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C.M. Pascual

S TATISTICS. Chapter 8 Hypothesis Testing. C.M. Pascual. 8-1 Overview 8-2 Fundamentals of Hypothesis Testing 8-3 Testing a Claim about a Mean: Large     Samples 8-4 Testing a Claim about a Mean: Small     Samples 8-5 Testing a Claim about a Proportion

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C.M. Pascual

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  1. STATISTICS Chapter 8 Hypothesis Testing C.M. Pascual

  2. 8-1 Overview 8-2 Fundamentals of Hypothesis Testing 8-3 Testing a Claim about a Mean: Large     Samples 8-4 Testing a Claim about a Mean: Small     Samples 8-5 Testing a Claim about a Proportion 8-6 Testing a Claim about a Standard     Deviation Chapter 8Hypothesis Testing

  3. Definition Hypothesis in statistics, is a claim or statement about a property of a population 8-1 Overview

  4. If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct. Rare Event Rule for Inferential Statistics

  5. 8-2 Fundamentals of Hypothesis Testing

  6. Figure 8-1 Central Limit Theorem

  7. Figure 8-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Likely sample means µx = 98.6

  8. z= - 1.96 x = 98.48 z = 1.96 x= 98.72 or or Figure 8-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Likely sample means µx = 98.6

  9. The Expected Distribution of Sample Means Assuming that = 98.6 z= - 1.96 x = 98.48 z = 1.96 x= 98.72 or or Figure 8-1 Central Limit Theorem Likely sample means Sample data: z= - 6.64 x= 98.20 or µx = 98.6

  10. Components of aFormal Hypothesis Test

  11. Statement about value of    population parameter Must contain condition of equality =, , or Test the Null Hypothesis directly RejectH0 or fail to rejectH0 Null Hypothesis: H0

  12. Must be true if H0 is false , <, > ‘opposite’ of Null Alternative Hypothesis: H1

  13. If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis. Note about Forming Your Own Claims (Hypotheses)

  14. Someone else’s claim may become the null hypothesis (because it contains equality), and it sometimes becomes the alternative hypothesis (because it does not contain equality). Note about Testing the Validity of Someone Else’s Claim

  15. a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis Test Statistic

  16. a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis For large samples, testing claims about population means Test Statistic x - µx z=  n

  17. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region

  18. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Critical Region

  19. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Critical Region

  20. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Critical Regions

  21. denoted by  the probability that the test   statistic will fall in the critical   region when the null hypothesis   is actually true. common choices are 0.05, 0.01,   and 0.10 Significance Level

  22. Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value

  23. Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value Critical Value ( z score )

  24. Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value Reject H0 Fail to reject H0 Critical Value ( z score )

  25. Two-tailed,Right-tailed,Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

  26. H0: µ = 100 H1: µ  100 Two-tailed Test

  27. H0: µ = 100 H1: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region

  28. H0: µ = 100 H1: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region Means less than or greater than

  29. H0: µ = 100 H1: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region Means less than or greater than Reject H0 Fail to reject H0 Reject H0 100 Values that differ significantly from 100

  30. H0: µ  100 H1: µ > 100 Right-tailed Test

  31. H0: µ  100 H1: µ > 100 Right-tailed Test Points Right

  32. H0: µ  100 H1: µ > 100 Fail to reject H0 Reject H0 Right-tailed Test Points Right Values that differ significantly from 100 100

  33. H0: µ  100 H1: µ < 100 Left-tailed Test

  34. H0: µ  100 H1: µ < 100 Left-tailed Test Points Left

  35. H0: µ  100 H1: µ < 100 Left-tailed Test Points Left Reject H0 Fail to reject H0 Values that differ significantly from 100 100

  36. always test the null hypothesis 1. Reject the H0 2. Fail to reject the H0 need to formulate correct wording of finalconclusion See Figure 8-4 Conclusions in Hypothesis Testing

  37. FIGURE 8-4 Wording of Final Conclusion Start Does the original claim contain the condition of equality (This is the only case in which the original claim is rejected). “There is sufficient evidence to warrant rejection of the claim that. . . (original claim).” Yes (Reject H0) Yes (Original claim contains equality and becomes H0) Do you reject H0?. No (Fail to reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” No (Original claim does not contain equality and becomes H1) (This is the only case in which the original claim is supported). Yes (Reject H0) “The sample data supports the claim that . . . (original claim).” Do you reject H0? No (Fail to reject H0) “There is not sufficient evidence to support the claim that. . . (original claim).”

  38. some texts use “accept the null hypothesis we are not proving the null hypothesis sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect) Accept versus Fail to Reject

  39. The mistake of rejecting the null hypothesis when it is true. (alpha) is used to represent the probability of a type I error Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6 Type I Error

  40. the mistake of failing to reject the null hypothesis when it is false. ß (beta) is used to represent the probability of a type II error Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6 Type II Error

  41. Table 8-2 Type I and Type II Errors True State of Nature The null hypothesis is true The null hypothesis is false Type I error (rejecting a true null hypothesis)  We decide to reject the null hypothesis Correct decision Decision Type II error (rejecting a false null hypothesis)  We fail to reject the null hypothesis Correct decision

  42. For any fixed , an increase in the sample size nwill cause a decrease in  For any fixed sample size n, a decrease in  will cause an increase in . Conversely, an increase in  will cause a decrease in  . To decrease both  and , increase the sample size. Controlling Type I and Type II Errors

  43. Power of a Hypothesis Test is the probability (1 - ) of rejecting a false null hypothesis, which is computed by using a particular significance level  and a particular value of the mean that is an alternative to the value assumed true in the null hypothesis. Definition

  44. Steps in Hypothesis Testing • State the null and alternative hypothesis; • Select the level of significance; • Determine the critical value and the rejection region/s; • State the decision rule; • Compute the test statistics; and • Make a decision, whether to reject or not to reject the null hypothesis.

  45. Example 1 • A manufacturer claims that the average lifetime of his lightbulbs is 3 years or 36 months. The stabdard deviation is 8 months. Fifty (50) bulbs are selected, and the average lifetime is found to be 32 months. Should the manufacturer’s statement be rejected at = 0.01?

  46. Example 1 • Solution: • Step 1. State the hypothesis: • Ho: µ = 36 months • Ha : µ  36 months • Step 2. Level of significance = 0.01 • Step 3. Determine critical values and rejection region

  47. Example 1 • Solution: • Step 3. Determine critical values and rejection region • Z = +/- 2.575 (from Appendix B of z values) • Step 4. State the decision rule • Reject the null hypothesis if Zc > 2.575 or Zc = - 2.575

  48. Example 1 • Solution: • Step 5. Compute the test statistic. Zc = (32-36)/ (8/(50)0.5 = - 3.54 x - µx zc=  n

  49. Example 1 • Solution: • Step 6. Make a decision. Zc = - 3.54 is less than Z = -2.575 And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months.

  50. Example 1 • Solution: • Step 6. Make a decision. Zc = - 3.54 is less than Z = -2.575 And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months.

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