Fundamentals of Hypothesis Testing: OneSample Tests

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Chapter 9

• Fundamentals of Hypothesis Testing One-Sample
  Tests

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9.1 Hypothesis Testing Methodology

• Confidence Intervals were our first Inference
• Hypothesis Tests are our second Inference
• Methodology implies a series of steps
• 1. Develop hypotheses
• 2. Determine decision rule
• 3. Calculate test statistic
• 4. Compare results from 2 and 3 make a decision
• 5. Write conclusion

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Step 1 Develop hypotheses

• You will need to develop 2 hypotheses
• Null hypothesis
• Alternative hypothesis
• Hypotheses concern the population parameter in
  question (ie µ or p or other)

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The Null Hypothesis

• A theory or idea about the population parameter.
• Always contains some sort of equality.
• Very often described as the hypothesis of no
  difference or status quo.
• H0 µ 368

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The Alternative Hypothesis

• An idea about a population parameter that is the
  opposite of the idea in the null hypothesis
• NEVER contains any sort of equality!
• H1 µ ? 368 (sometimes use Ha)

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Hypotheses

• Null and alternative hypotheses are mutually
  exclusive and collectively exhaustive.
• Our sample either contains enough information to
  reject the NULL hypothesis OR the sample does not
  contain enough information to reject the null
  hypothesis.

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Proof

• There is no proof.
• There is only supporting information.

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Step 2 Decision Rule or Rejection Region

• Always says something like we shall reject H0
  for some extreme value of the test statistic.
• The Rejection Region is the range of the test
  statistic that is extreme enoughso extreme that
  the test statistic probably would not occur IF
  the null hypothesis is true.
• Figure 9.1 shows the rejection region for a
  hypothesis test of the mean.
• The critical value is looked up based on the
  error rate that you are comfortable with.

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Step 3 Calculating the Test Statistic

• The test statistic depends on the sampling
  distribution in use. This depends on the
  parameter.
• This will be determined the same way it was in
  chapter 8.

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Step 4 5 Decision and Conclusion

• The decision is always either (1) reject H0 or
  (2) fail to reject H0. This is determined by
  evaluating the decision rule in step 2.
• The conclusion always says At a 0.05, there is
  (in)sufficient information to say H1

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Alpha

• a is the probability of committing a Type I
  error erroneously rejecting a true Null
  Hypothesis.
• a is called The Level of Significance
• a is determined before the sample results are
  examined.
• a determines the critical value and rejection
  region(s).
• a is set at an acceptably low level.

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Beta

• Beta is the probability of committing a Type II
  error erroneously failing to reject a false
  null hypothesis.
• Beta depends on several factors and it cannot be
  arbitrarily set. Beta can be indirectly
  influenced.

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Compliments of Alpha and Beta

• (1-a) is called the confidence coefficient. This
  is what we used in Chapter 8.
• (1-beta) is called the Power of the test. Power
  is the chance of rejecting a null hypothesis that
  ought to be rejected, ie a false null. Bigger is
  better. Power cannot be set directly.

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9.2 z Test of Hypothesis for the mean

• Use this test ONLY for the mean and only when s
  is known.
• There are two approaches
• critical value approach
• p value approach

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Critical Value Approach

• Remember your methodology (steps)
• Create hypotheses
• Create decision rule
• depends on a
• depends on distribution
• Calculate test statistic
• State the result
• State the managerial conclusion

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Hypotheses

• The discussion in 7.2 assumes a two-tail test
  because the sample mean might be extremely large
  or extremely small.
• Either one would make you think the null
  hypothesis is wrong.

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Rejection Region

• The standard approach requires that the value of
  a be divided evenly between the tail areas.
• These tail areas are called the rejection
  region.

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Conclusion

• See Step 6 on pages 308-309.

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p-value approach

• Rewrite the decision rule to say, we will reject
  the null hypothesis if the p-value is less than
  the value of a .
• p-value definition, page 309.
• p-value is called the observed level of
  significance.
• Excel--most statistical software--does a good job
  of this (thats why its a popular approach).

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Estimation and Hypothesis Testing

• The two inferences are closely related.
• Estimation answers the question what is it?
• Hypothesis Testing answers the question is it
  ______ than some number?
• See page 312.

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9.3 One-Tail Tests

• The rejection region is one single area.
• Sometimes called a directional test.
• Mechanics
• see the text example
• Problem identification
• hypothesis test or interval estimation?
• One-tail or Two-tail?
• If One-tail, which is the null?

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Text Example

• Page 314, the milk problem.
• Are we buying watered-down milk ?
• Watered-down milk freezes at a colder temperature
  than normal milk.
• What are the null and alternative hypotheses?
• Hint what do you want to conclude?
• Hint what is the hypothesis of action?
• Hint what is the hypothesis of status quo?

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Mechanics of the One-tailed test

• Different hypotheses.
• Different decision rule/rejection region.
• Different p-value or observed significance of
  observed level of significance.

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Consider Problem 9.44, page 317

• Reading only the context, not the steps (a, b,
  etc.), can you tell that the problem calls for a
  hypothesis test?
• Knowing that a test of hypothesis is called for,
  can you determine that a one-tail test is
  appropriate?
• Knowing that a one-tail test is to be used, can
  you set up the hypotheses?

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9.4 t test of Hypothesis for the Mean (s
Unknown)

• When s is unknown, the distribution for x-bar is
  a t distribution with n-1 degrees of freedom.
• Use sample standard deviation s to estimate s.
• This test is more commonly used than the z test.

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Assumptions

• Random Sample.
• You must assume that the underlying population,
  i.e. the underlying random variable x is
  distributed normally.
• This test is very robust in that it does not
  lose power for small violations of the above
  assumption.

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Methodology

• You still need your 5 step Hypothesis Test
  Methodology.
• The critical value approach is the same as that
  for the z test.
• The p-value method does not work as well when
  done by hand because of limitations in the t
  table.
• One- and Two-tail tests are possible.
• You could add another step to check assumptions.

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EXAMPLE

• 9.54, page 323

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9.5 z Test of Hypothesis for the Proportion

• For the nominal variablevariable values are
  categories and you tend to describe the data set
  in terms of proportions.
• Both one- and two-tail tests are possible.
• Problem 9.72 on page 329 is a good example.

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Assumptions

• The number of observations of interest
  (successes) and the number of uninteresting
  observations (failures) are both at least 5.