Hypothesis Test

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Chapter 8

• Hypothesis Test

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Steps to a Hypothesis Test

• Hypotheses
• Null Hypothesis (Ho)
• Alternative Hypothesis (Ha)
• Alpha
• Distribution (aka model)
• Test Statistics and P-value
• Decision
• Conclusion

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Steps to a Hypothesis Test

• Can remember the steps by the sentence
• Happy Aunts Make The Darndest Cookies

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Example 1 Hypothesis Testing

• An attorney claims that more than 25 of all
  lawyers advertise. A sample of 200 lawyers in a
  certain city showed that 63 had used some form of
  advertising. At a 0.05, is there enough
  evidence to support the attorneys claim?

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Hypotheses (Sets up the two sides of the test)

• Build the Alternative Hypothesis (Ha) first.
• based on the claim you are testing (you get this
  from the words in the problem)
• Three choices
• Ha parameter ? hypothesized value
• Ha parameter lt hypothesized value
• Ha parameter gt hypothesized value
• Build Null Hypothesis (Ho) next.
• opposite of the Ha (i.e. , , )

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Example 1 Constructing Hypotheses

• We need to know what parameter we are testing and
  which of the three choices for alternative
  hypothesis we are going to use.
• An attorney claims that more than 25 of all
  lawyers advertise tells us that this is a test
  for proportions so our parameter is p.
• claims that more than 25 tells us that
• Ha p gt .25 and therefore Ho p .25

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Alpha

• Alpha a significance level
• How much proof we are requiring in order to
  reject the null hypothesis.
• The complement of the confidence level that we
  learned in the last chapter
• Usually given to you in the problem, if not, you
  can choose.
• Most popular alphas 0.05, 0.01, and 0.10

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Example 1 Alpha

• At a 0.05 is given to us in the problem so
  we just copy a 0.05

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Model

• The model is the distribution used for the
  parameter that you are testing. These are just
  the same as we used in the confidence intervals.
• p and µ (n 30) use the normal distribution
• µ (n lt 30) uses the t-distribution
• uses the chi-squared distribution

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Example 1 - Model

• The model used for a proportion is the normal.

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Test Statistic

• You will have a different test statistic for each
  of the four different parameters that we have
  learned about.
• p
• µ (n 30)

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Test Statistic

• You will have a different test statistic for each
  of the four different parameters that we have
  learned about.
• µ (n lt 30)

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p-value

• This is the evidence (probability) that you will
  get off of your chart and then compare against
  your criteria (alpha).
• You will need to find the appropriate probability
  that goes with your Ha.
• gt and lt Has are called one-tailed tests.
• ? Has are called two-tailed tests.
• For z and ?2 you have to take the gt probability X2

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Example 1 Test Statistic and p-value

• The formula for a test statistic for proportions
  is
• So, from our problem we need a proportion from a
  sample (p-hat), the proportion from our
  hypothesis (po), and a sample size (n).

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Example 1 Test Statistic and p-value

• A sample of 200 lawyers in a certain city showed
  that 63 had used some form of advertising tells
  us that
• p-hat 63/200 or 0.315
• From our hypothesis we know
• po 0.25 (which means that qo 0.75)
• sample of 200 tells us that
• n 200

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Example 1 Test Statistic and p-value

• So our test statistic and p-value are

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Decision (always about Ho)

• We have two choices for decision
• Reject Ho
• Do Not Reject Ho
• If our evidence (p-value) is less than a we
  REJECT Ho.
• If our evidence (p-value) is greater than a we DO
  NOT REJECT Ho.

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Example 1 - Decision

• Our p-value is 0.0170 and our alpha is 0.05
• So, since our p-value is less than our alpha our
  decision is REJECT Ho.

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Conclusion (always in terms of Ha)

• Conclusions
• Reject Ho
• There is enough evidence to suggest (Ha).
• Do Not Reject
• There is not enough evidence to suggest (Ha).

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Example 1 - Conclusion

• Our decision to was to reject Ho, so our
  conclusion is
• There is enough evidence to suggest that
  pgt0.25

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Example 1 - Summary

• Ho p 0.25
• Ha p gt 0.25
• a 0.05
• Model Normal
• z 2.12 and p-value 0.0170
• Reject Ho
• There is enough evidence to suggest that pgt0.25.

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Example 2 Hypothesis Testing

• A researcher reports that the average salary of
  assistant professors is more than 42,000. A
  sample of 30 assistant professors has a mean of
  43,260. At a 0.05, test the claim that
  assistant professors earn more than 42,000 a
  year. The standard deviation of the population is
  5230.

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Example 2 (cont.)

• Hypotheses
• Ho µ 42,000
• Ha µ gt 42,000 (given claim is more than)
• Alpha
• a 0.05 (given)
• Model
• Normal (n 30 and its a mean)

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Example 2 (cont.)

• Test statistic and p-value

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Example 2 (cont.)

• Decision
• 0.0934 gt 0.05 (p-value gt alpha)
• DO NOT REJECT Ho
• Conclusion
• We do not have evidence to suggest that
• µ gt 42,000.

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Example 3 Hypothesis Testing

• A physician claims that joggers maximal volume
  oxygen uptake is greater than the average of all
  adults. A sample of 15 joggers has a mean of 40.6
  milliliters per kilogram (ml/kg) and a standard
  deviation of 6 ml/kg. If the average of all
  adults is 36.7 ml/kg, is there enough evidence to
  support the physicians claim at a 0.05?

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Example 3 (cont.)

• Hypotheses
• Ho µ 36.7
• Ha µ gt 36.7
• Alpha
• a 0.05 (given)
• Model
• t(14)

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Example 3 (cont.)

• Test statistic and p-value

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Example 3 (cont.)

• Decision
• (0.01,0.025) lt 0.05 (p-value lt alpha)
• REJECT Ho
• Conclusion
• There is evidence to suggest that µ gt 36.7.

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Example 4 Hypothesis Testing

• A researcher knows from past studies that the
  standard deviation of the time it takes to
  inspect a car is 16.8 minutes. A sample of 24
  cars is selected and inspected. The standard
  deviation was 12.5 minutes. At a0.05, can it be
  concluded that the standard deviation has changed?

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Example 4 (cont.)

• Hypotheses
• Ho s 16.8
• Ha s ? 16.8
• Alpha
• a 0.05 (given)
• Model
• ?2(23)

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Example 4 (cont.)

• Test statistic and p-value

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Example 4 (cont.)

• Decision
• (0.05,0.10) gt 0.05 (p-value gt alpha)
• DO NOT REJECT Ho
• Conclusion
• There is not enough evidence to suggest that
• s ? 16.8.