REVIEW

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• REVIEW
• Hypothesis Tests of Means

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5 Steps for Hypothesis TestingTest Value Method

1. Develop null and alternative hypotheses
2. Specify the level of significance, ?
3. Use the level of significance to determine the
   critical values for the test statistic and state
   the rejection rule for H0
4. Collect sample data and compute the value of test
   statistic
5. Use the value of test statistic and the rejection
   rule to determine whether to reject H0

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When to use z and When to use t

• z and t distributions are used in hypothesis
  testing.
• 
  _
  
  These are determined by the
  distribution of X.

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General Form ofTest Statistics for Hypothesis
Tests

• A test statistic is nothing more than a
  measurement of how far away the observed value
  from your sample is from some hypothesized value,
  v.
• It is measured in terms of standard errors
• s known z-statistic with standard error
• s unknown t-statistic with standard error
• The general form of a test statistic is

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Example

• The average cost of all required texts for
  introductory college English courses seems to
  have gone up substantially as the professors are
  assigning several texts.
• A sample of 41 courses was taken
• The average cost of texts for these 41 courses is
  86.15
• Can we conclude the average cost
• Exceeds 80?
• Is less than 90?
• Differs from last years average of 95?
• Differs from two years ago average of 78?

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CASE 1 z-tests for s Known

• Assume the standard deviation is 22.
• Because the sample size gt 30, it is not necessary
  to assume that the costs follow a normal
  distribution to determine the z-statistic.
• In this case because it is assumed that s is
  known (to be 22), these will be z-tests.

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Example 1 Can we conclude µ gt 80?
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• H0 µ 80
• HA µ gt 80
• Select a .05
• TEST Reject H0 (Accept HA) if z gt z.05
  1.645
• z calculation
• Conclusion 1.790 gt 1.645
• There is enough evidence to conclude µ gt 80.

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Example 2 Can we conclude µ lt 90?
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• H0 µ 90
• HA µ lt 90
• Select a .05
• TEST Reject H0 (Accept HA) if z lt-z.05 -1.645
• z calculation
• Conclusion -1.121 gt -1.645
• There is not enough evidence to conclude µ lt 90.

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Example 3 Can we conclude µ ? 95?

• H0 µ 95
• HA µ ? 95
• Select a .05
• TEST Reject H0 (Accept HA) if z lt-z.025 -1.96
  or if z gt z.025 1.96
• z calculation
• Conclusion -2.578 lt -1.96
• There is enough evidence to conclude µ ? 95.

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Example 4 Can we conclude µ ? 78?

• H0 µ 78
• HA µ ? 78
• Select a .05
• TEST Reject H0 (Accept HA) if z lt-z.025 -1.96
  or if z gt z.025 1.96
• z calculation
• Conclusion 2.372 gt 1.96
• There is enough evidence to conclude µ ? 78.

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P-values

• P-values are a very important concept in
  hypothesis testing.
• A p-value is a measure of how sure you are that
  the alternate hypothesis HA, is true.
• The lower the p-value, the more sure you are that
  the alternate hypothesis, the thing you are
  trying to show, is true. So
• A p-value is compared to a.
• If the p-value lt a accept HA you proved your
  conjecture
• If the p-value gt a do not accept HA you failed
  to prove your conjecture

Low p-values Are Good!
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5 Steps for Hypothesis TestingP-value Method

• Develop null and alternative hypotheses
• Specify the level of significance, ?
• Collect sample data and compute the value of test
  statistic
• Calculate p-value Determine the probability for
  the test statistic
• Compare p-value and ?
• Reject H0 (Accept HA), if p-value lt ?

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Calculating p-values

• A p-value is the probability that, if H0 were
  really true, you would have gotten a value
• as least as great as the sample value for gt
  tests
• at most as great as the sample value for lt
  tests
• at least as far away from the sample value for
  ? tests
• First calculate the z-value for the test.
• The p-value is calculated as follows

TEST P-value EXCEL
gt P(Zgtz) Area to the right of z 1-NORMSDIST(z)
lt P(Zltz) Area to the left of z NORMSDIST(z)
? For z lt 0 2(Area to the left of z) For z gt 0 2(Area to the right of z) 2NORMSDIST(z) 2(1-NORMSDIST(z))
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Examples p-Values

• Example 1 Can we conclude µ gt 80?
• z 1.79
• P-value 1 - .9633 .0367 (lt a .05).
• Can conclude µ gt 80.
• Example 2 Can we conclude µ lt 90?
• z -1.12
• P-value .1314 (gt a .05).
• Cannot conclude µ lt 90.
• Example 3 Can we conclude µ ? 95?
• z -2.58
• P-value 2(.0049) .0098 (lt a .05).
• Can conclude µ ? 95.
• Example 4 Can we conclude µ ? 78?
• z 2.37
• P-value 2(1-.9911) .0178 (lt a .05).
• Can conclude µ ? 78.

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CASE 2 t-tests for s Unknown

• Because the sample size gt 30, it is not necessary
  to assume that the costs follow a normal
  distribution to determine the t-statistic.
• In this case because it is assumed that s is
  unknown, these will be t-tests with 41-1 40
  degrees of freedom.
• Assume s 24.77.

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Example 1 Can we conclude µ gt 80?
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• H0 µ 80
• HA µ gt 80
• Select a .05
• TEST Reject H0 (Accept HA) if t gtt.05,40
  1.684
• t calculation
• Conclusion 1.590 lt 1.684
• Cannot conclude µ gt 80.

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Example 2 Can we conclude µ lt 90?
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• H0 µ 90
• HA µ lt 90
• Select a .05
• TEST Reject H0(Accept HA) if tlt-t.05,40
  -1.684
• t calculation
• Conclusion -0.995 gt -1.684
• Cannot conclude µ lt 90.

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Example 3 Can we conclude µ ? 95?

• H0 µ 95
• HA µ ? 95
• Select a .05
• TEST Reject H0 (Accept HA) if t lt-t.025,40
  -2.021 or if t gt t.025,40 2.021
• t calculation
• Conclusion -2.288 lt -2.021
• Can conclude µ ? 95.

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Example 4 Can we conclude µ ? 78?

• H0 µ 78
• HA µ ? 78
• Select a .05
• TEST Reject H0 (Accept HA) if t lt-t.025,40
  -2.021 or if t gt t.025,40 2.021
• t calculation
• Conclusion 2.107 gt 2.012
• Can conclude µ ? 78.

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The TDIST Function in Excel

• TDIST(t,degrees of freedom,1) gives the area to
  the right of a positive value of t.
• 1-TDIST(t,degrees of freedom,1) gives the area to
  the left of a positive value of t.
• Excel does not work for negative vales of t.
• But the t-distribution is symmetric. Thus,
• The area to the left of a negative value of t
  area to the right of the corresponding positive
  value of t.
• TDIST(-t,degrees of freedom,1) gives the area to
  the left of a negative value of t.
• 1-TDIST(-t,degrees of freedom,1) gives the area
  to the right of a negative value of t.
• TDIST(t,degrees of freedom,2) gives twice the
  area to the right of a positive value of t.
• TDIST(-t,degrees of freedom,2) gives twice the
  area to the right of a negative value of t.

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p-Values for t-Tests Using Excel

• P-values for t-tests are calculated as follows

HA TEST Sign of t EXCEL P-value
gt gt0 lt0 TDIST(t,degrees of freedom,1) Usual case 1-TDIST(-t,degrees of freedom,1)
lt lt0 gt0 TDIST(-t,degrees of freedom,1) Usual case 1-TDIST(t,degrees of freedom,1)
? lt0 gt0 TDIST(-t,degrees of freedom,2) TDIST(t,degrees of freedom,2)
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Test Value vs P-Value

• Example of one-tailed, positive test value

? is known ? is unknown
Test Value Compare ztest with zcritical Accept HA if ztest gt zcritical Compare ttest with tcritical Accept HA if ttest gt tcritical
P-value Compare p-value (based on normal distribution) with a Accept HA if p-value lt a Compare p-value (based on t distribution) with a Accept HA if p-value lt a
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Review

• When to use z and when to use t in hypothesis
  testing
• s known z
• s unknown t
• z and t statistics measure how many standard
  errors the observed value is from the
  hypothesized value
• Form of the z or t statistic
• Meaning of a p-value
• z-tests and t-tests
• By hand
• Excel