Math 10 Chapter 9 Notes: Hypothesis Testing

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Math 10 Chapter 9 Notes Hypothesis Testing

• done constantly in medicine, business,
  polling, education, etc.
• to do set up 2 contradictory statements
• first statement - often the accepted belief
• conduct a test to see whether our data supports
  or does not support the first hypothesis

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Math 10 Chapter 9 Notes Hypothesis Testing

• Null hypothesis Ho
• Alternate hypothesis Ha
• Example Ho John loves Marcia
• Ha John does not love Marcia
• Example Ho m 6
• Ha m lt 6

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Math 10 Chapter 9 Notes Hypothesis Testing

• Decision Ho is
• True False
• Do not reject Ho Correct Type II
  Decision Error
• Reject Ho Type I Correct Error Decision

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Math 10 Chapter 9 Notes Hypothesis Testing

• Type I error Reject the null hypothesis when
  the null is TRUE.
• P(Type I error) a
• Type II error Do not reject the null
  hypothesis when the null is FALSE.
• P(Type II error) b
• Goal Minimize a and b.

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Math 10 Chapter 9 Notes Hypothesis Testing

• , , or ³ are ALWAYS in the null
  hypothesis Ho.
• ¹ , gt , or lt are ALWAYS in the alternate
  hypothesis Ha.

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Math 10 Chapter 9 Notes Hypothesis Testing

• Examples State the null hypothesis, Ho, and the
  alternative hypothesis, Ha, in terms of the
  appropriate parameter (m or p).
• At most 60 of Americans vote in presidential
  elections. (Right-Tailed)
• Ho p 0.60
• Ha p gt 0.60

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Math 10 Chapter 9 Notes Hypothesis Testing

• Fewer than 5 of adults ride the bus to work in
  New York City. (Left-tailed)
• Ho p ³ 0.05
• Ha p lt 0.05

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Math 10 Chapter 9 Notes Hypothesis Testing

• The average number of cars a person owns in
  his/her lifetime is not more than 10.
  (Right-tailed)
• Ho m 10
• Ha m gt 10

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Math 10 Chapter 9 Notes Hypothesis Testing

• Europeans have an average paid vacation each
  year of six weeks. (Two-tailed)
• Ho m 10
• Ha m ¹ 10

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Math 10 Chapter 9 Notes Hypothesis Testing

• Private universities cost, on average, more than
  20,000 per year for tuition, room, and board.
  (Right-tailed)
• Ho m 20,000
• Ha m gt 20,000

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Math 10 Chapter 9 Notes Hypothesis Testing

• What are Type I and Type II errors for some of
  these problems?
• Ho m 20,000 Ha m gt 20,000
• Type I We believe that private universities
  cost, on average, more than 20,000 per year for
  tuition, room, and board when, in fact, the
  average cost is no more than 20,000.

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Math 10 Chapter 9 Notes Hypothesis Testing

• Ho m 20,000 Ha m gt 20,000
• Type II We believe that private universities
  cost, on average, no more than 20,000 per year
  for tuition, room, and board when, in fact, the
  average cost is more than 20,000.
• no more than at most less than or equal to

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Math 10 Chapter 9 Notes Hypothesis Testing

• To perform a hypothesis test
• sample data is gathered
• data typically favors one of the hypotheses
• Decisions
• if data favors the null hypothesis (Ho), we do
  not reject the null
• if data favors the alternate hypothesis (Ha), we
  reject the null

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Math 10 Chapter 9 Notes Hypothesis Testing

• NOTE We are ALWAYS testing the null hypothesis,
  never the alternate. Our conclusion is ALWAYS in
  regards to the null (Ho).
• after a decision is made, an appropriate
  CONCLUSION is made regarding the null hypothesis

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Math 10 Chapter 9 Notes Hypothesis Testing

• sometimes, data favors neither hypothesis (this
  implies an inconclusive test result)
• a test may be left-tailed, right-tailed, or
  two-tailed depending upon the null hypothesis
• associated with the null hypothesis is a
  pre-conceived a. P(Type I error ) a

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Math 10 Chapter 9 Notes Hypothesis Testing

• If no pre-conceived a is given, it is common
  practice to use a 0.05. A test may be
  left-tailed, right-tailed, or two-tailed
  depending upon the null hypothesis
• data is collected to calculate what is called the
  p-value, or level of significance, or calculated a

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Math 10 Chapter 9 Notes Hypothesis Testing

• p-value P(the information/data will happen
  purely by chance GIVEN that the null hypothes is
  true)
• decision to reject or to not reject the null is
  based upon whether a gt p-value or a lt p-value
• Reject Ho if a gt p-value
• Do not Reject Ho if a lt p-value

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Math 10 Chapter 9 Notes Hypothesis Testing

• Random Variable
• Xbar average
• P proportion
• Examples
• Xbar the average tuition for a private college
• P the proportion of voters who voted for the
  winning candidate