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Section 8.4

Section 8.4. Day 2. Page 539, E74. Page 539, E74. Two populations: Bus A arrivals and Bus B arrivals Let n 1 be sample for Bus B and n 2 be sample for Bus A. Page 539, E74. Name: One-sided significance test for the difference of two proportions

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Section 8.4

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  1. Section 8.4 Day 2

  2. Page 539, E74

  3. Page 539, E74 Two populations: Bus A arrivals and Bus B arrivals Let n1 be sample for Bus B and n2 be sample for Bus A.

  4. Page 539, E74 Name: One-sided significance test for the difference of two proportions One-sided because our decision will be based on if there is a difference of more than 2%.

  5. Conditions Two populations: Bus A arrivals and Bus B arrivals Random samples as problem states mornings to check arrivals were randomly selected Independent samples as Bus A arrivals do not affect Bus B arrivals and vice versa

  6. Conditions Let n1 be sample for Bus B and n2 be sample for Bus A. n1p1 = 5 n1(1 – p1) = 59 n2p2 = 5 n2(1 – p2) = 95 All are at least 5

  7. Conditions Population sizes are at least 10(100) = 1000 as there are over 1000 mornings to check out bus arrivals. (Can check historical records)

  8. Hypotheses Ho:

  9. Hypotheses Ho: Will use Bus A only if you conclude that the difference in the proportions of late arrivals is more than 2%

  10. Hypotheses Ho: p1 – p2 =0.02 where p1 is the proportion of all mornings Bus B is late and p2 is the proportion of all mornings Bus A is late Will use Bus A only if you conclude that the difference in the proportions of late arrivals is more than 2%

  11. Hypotheses Ho: p1 – p2 = 0.02 where p1 is the proportion of all mornings Bus B is late and p2 is the proportion of all mornings Bus A is late Ha: p1 – p2> 0.02 Will use Bus A only if you conclude that the difference in the proportions of late arrivals is more than 2%

  12. Compute Test Statistic and P-value If do notassume p1 = p2, then use:

  13. Compute Test Statistic and P-value

  14. P-value

  15. P-value 2nd DISTR normalcdf(.2031, 1EE99) = 0.4195

  16. Conclusion

  17. Conclusion I do not reject the null hypothesis because the P-value of 0.4195 is greater than the significance level of α = 0.05.

  18. Conclusion I do not reject the null hypothesis because the P-value of 0.4195 is greater than the significance level of α = 0.05. There is insufficient evidence to conclude that the difference in late arrivals between Bus B and Bus A is more than 2%.

  19. Conclusion I do not reject the null hypothesis because the P-value of 0.4195 is greater than the significance level of α = 0.05. There is insufficient evidence to conclude that the difference in late arrivals between Bus B and Bus A is more than 2%. Therefore, I will use Bus B.

  20. Page 536, P49 Answer true or false for each statement. Be able to justify your choice.

  21. Page 536, P49 a. False; the values of and vary from sample to sample and rarely are equal. Think variation in sampling.

  22. Page 536, P49 b. True; for this problem you are told that the proportions of successes in the two populations are equal.

  23. Page 536, P49 c. True; consider effects of variation in sampling

  24. Page 536, P49

  25. Page 536, P49 e. True; as sample size increases, the spread decreases.

  26. Problem # 1 If the P-value of a test is less than the level of significance, then which of these conclusions is correct?

  27. If the P-value of a test is less than the level of significance, then which of these conclusions is correct? A. The value of the test statistic is in the rejection region for this test. B. The sample size should be increased to decrease the margin of error. C. The null hypothesis is true. D. The corresponding confidence interval will contain the hypothesized value of the parameter in the null hypothesis. E. None of these is a valid conclusion.

  28. If the P-value of a test is less than the level of significance, then which of these conclusions is correct? A.The value of the test statistic is in the rejection region for this test. B. The sample size should be increased to decrease the margin of error. C. The null hypothesis is true. D. The corresponding confidence interval will contain the hypothesized value of the parameter in the null hypothesis. E. None of these is a valid conclusion.

  29. Problem # 1 A. If the P-value is less than α, then the result is “statistically significant.” Reject H0. This corresponds to the test statistic falling in the rejection region.

  30. Problem # 2 A 2003 Harris Poll of 993 randomly selected American adults found that 69% believed in capital punishment (death penalty). A sample size of 1010 American adults in 2000 found that 64% believed in capital punishment. Write the hypotheses for a significance test to determine whether a significantly higher proportion of American adults believed in capital punishment in 2003 than in 2000.

  31. Problem # 2 H0: p1 = p2, where p1 is the proportion of all American adults who believed in capital punishment in 2003 and p2 is the proportion who believed in capital punishment in 2000.

  32. Problem # 2 H0: p1 = p2, where p1 is the proportion of all American adults who believed in capital punishment in 2003 and p2 is the proportion who believed in capital punishment in 2000. Ha: p1 > p2

  33. Problem # 2 H0: p1 = p2, where p1 is the proportion of all American adults who believed in capital punishment in 2003 and p2 is the proportion who believed in capital punishment in 2000. Ha: p1 > p2 determine whether a significantly higher proportion of American adults believed

  34. Problem # 3 If all else remains the same, which of these will make a confidence interval for the difference of two proportions wider? I. Increase the confidence level II. Increase the sample size III. Increase the margin of error IV. Increase the probability of a Type I error

  35. If all else remains the same, which of these will make a confidence interval for the difference of two proportions wider? I. Increase the confidence level II. Increase the sample size III. Increase the margin of error IV. Increase the probability of a Type I error

  36. Page 537, E65

  37. Page 537, E65 To answer the question “is this a significant difference?” what do you have to do?

  38. Page 537, E65 To answer the question “is this a significant difference?” what do you have to do? 1) Name test and check conditions 2) State hypothesis 3) Compute test statistic and P-value; draw sketch. 4) Write conclusion in context.

  39. Page 537, E65 Name: One-sided significance test for the difference of two proportions

  40. Page 537, E65 We are told we can assume the samples were selected randomly. Can also assume they were selected independently because one random sample is boys and the other random sample is girls.

  41. Page 537, E65 Let n1 be sample of girls and n2 be sample of boys.

  42. Page 537, E65 There are more than 2560 boys and 2570 girls aged 12 to 17 in the United States so each population is more than 10 times its sample size.

  43. Page 537, E65 Ho: p1 = p2, where p1 is the proportion of girls who would answer “yes” and p2 is the proportion of boys who would answer “yes”

  44. Page 537, E65 Ho: p1 = p2, where p1 is the proportion of girls who would answer “yes” and p2 is the proportion of boys who would answer “yes” Ha: p1 > p2 (Are you confident larger proportion of girls than boys?)

  45. Page 537, E65 2-PropZTest x1: 195 n1: 257 x2: 161 n2: 256 p1 > p2

  46. Page 537, E65 2-PropZTest x1: 195 n1: 257 x2: 161 n2: 256 p1 > p2 z = 3.19 and P-value = 0.0007

  47. Write Conclusion in Context I reject the null hypothesis because the P-value of 0.00071 is less than the significance level of α = 0.05. There is sufficient evidence to conclude that if I could have asked all girls and all boys ages 12 to 17, a larger proportion of girls than boys would answer they feel they are making a positive difference in their community.

  48. Page 537, E65

  49. Page 537, E65 1-PropZInt x: 356 n: 513 C-level: .95 Calculate (0.654, 0.734)

  50. Page 537, E65

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