Hypothesis Testing

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Hypothesis Testing

• A hypothesis is a claim or statement about a
  property of a population (in our case, about the
  mean or a proportion of the population)
•  
• A hypothesis test (or test of significance) is a
  standard procedure for testing a claim or
  statement about a property of a population.
•  
•  
• It is extremely important to realize that we are
  not making definitive conclusions. We are giving
  probabilistic conclusions. We are either
  concluding that the results we get are likely due
  to chance, or unlikely.

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Examples

• If we flip a coin 100 times, and 52 come up
  heads, this could easily occur by chance. There
  is not sufficient evidence to suggest that the
  coin is unfair.
•  
• If we flip a coin 100 times, and 75 come up
  heads, this would be an extremely rare event if
  the coin was fair. The extremely low probability
  is evidence that the coin may not be fair.
•  
• Note If would be very sloppy of us to conclude
  in the second example that the coin is definitely
  unfair. Although extremely rare, 75 heads is
  still possible by chance from a fair coin.

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Another Example

• A light bulb is advertised as having a mean life
  of 1000 hours. From a sample, we find the mean
  life of our sample to be 900 hours. The 95
  confidence interval for the population mean is
  850 lt µ lt 1050 hours.
• We CANNOT conclude
• That the actual mean life of light bulbs is 900
  hours
• That the advertised life is wrong
• That the advertised life is correct
• We CAN conclude
• From our sample, we are 95 confident that the
  population mean is between 850 hours and 1050
  hours. Since 1000 hours is included in that
  interval, we do not have sufficient evidence to
  say that the advertised life is wrong.

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Another approach

• Claim The mean life of light bulbs is less than
  1000
• Working Assumption The mean life of light bulbs
  is 1000
• The sample resulted in a mean life of 900
• Assuming that µ 1000, the probability that the
  mean of our sample would be less than 900 is P(
  lt 900) 0.0951
• There are two possible explanations for why our
  sample came out with a mean life of 900 hours.
  Either this occurred by chance (with probability
  9.5), or the actual mean life of light bulbs is
  less than 900. Since the probability (9.5)
  isnt horribly small, we decide that random
  chance is a reasonable explanation. There isnt
  sufficient evidence to support the claim that the
  mean life of light bulbs is less than 1000 hours.

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Formal Hypothesis TestingThe brief process

• Convert your claim into a symbolic null and
  alternative hypothesis
• Calculate a test statistic
• Compare the test statistic to critical values OR
  Find a probability
• Write a conclusion

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Components of a Formal Hypothesis Test

• The Null hypothesis (denoted H0) is a statement
  that the value of a population parameter (such as
  proportion or mean) is equal to some claimed
  value.
•  
• The alternative hypothesis (denoted H1 or Ha) is
  a statement that the value of a population
  parameter somehow differs from the null
  hypothesis. The symbolic form must be a gt, lt or
  ? statement.

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• We will be testing the null hypothesis directly
  (by assuming its true) to reach a conclusion to
  either reject H0 or fail to reject H0.
•  
• Note We cannot support a claim that a parameter
  is equal to a value. So, the null hypothesis
  must always include equality, and the alternative
  hypothesis must be inequality.

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Process

• Identify the claim to be tested and express it in
  symbolic form.
• Give the symbolic form that must be true when the
  original claim is false
• Pick the one not including equality to be H1, and
  let the null hypotheses be that the parameter
  equals the value being considered.

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Example

• Claim The mean IQ of statistics students is
  greater than 110.
• Symbolic form µ gt 110
• Opposite µ 110
• H0 µ 110
• H1 µ gt 110
• Note While often your claim will be the
  alternative hypothesis, it wont always be.

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Test Statistics

• A test statistic is a value computed from the
  sample data, used in making the decision whether
  or not to reject the null hypothesis.
•  
• Z value for proportion
• Z value for mean (sigma known)
• T value for mean (sigma unknown)
• The test statistic indicates how far our sample
  deviates from the assumed population parameter.

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Critical region and significance

• Critical region (or rejection region) is the set
  of all values of the test statistic that cause us
  to reject the null hypothesis.
•  
• Significance level (a) is the probability that
  the test statistic will fall in the critical
  region when the null hypothesis is actually true.
  Common values are 0.01, 0.05 and 0.10
•  
• A Critical value is any value that separates the
  critical region from values of the test statistic
  that would not cause us to reject the null
  hypothesis

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Example

• Using a significance level of a 0.05, lets find
  the critical value for each of these alternative
  hypotheses
• P ? 0.5 Critical region is in two tails of the
  normal distribution. Using the same method we
  used in chapter 6, we find the critical values to
  be z -1.96 and z1.96
• P lt 0.5 The critical region is in the left tail
  of the normal distribution. Using the methods
  from 5.2, we find c so P(z lt c) 0.05. The
  critical value is -1.645
• P gt 0.5 The critical region is in the left tail
  of the normal distribution. Using the methods
  from 5.2, we find c so P(z lt c) 0.95. The
  critical value is 1.645

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P-Value

• The P-value is the probability of getting a value
  of the test statistic that is at least as extreme
  as the one obtained for the sample data. If the
  P-value is very small (such as less than 0.05),
  we will reject the null hypothesis.
• See pullout for help on how to calculate P-value.
  The exact process depends on your alternative
  hypothesis.

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Decisions and Conclusions

• Our final conclusion will always be one of these
• Reject the null hypothesis
• Fail to reject the null hypothesis
• Traditional Method
• Reject H0 if the test statistic falls within the
  critical region
• Otherwise fail to reject the null hypothesis

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Decisions and Conclusions

• P-value method
• Reject H0 if P-value a
• Fail to reject if H0 gt a
•  
• Less common methods
• Find P-value, and leave conclusion to the reader
• Look at whether population parameter falls in
  confidence interval estimate

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Final Wording

• If your original claim contains equality (became
  H0)
• Reject H0 There is sufficient evidence to
  warrant rejection of the claim that
• Fail to Reject H0 There is not sufficient
  evidence to warrant rejection of the claim that
•  
• If your original claim does not contain equality
  (was H1)
• Reject H0 The sample data support the claim
  that
• Fail to Reject H0 There is not sufficient
  sample evidence to support the claim that

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Homework

• 7-2 1-35 every other odd
• Every odd recommended.