Hypothesis Testing For ? With ? Known

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• Hypothesis Testing For ? With ? Known

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HYPOTHESIS TESTING

• Basic idea You want to see whether or not your
  data supports a statement about a parameter of
  the population.
• The statement might be The average age of night
  students is greater than 25 (? gt25)
• To do this you take a sample and compute

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• If lt 25
• You did not prove ? gt25.
• If gtgt25
• You are probably satisfied that ? gt25.
• If is slightly gt 25
• You are probably not convinced that ? gt25.
  (although there is some evidence to support this)

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Being Convinced

• Hypothesis testing is like serving on a jury.
• The prosecution is presenting evidence (the data)
  to show that a defendant is guilty (a hypothesis
  is true).
• But even though the evidence may indicate that
  the defendant might be guilty (the data may
  indicate that the hypothesis might be true), you
  must be convinced beyond a reasonable doubt.
• Otherwise you find the defendant not guilty
  this does not mean he was innocent, (there is not
  enough evidence to support the hypothesis this
  does not mean that the hypothesis is not true,
  just that there was not enough evidence to say it
  was true beyond a reasonable doubt).

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When Can You Conclude ? gt 25?

• You are convinced ? gt25 if you get an
  that is a lot greater than 25.
• How much is a lot ?
• This is hypothesis testing.

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lt Tests

• Can you conclude that the average age of night
  students is less than 27? (? lt 27)

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? Tests

• Can we conclude that the average age of students
  is different from 26? (? ? 26)

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Hypothesis Testing

• Five Step Procedure
• 1. Define Opposing Hypotheses.
• 2. Choose a level of risk (?) for making the
  mistake of concluding something is true when its
  not.
• 3. Set up test (Define Rejection Region).
• 4. Take a random sample.
• 5. Calculate statistics and draw a conclusion.

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STEP 1 Defining Hypotheses

• H0 (null hypothesis -- status quo)
• HA (alternate hypothesis -- what you are trying
  to show)
• Can we conclude that the
  average age exceeds 25?
• H0 ?? ? 25
• HA ?? gt 25

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Test H0 At the break point ? 25

• Test the hypothesis at the point that would give
  us the most problem in deciding if ? gt25.
• If ? really were 15, it would be very unlikely
  that we would draw a sample of students that
  would lead us to the false conclusion that ? gt25.
• If ? really were 22, it is more likely that we
  would draw a sample with a large enough sample
  mean to lead us to the false conclusion that ?
  gt25 if ? really were 24, it would be even more
  likely, 24.9 even more likely.
• ? 25 is the most likely value in H0 of
  generating a sample mean that would lead us to
  the false conclusion that ? gt25.

So H0 is tested at the breakpoint, ? 25
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STEP 2 Choosing a Measure of Risk(Selecting ?)

• ? P(concluding HA is true when it is not)
• Typical ? values are .10, .05, .01, but ? can be
  anything.
• Values of ? are often specified by professional
  organizations (e.g. audit sampling normally uses
  values of .05 or .10).

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STEP 3 How to Set Up the TestDefining the
Rejection Region

• Depends on whether HA is a gt, lt, or ? hypothesis.
• The gt Case
• Suppose we are hypothesizing ?? gt 25.
• When a random sample of size n is taken
• If ? really 25, then there is only a
  probability ? that we would get an value
  that is more than z? standard errors above 25.
• Thus if we get an value that is greater
  than z? standard errors above 25, we are willing
  to conclude ? gt 25.
• We call this critical value xcrit.

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Rejecting H0 (Accepting HA)
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How many standard errors away is ?

• This is called the test statistic, z

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A ONE-TAILED gt TEST

• Reject H0 (Accept HA) if
• z gt z?
• Values of z?
• ? z?
• .10 1.282
• .05 1.645
• .01 2.326

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A ONE-TAILED lt TEST

• If HA were, HA ? lt 27
• The test would be Reject H0 (Accept HA) if
• z lt -z?
• Values of -z?
• ? -z?
• .10 -1.282
• .05 -1.645
• .01 -2.326

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A TWO-TAILED ? TEST

• If HA were, HA ? ? 26
• The test would be Reject H0 (Accept HA) if
• z lt -z?/2 or z gt z?/2
• Values of -z?
• ? -z?/2 z?/2
• .10 -1.645 1.645
• .05 -1.96 1.96
• .01 -2.576 2.576

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STEP 4 TAKE SAMPLE

• After designing the test, we would take the
  sample according to a random sampling procedure.

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STEP 5 CALCULATE STATISTISTICS

• From the sample we would calculate
• Then calculate

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DRAWING CONCLUSIONSOne Tail Tests

• gt TEST HA ? gt 25
• If z gt z? -- Conclude HA is true (Reject H0)
• If z lt z? -- Cannot conclude HA is true
  (Do not reject H0)
• lt TEST HA ? lt 27
• If z lt -z? -- Conclude HA is true (Reject H0)
• If z gt -z? -- Cannot conclude HA is true
  (Do not reject H0)

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DRAWING CONCLUSIONSTwo Tail Tests

• HA ? ? 26
• Case 1 -- When z gt 0 compare z to z?/2
• If z gt z?/2 -- Conclude HA is true (Reject H0)
• If z lt z?/2 -- Cannot conclude HA is true
  (Do not reject H0)
• Case 2 -- When z lt 0 compare z to -z?/2
• If z lt -z?/2 -- Conclude HA is true (Reject H0)
• If z gt -z?/2 -- Cannot conclude HA is true
  (Do not reject H0)

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DRAWING CONCLUSIONSTwo Tail Tests (Alternative)

• HA ? ? 26
• Cases 1 and 2 can be combined by simply looking
  at z. The test becomes Compare
  z to za/2
• If z gt z?/2 -- Conclude HA is true (Reject H0)
• If zlt z?/2 -- Cannot conclude HA is true
  (Do not reject H0)

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Examples

• Suppose
• We know from long experience that ? 4.2
• We take a sample of n 49 students
• We are willing to take an ? .05 chance of
  concluding that HA is true when it is not (Note
  z.05 1.645)
• Because our sample is large, a normal
  distribution approximates the distribution of

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Example 1 Can we conclude ? gt 25?

• 1. H0 ? 25
• HA ? gt 25
• 2. ? .05
• 3. Reject H0 (Accept HA) if
• 4. Take sample 25,21, 33.

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Example 2 Can we conclude ? lt 27?

• 1. H0 ? 27
• HA ? lt 27
• 2. ? .05
• 3. Reject H0 (Accept HA) if
• 4. Take sample 22,28, 33.

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Example 3 Can we conclude ? ? 26?

• 1. H0 ? 26
• HA ? ? 26 (This is a two-tail test)
• 2. ? .05
• 3. Reject H0 (Accept HA) if
• 4. Take sample 25,21, 33.

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gt TESTS
2.05102 gt 1.644853 Can conclude mu gt 25
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lt TESTS
-1.28231 gt -1.64485 Cannot conclude mu lt 27
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? TESTS
.384354 lt 1.959963 Cannot conclude mu ? 26
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REVIEW

• Common sense concept of hypothesis testing
• 5 Step Approach
• 1. Define H0 (the status quo), and HA (what you
  are trying to show.)
• 2. Choose ?? Probability of concluding HA is
  true when its not.
• 3. Define the rejection region and how to
  calculate the test statistic.
• 4. Take a random sample.
• 5. Calculate the required statistics and draw
  conclusion.
• There is enough evidence to conclude HA is true
  (Reject H0)
• There is not enough evidence to conclude HA is
  true (Do not reject H0).