MATH 401 Probability and Statistics

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MATH 401Probability and Statistics

• Spring 2009

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Example

• A manufacturer claims that television picture
  tubes have an average lifetime of 6.1 years.
• We want to test this claim, and either accept or
  reject it.
• Suppose for certainty that a random sample
  produced a mean of 5.9 years.
• How should we approach this problem?

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Reminder Parameters and Statistics

• A parameter is a population measure (e.g. ?, ?2).
• A statistic is a sample function (e.g. sample
  mean, sample variance).
• Hence, statistics are random variables.
• A point estimator is a statistic used to estimate
  a parameter.
• A point estimate of a parameter is a single
  numerical value of a respective estimator.
• The standard error is the standard deviation of
  an estimator.

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Example Analysis

• In the example, a claim about a parameter is made
  (average life-time 6.1 years).
• Available is a sample with the sample mean.
• From experience, statistics are useful for
  parameter estimation.
• We shall try and employ them for checking claims
  about parameters.

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Hypothesis Testing

• Lecture 11

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Observation 1

• Testing a hypothesis is like putting someone on
  trial, in the sense that there can be exactly two
  outcomes
• conviction or acquittal.
• Let us recall how a trial works.

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Reminder Court hearing

• When a person is accused of committing a crime
  there are two ways to proceed
• He/she has to prove his/her innocence, otherwise
  the person will be convicted.
• The court has to prove the persons guilt,
  otherwise the person will be acquitted.

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Reminder Court hearing

• Since the Romans, the main principle of almost
  all judicial systems says
• In dubio pro reo,
• That is, the second approach is adopted.
• A persons guilt should be proved, otherwise the
  person will be acquitted.

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Court hearing Summary

• Hence, at a court hearing two hypotheses are
  tested
• Not Guilty vs. Guilty
• An evidence is searched for to prove that the
  person is guilty beyond any doubt.
• If such an evidence can not be found, then the
  person is found Not Guilty.
• We shall now try and place this idea in a
  statistical framework.

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Statistical Hypotheses

• In Statistics these hypotheses are called
• H0 null hypothesis
• (hypothesis we want to test)
• versus
• H1 alternative hypothesis
• (i.e. that H0 is not true).

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Error Types

• When making a decision two errors can occur.
• Type I Error H0 is rejected although it is true
  (conviction of an innocent). The probability of
  type I error is called the avalue or the level
  of significance of the test.
• Type II Error H0 is accepted although not true
  (acquittal of a guilty). The evidence is not
  enough to reject the hypothesis. The probability
  of type II error is called the b-value of the
  test.

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Observation 2

• A judge at a court has to accept a certain Type I
  Error - otherwise he would not send anyone to
  jail.
• Similarly, in Statistics we have to choose an
  a-value the significance level for each test.

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General Approach

• We formulate our hypotheses in the form
• H0
• H1
• and try finding statistical evidence that the
  null hypothesis is not true.

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Test Procedure

• In order to test a population mean m we take a
  sample, compute the sample mean, and compare this
  value with m0.
• If they happen be far away from each other we
  regard it as statistical evidence against the
  null hypothesis, and reject it. Otherwise, H0
  will be accepted.
• But... What is far away?

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Example (revisited)

• In our example with TV tubes
• H0
• H1
• Suppose for certainty that a random sample
  produced a mean of 5.9 years. Is it far away?

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Test Procedure

• Far away is, of course, dependent on the
  significance level a.
• Suppose the population is normal, with known
  variance (or the sample is large enough for the
  Central Limit Theorem to work).
• Then, if the claim is true,

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Reminder

• If m m0, i.e. if H0 is true, then
• Clearly,

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Significance Level of Test

• So the hypothesis H0 will be rejected, if
• Otherwise, H0 will be accepted.

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Practical Part

• In practice, we compute the value of
• and check whether it lies in
• If it does the hypothesis is accepted, otherwise
  it is rejected. See figure.

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p-value

• The probability
• is called the p-value of the test.
• The p-value gives the critical significance level
  in the sense that H0 is accepted if a is less
  than or equal to the p-value, and rejected if a
  is greater than this number.

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Return to TV Tubes

• A manufacturer claims that television tubes it
  produces have an average lifetime of 6.1 years.
• A sample of 49 tubes has produced the sample mean
  of 5.9, with s 0.7 years.
• Do we have enough evidence to reject the claim at
  the significance level of 0.05?

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Solution of Example

• We set a 0.05.
• From the table, za/21.96
• Compute z0
• So the claim is rejected if we allow for a 5 of
  type I error.
• The p-value of the test equals 0.0457.
• Hence, if the significance level is reduced to
  0.04, then the null hypothesis must be accepted.

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Two-sided Test

• In the preceding example, we chose a test that
  calls for a rejection when a sample mean is far
  from m0 , regardless whether it is smaller or
  greater. Such a test is called two-sided.
• Suppose the manufacturer claims that the average
  lifetime of TV tubes to be at least 6.1 years.
• It is clear that the situation has changed.

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Example (new situation)

• The new hypotheses are
• H0
• H1
• As before, we accept H0 unless we find enough
  evidence against it.

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General Approach

• We formulate our hypothesis in the form
• H0
• H1
• set the significance level, take a large sample,
  compute the mean and compare its value with m0.

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One-sided Test Procedure

• The difference is that we will only reject the
  null hypothesis if the sample mean appears to be
  much less than m0.
• The exact meaning of much less is, of course,
  determined by the significance level a.

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Reminder

• If m is greater or equal than m0, i.e. if H0 is
  true, then
• Clearly,

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One-sided Significance Level Test

• So H0 will be accepted if
• Otherwise, H0 will be rejected. See figure.

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Practical Part

• In practice, we simply compute
• and check whether
• If it is, then the hypothesis is accepted,
  otherwise, it is rejected.

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p-value (one-sided test)

• The p-value of a one-sided test on mean is given
  by

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Solution of Example

• A manufacturer claims that television tubes it
  produces have an average lifetime of at least 6.1
  years.
• We take a sample of 49 tubes and discover that
  the sample mean is 5.9 and s 0.7 years.
• We set the significance level a 0.05.

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Solution of Example

• The SND table gives za1.645
• Compute z0
• So the hypothesis is rejected at 0.05
  significance level.
• The p-value approx. equals 0.02285.
• Hence, the claim would be accepted if the level
  of significance was set to be 0.02.

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Food for Thought

• In the case of
• H0
• H1
• we proceed in a similar way.
• Do this at home to make sure that you understand
  how one-sided test works!

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Another Problem

• Suppose only 16 tubes are available for testing.
  Is it still possible to test the manufacturers
  claim?

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Another Problem

• Suppose only 16 tubes are available for testing.
  Is it still possible to test the manufacturers
  claim?
• Yes, if the population is known to be normal.

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Reminder

• If m m0, i.e. if H0 is true, then
• Clearly,

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Two-sided and One-sided Tests for Small Sample.

• It is clear that the procedure remains identical.
  One simply uses a statistic that has a
  t-distribution.
• Develop the criteria for the rejection of a null
  hypothesis in one-sided tests for small samples!
  See figure.
• What can you say about the p-value, if only a
  table is available?

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Another Problem

• Suppose the manufacturer claims that the
  standard deviation of life time of picture tubes
  does not exceed 6 months. How could we test this
  claim? What are additional conditions?
• Develop rejection criteria in this case to make
  sure you really understand how it works! See
  figure.
• What can you say about the p-value if only a
  table c2-distribution is available?

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Reminder Error Types

• Type I Error H0 is rejected although it is true
  (conviction of an innocent). The probability of
  type I error is called the avalue or the level
  of significance of the test.
• Type II Error H0 is accepted although not true
  (acquittal of a guilty). The evidence is not
  enough to reject the hypothesis. The probability
  of type II error is called the b-value of the
  test.

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Type II Error

• Suppose that the true mean value is different
  from the one stated in H0-hypothesis. The
  evidence is, however, not strong enough to reject
  the claim.
• What is the probability of this happening?
• What are the factors involved?

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Type II Error

• Let d m - m0. We notice that

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Type II Error

• Clearly, a type II error occurs only if Z0
• takes on a value between za/2 and za/2.
• The probability of this happening is, clearly,
  (see figure).

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Practical Part

• Suppose for certainty that d gt 0. Then
• the second term in the last expression is
  approximately 0, so

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Sample Size

• Let zb be the 100(1-b) percentage point of a SND
  random variable. Then
• Solving for n we get

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Sample Size. Conclusion

• Hence, given a and d, and b, we can determine a
  minimal sample size that guarantees that the
  probability of wrong acquittal is at most b

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Food for Thought

• What if a true mean value is smaller than the one
  stated in H0, i.e. if d lt 0?
• Work out an expression for b !
• Show that the formula for the sample size is not
  affected, i.e.

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Food for Thought

• What if the hypotheses are as follows?
• H0
• H1
• Work out expressions for b and n in this case.
  
• What is going to change, if H0 states that a
  true mean does not exceed m0?

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Future Plans

• In our last meeting we are going to see how some
  other Statistical Hypotheses are tested.
• Needless to say, these techniques are dependent
  on a proper choice of sample functions.
• So the next lecture is on
• FURTHER HYPOTHESIS TESTING.

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Thank you