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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Chapter 7 Hypothesis Testing . M ARIO F . T RIOLA. E IGHTH. E DITION. 7-2 Fundamentals of Hypothesis Testing 7-3 Testing a Claim about a Mean: Large Samples 7-4 Testing a Claim about a Mean: Small Samples

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Chapter 7 Hypothesis Testing MARIO F. TRIOLA EIGHTH EDITION

  2. 7-2 Fundamentals of Hypothesis Testing 7-3 Testing a Claim about a Mean: Large Samples 7-4 Testing a Claim about a Mean: Small Samples 7-5 Testing a Claim about a Proportion Chapter 7Hypothesis Testing

  3. Definition Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population.

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

  5. 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 7-1 Central Limit Theorem Likely sample means or z= - 6.64 Sample data:x= 98.20 µx = 98.6

  6. Components of aFormal Hypothesis Test

  7. Statement about the value of a POPULATION PARAMETER Must contain condition of EQUALITY: = ,≤ , or ≥ Test the Null Hypothesis directly RejectH0 or fail to rejectH0 Null Hypothesis: H0

  8. Must be true if H0 is false Must contain condition of INEQUALITY:  , < ,or > ‘Opposite’ of Null Hypothesis Alternative Hypothesis: H1

  9. Test Statistic • A value computed from the sample data that is used in making the decision about whether to reject the null hypothesis • For large samples, when testing claims about population means, the test statistic is a z-score corresponding to the sample mean.

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

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

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

  13. 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

  14. 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 )

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

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

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

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

  19. Always test the NULL hypothesis: Reject H0 Fail to rejectH0 Be careful to include the correct wording of the final conclusion Conclusions in Hypothesis Testing or

  20. Wording of Final Conclusion Start Only case in which original claim is rejected Claim contains equality? “There is sufficient evidence to reject the claim that. . . (original claim).” Yes Yes Claim becomes H0 Reject H0? No “There is not sufficient evidence to reject the claim that (original claim).” No Claim becomes H1 Only case in which original claim is supported “There issufficient evidence to support the claim that . . . (original claim).” Yes Reject H0? No “There is not sufficient evidence to support the claim that (original claim).”

  21. Some texts use “accept the null hypothesis” We are not proving the null hypothesis (can’t PROVE equality) If the sample evidence is not strong enough to warrant rejection, then the null hypothesis may or may not be true (just as a defendant found NOT GUILTY may or may not be innocent) “Fail to Reject” versus “Accept”

  22. Rejecting the null hypothesis when it is true. (alpha) represents the probability of a type I error Example:Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really is 98.6 Type I Error

  23. Failing to reject the null hypothesis when it is false. β (beta) represents 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 really isn’t 98.6 Type II Error

  24. Type I and Type II Errors NULL HYPOTHESIS TRUE FALSE Type I error α Rejecting a true null hypothesis Reject the null hypothesis CORRECT DECISION Type II error β Failing to reject a false null hypothesis Fail to reject the null hypothesis CORRECT

  25. , , and nare interrelated. If one is kept constant, then an increase in one of the remaining two will cause a decrease in the other. For any fixed , an increase in the sample size nwill cause a ??????? in  For any fixed sample size n, a decrease in  will cause a ??????? in . Conversely, an increase in  will cause a ??????? in  . To decrease both  and , ??????? the sample size n. Controlling Type I and Type II Errors

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