember the critical value used is always
  2-tailed even if our original test was one-tailed

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Confidence Limits on the Mean

• Step 3
• We need to compute the upper CI limit and the
  lower CI limit
• CIupper X (t.025 Sx)
• CIlower X - (t.025 Sx)
• Step 4
• We need to put our answer into the following
  form
• CIlower lt ? lt CIupper
• Step 5
• Interpret There is a 95 chance that this
  interval will include the population mean
• This would be a 99 chance if the critical value
  was based on a critical t with alpha.01