APPENDIX D 803 11. Test statistic: x2 = 11.161. P-value = 0.011 (Table: 60.025). Critical value: x2 = 7.815. There is sufficient evidence to support the claim that the results contradict Mendel’s theory. 13. Test statistic: x2 = 13.689. P@value = 0.134 (Table: 7 0.10). Critical value: x2 = 16.919. There is not sufficient evidence to warrant rejection of the claim that the likelihood of winning is the same for the different post positions. Based on these results, post position should not be considered when betting on the Kentucky Derby. 15. Test statistic: x2 = 7.658. P@value = 0.054 (Table: 7 0.05). Critical value: x2 = 7.815. There is not sufficient evidence to warrant rejection of the claim that the actual numbers of games fit the distribution indicated by the proportions listed in the given table. It appears that the actual games conform reasonably well with the results expected by theory. 17. Test statistic: x2 = 9.500. P-value = 0.147 (Table: 70.10). Critical value: x2 = 16.812. There is not sufficient evidence to support the claim that births do not occur on the seven different days of the week with equal frequency. 19. Test statistic: x2 = 5.714. P@value = 0.335 (Table: 7 0.10). Critical value: x2 = 11.071. There is not sufficient evidence to warrant rejection of the claim that the color distribution is as claimed. 21. Test statistic: x2 = 3650.251. P-value = 0.000 (Table: 60.005). Critical value: x2 = 20.090. There is sufficient evidence to warrant rejection of the claim that the leading digits are from a population with a distribution that conforms to Benford’s law. It does appear that the checks are the result of fraud (although the results cannot confirm that fraud is the cause of the discrepancy between the observed results and the expected results). 23. Test statistic: x2 = 1.762. P-value = 0.988 (Table: 70.10). Critical value: x2 = 15.507. There is not sufficient evidence to warrant rejection of the claim that the leading digits are from a population with a distribution that conforms to Benford’s law. The tax entries do appear to be legitimate. 25. a. 26, 46, 49, 26 b. 0.2023, 0.3171, 0.3046, 0.1761 (Table: 0.2033, 0.3166, 0.3039, 0.1762) c. 29.7381, 46.6137, 44.7762, 25.8867 (Table: 29.8851, 46.5402, 44.6733, 25.9014) d. Test statistic: x2 = 0.877 (Using probabilities from table: 0.931). P-value = 0.831 (Table: 70.10). Critical value: x2 = 11.345. There is not sufficient evidence to warrant rejection of the claim that heights were randomly selected from a normally distributed population. The test suggests that we cannot rule out the possibility that the data are from a normally distributed population. Section 11-2 1. E = 138.906 3. a. Test statistic: x2 = 19.490. P-value: 0.000. Reject the null hypothesis of independence between whether the dog is correct and whether malaria is present. b. No. It is possible that the dogs are wrong significantly more than they are correct. However, with the given data, dogs were correct 70.3% of the time when malaria was present, and they were correct 90.3% of the time when malaria was not present, so they do appear to be effective in their identifications. c. H0: m = 16 oz. H1: m ≠ 16 oz. Test statistic: t = 0.998. P@value = 0.3363 (Table: P@value 7 0.20). Critical values (assuming a 0.05 significance level): t = {2.160. Fail to reject H0. There is not sufficient evidence to warrant rejection of the claim that the mean is equal to 16 oz. (This 95% confidence interval could also be used: 15.42 oz 6 m 6 17.58 oz. Because the confidence interval includes 16 oz, there is not sufficient evidence to reject the claim that the mean is equal to 16 oz.). Although the mean appears to be OK, exploring the data reveals that the variation appears to be too high. The minimum sample value is 12.6 oz and the maximum is 19.2 oz, and they show that the amount of variation is far too high. Some containers would be dramatically underfilled while others would be overflowing. The filling device must be modified to correct the unacceptably high amount of variation. 8. a. What is an estimate of the proportion of wrong results in the population of all of the drug tests? b. Use the methods of Section 7-1 to construct a confidence interval estimate of the proportion p of wrong results in the population of all of the drug tests. Decide whether the proportion of wrong results is acceptable. c. 95% confidence interval estimate: 0.0576 6 p 6 0.122. This shows that we have 95% confidence that the percentage of wrong test results is between 5.76% and 12.2%. Because wrong test results could possibly have adverse implications, such as unfair rejection of job applicants, it appears that the proportion of wrong results is unacceptably high. Chapter 11 Answers Section 11-1 1. a. Observed values are represented by O and expected values are represented by E. b. For the leading digit of 2, O = 62 and E = 13172 10.1762 = 55.792. c. For the leading digit of 2, (O - E)2>E = 0.691. 3. There is sufficient evidence to warrant rejection of the claim that the leading digits have a distribution that fits well with Benford’s law. 5. Test statistic: x2 = 4.050. P@value = 0.908 (Table: 7 0.10). Critical value: x2 = 16.919. There is not sufficient evidence to warrant rejection of the claim that the last digits of the reported heights occur with about the same frequency. There is not sufficient evidence to conclude that the heights were reported instead of being measured. 7. P-value = 0.516 (Table: 70.10). Critical value: x2 = 16.919. There is not sufficient evidence to warrant rejection of the claim that the observed outcomes agree with the expected frequencies. The slot machine appears to be functioning as expected. 9. Test statistic: x2 = 524.713. P@value = 0.0000 (Table: 60.005). Critical value: x2 = 13.277. There is sufficient evidence to warrant rejection of the claim that the distribution of clinical trial participants fits well with the population distribution. Hispanics have an observed frequency of 60 and an expected frequency of 391.027, so they are very underrepresented. Also, the Asian> Pacific Islander subjects have an observed frequency of 54 and an expected frequency of 163.286, so they are also underrepresented.
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