Chapter 10 One and TwoSample Tests of Hypotheses

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Chapter 10 One- and Two-Sample Tests of
Hypotheses

• Wen-Hsiang Lu (???)
• Department of Computer Science and Information
  Engineering,
• National Cheng Kung University
• 2005/06/06

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Statistical Hypotheses

• Decision based on experimental evidence whether
• Coffee drinking increases the risk of cancer in
  humans.
• A persons blood type or eye color are
  independent variables.
• Definition 10.1 A statistical hypothesis is an
  assertion or conjecture concerning one or more
  populations.

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Role of Probability in Hypothesis Testing

• Awareness of the probability of a wrong
  conclusion.
• The acceptance of a hypothesis merely implies
  that the data do not give sufficient evidence to
  refute it.
• Rejection means that there is a small probability
  of obtaining the sample information observed when
  the hypothesis is true.
• E.g., for the conjecture of the fraction
  defective p 0.10, a sample of 100 revealing 20
  defective items is certainly evidence of
  rejection. (probability 0.002)

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Supporting a Contention

• Contention (??) coffee drinking increases the
  risk of cancer
• ?Hypothesis there is no increase in cancer risk
  produced by drinking coffee
• Contention one kind of gauge (????) is more
  accurate than another
• ?Hypothesis there is no difference in the
  accuracy of the two kinds of gauges

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Null and Alternative Hypotheses

• Structure of hypothesis
• Null hypothesis, H0 any hypothesis we wish to
  test
• Alternative hypothesis, H1 the opposite
  hypothesis to reject H0
• Example
• H0 is the null hypothesis p 0.5 for a binomial
  population,
• H1 would be one of the following p gt 0.5, p lt
  0.5, or p ? 0.5

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Testing a Statistical Hypothesis

• Definition 10.2 Rejection of the null hypothesis
  when it is true is called a type I error (level
  of significance).
• Definition 10.3 Acceptance of the null
  hypothesis when it is false is called a type II
  error.

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Testing a Statistical Hypothesis

• Critical value the last number passing from the
  acceptance region into the critical region.
• H0 p 1/4 H1 p gt 1/4 (p 1/4, n 20,
  binomial)

(Unlikely to commit a type I error)
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Testing a Statistical Hypothesis

• The probability of committing both types of error
  can be reduced by increasing sample size
• Example H0 p 1/4 H1 p gt ¼ (n 100,
  critical value 36) sol use the normal-curve
  approximation with n gt 30.

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Hypothesis Testing with a Continuous Random
Variable

• Consider the null hypothesis that the average
  weight of male students in a certain college is
  68 kilograms against the alternative hypothesis
  that it is unequal to 68.
• H0 ? 68 H1 ? ? 68

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Hypothesis Testing with a Continuous Random
Variable
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Hypothesis Testing with a Continuous Random
Variable
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Hypothesis Testing with a Continuous Random
Variable
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One- and Two-Tailed Tests
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One- and Two-Tailed Tests
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One- and Two-Tailed Tests
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The Use of P-Values for Decision Making
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The Use of P-Values for Decision Making
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Single Sample Tests Concerning a Single Mean
(Variance Known)
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Single Sample Tests Concerning a Single Mean
(Variance Known)
P P(Z gt 2.02) 0.0217
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Single Sample Tests Concerning a Single Mean
(Variance Known)
P P(Z gt 2.83) 2 P(Z lt -2.83)
0.0046
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Single Sample Tests on a Single Mean (Variance
Unknown)