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Chapter Nine Part 3 (Section 9.4) Hypothesis Testing

Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College. Chapter Nine Part 3 (Section 9.4) Hypothesis Testing. Hypothesis Testing About a Population Mean  when Sample Evidence Comes From a Small (n < 30) Sample.

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Chapter Nine Part 3 (Section 9.4) Hypothesis Testing

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  1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County College Chapter Nine Part 3 (Section 9.4) Hypothesis Testing

  2. Hypothesis Testing About a Population Mean when Sample Evidence Comes From a Small (n < 30) Sample Use the Student’s t distribution with n – 1 degrees of freedom.

  3. Student’s t Variable Wen we draw a random sample from a population that has a mound-shaped distribution with mean , then:

  4. C represents the level of confidence

  5. ' is the significance level for a one-tailed test

  6. ' is the significance level for a right-tailed test  ' = area to the right of t ' 0 t

  7. ' is the significance level for a left-tailed test  ' = area to the left of – t ' – t 0

  8. '' is the significance level for a two-tailed test

  9. '' is the significance level for a two-tailed test  ' ' = sum of the areas in the two tails ' ' – t 0 t

  10. '' = 2 '

  11. Find the critical value t0 for a left-tailed test of  with n = 4 and level of significance 0.05.

  12. Find the critical value t0 for a left-tailed test of  with n = 4 and level of significance 0.05.

  13. Find the critical value t0 for a left-tailed test of  with n = 4 and level of significance 0.05.

  14. Find the critical value t0 for a left-tailed test of  with n = 4 and level of significance 0.05.

  15. Find the critical value t0 for a left-tailed test of  with n = 4 and level of significance 0.05. t = – 2.353

  16. The Critical Region for the Left-Tailed Test ' = 0.05 – 2.353 0

  17. Find the critical values t0 for a two-tailed test of  with n = 4 and level of significance 0.05.

  18. Find the critical value t0 for a two-tailed test of  with n = 4 and level of significance 0.05.

  19. Find the critical value t0 for a two-tailed test of  with n = 4 and level of significance 0.05.

  20. Find the critical value t0 for a two-tailed test of  with n = 4 and level of significance 0.05.

  21. Find the critical value t0 for a two-tailed test of  with n = 4 and level of significance 0.05. t =  3.182

  22. The Critical Region for the Two-Tailed Test  ' ' = sum of the areas in the two tails = 0.05 ' = 0.025 ' = 0.025 – 3.182 0 3.182

  23. To Complete a t Test • Find the critical value(s) and critical region. • Convert the sample test statistic to a t value. • Locate the t value on a diagram showing the critical region. • If the sample t value falls in the critical region, reject H0. • If the sample t value falls outside the critical region, do not reject H0.

  24. Use a 10% level of significance to test the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower). A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

  25. … test the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower). ... H0:  = 2.1 H1:  < 2.1

  26. A sample of five fish weighed... d.f. = 5 – 1 = 4

  27. Find the critical value(s) and critical region. For a left-tailed test with  ' = 0.10 and d.f. = 4, Table 6 indicates that the critical value of t = – 1.533

  28. The Critical Region for the Left-Tailed Test ' = 0.10 – 1.533 0

  29. … A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

  30. When t falls within the critical region reject the null hypothesis. – 2.73 – 1.533 0

  31. We conclude (at 10% level of significance) that the true weight of the fish in the lake is less than 2.1 kg.

  32. P Values for Tests of  for Small Samples • The probability of getting a sample statistic as far (or farther) into the tails of the sampling distribution as the observed sample statistic. • The smaller the P value, the stronger the evidence to reject H0. • Using Table 6 we find an interval containing the P value.

  33. Determine the P value when testing the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower). A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

  34. We completed a left-tailed test with:

  35. When working with a left-tailed test, use '.

  36. For t = –2.73 and d.f = 4 Sample t = 2.73

  37. 0.025 < P value < 0.050 Sample t = 2.73

  38. 0.025 < P value < 0.050 Since the range of P values was less than  (10%), we rejected the null hypothesis.

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