Hypothesis Testing

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Hypothesis Testing

• Chapter 8

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The Elements of a Test of a Hypothesis

• Null Hypothesis A theory about a population
  parameter. The theory generally represents the
  status quo.
• Alternative Hypothesis A theory that contradicts
  the Null Hypothesis. It is a theory that will be
  accepted only when sufficient evidence exists to
  establish its truth.

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The Elements of a Test of a Hypothesis-continued

• Test Statistic A sample statistic used to
  deicide whether or not to reject the null
  hypothesis.
• Rejection Region The numerical values of the
  test statistic for which the null hypothesis will
  be rejected. The rejection region is chosen so
  that the probability that it will contain the
  test statistic is
• Level of significance

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The Elements of a Test of a Hypothesis-continued

• Assumptions Clear statements of any assumptions
  made about the population(s) being sampled.
• Experiment and calculation of test statistic
  Performance of the sampling experiment and
  determination of the numerical value of the test
  statistic.

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The Elements of a Test of a Hypothesis-continued

• Conclusion If the numerical value of the test
  statistic falls in the rejection region, we
  reject the null hypothesis and conclude that the
  alternative hypothesis is true. We know that the
  hypothesis testing process will lead to this
  conclusion incorrectly ( Type I error) 100
  of the time when the null hypothesis is true.

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The Elements of a Test of a Hypothesis-continued

• Conclusion ( continued). If the test statistic
  does not fall in the rejection region, we do not
  reject the null hypothesis. Thus we reserve
  judgment about which hypothesis is true. We do
  NOT conclude that the null hypothesis is true.
  Concluding that the null hypothesis is true when
  it is not is called a Type II error and the
  probability of that happening is

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Section 8.2

• One Tailed Statistical Test

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Section 8.2 continued

• Two-Tailed Statistical Test

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Rejection Regions for Common Values of
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Test Statistic

• When endeavoring to prove that the mean of a
  population is some particular value, then the
  test statistic is the mean of the sample. After
  finding this mean, calculate its z-score and
  determine whether it is in the rejection region.

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Determining the Null and Alternative Hypothesis

• The claim is the statement that is being
  investigated. Once the claim has been identified,
  then write it in symbolic form. For instance A
  college professor claims that the mean age of
  college statistics students is 24.
• Symbolic form

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Complement of the Claim

• The complement of the claim is the opposite of
  the claim. In this case, the complement is The
  mean age of college statistics students is not
  24.

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Null Hypothesis

• The null hypothesis is the statement with the
  equal sign. In this case, the null hypothesis is

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Alternative Hypothesis

• The alternative hypothesis is the complement of
  the null hypothesis.

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Examples of Claims and Their Complements

• Claim The mean age of all college statistics
  students is less than 24.
• Claim
• Complement
• In this case , the complement is the null
  hypothesis.

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Another Example of Claims and Their Complements

• Claim The mean age of all college statistics
  students is at least 24.
• Claim
• Complement
• In this case, the null hypothesis is the claim.

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A Third Example of Claims and Their Complements

• Claim The mean age of all college statistics
  students is greater than 24.
• Claim
• Complement
• In this case the complement is the null
  hypothesis.

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8.24 Page 339

• The difference between the game outcome and point
  spread ( called point-spread error) was
  calculated for 240 NFL games. The mean and
  standard deviation of the point-spread errors
  are
• Use this information to test the null hypothesis
  that the true mean point-spread error is 0.
  Conduct the test at

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Conducting Hypothesis Testing by Using
p-values Section 8.3

• The p-value of a test statistic is the
  probability that test statistic could happen
  given that the null hypothesis is true.
• P-value is also called the observed significance
  level.

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Reporting Test Results as p-values How to Decide
Whether to Reject the Null Hypothesis

• 1) Choose the maximum value of that you are
  willing to tolerate
• 2) If the p-value of the test statistic is less
  than the chosen value of
• then reject the null hypothesis. Otherwise, do
  not reject the null hypothesis.

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8.32 Page 345

• An analyst tested
• The analyst reported a p-value of .06. What is
  the smallest level of significance for which
  the null hypothesis would be rejected?
• Ans gt.06

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8.34 Page 345

• In a test of
• The sample data yielded the test statistic
• z 2.17. Find the
• p-value for the test.
• Ans p.0300

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8.38 Page 346
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Small Sample Test of Hypothesis About a
Population Mean -Section 8.4

• When testing Hypotheses with small samples (
  nlt30), we must assume that the original
  population is normal. If

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8.58 Page 353

• It is known that the great white shark grows to
  a mean length of 21 feet however, one marine
  biologist believes that the great white sharks
  off the coast of Bermuda grow much longer. To
  test this claim, three full grown white sharks
  were measured. Their lengths were 24,20, and 22
  feet.
• A) Does the data present sufficient evidence to
  support the biologists claim?
• B) Give the p-value
• C) What assumptions must be made in order to
  carry on the test?
• D) Do you think these assumptions are likely to
  be satisfied in this sampling situation?

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8.58 continued
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8.58 Continued

• A) At a level of significance of .10,
• Then the data does not support the alternative
  hypothesis. t.87
• B) p.2388
• C) You must assume that the length of great white
  sharks is normally distributed.
• D) Yes

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Large-Sample Test of Hypothesis About of a
Population Proportion Section 8.5

• Hypothesis testing may be used to test the
  population proportion as well as the population
  mean.

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One-Tailed Test
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Two-Tailed Test
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8.68 Page 358

• Ponds Age-Defying Complex advertises that it can
  reduce wrinkles and improve the skin. In a study,
  33 women over age 40 used the cream for 22 weeks.
  At the end of the study period, 23 of the women
  exhibited skin improvement. Is this eviudence
  that the cream will improve the skin of more than
  60 of the women over age 40? Test at a
  significance level of .05

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8.68 continued
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Chapter 8 Homework

• A) Section 8.1,
• 8.1-8.7,8.13,8.14,8.15,8.16
• B) Section 8.2
• 8.17,8.18,8.19,8.24,8.25
• C) Section 8.3
• 8.30,8.31,8.32,8.33,8.34,8.38,8.42
• D) Section 8.4
• 8.45,8.46,8.47,8.49,8.57,8.58
• E) Section 8.5
• 8.61,8.63,8.66,8.68,8.70,8.73,8.75