Chapter 5 Inferences Based on a Single Sample: Estimation with Confidence Intervals α
Chapter 5 Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.1 5.2 5.3 a. For α = .10, α/2 = .10/2 = .05. zα/2 = z.05 is the z score with .05 of the area to the right of it. The area between 0 and z.05 is .5 − .05 = .4500. Using Table IV, Appendix B, z.05 = 1.645. b. For α = .01, /2 = .01/2 = .005. z /2 = z.005 is the z score with .005 of the area to the right of it. The area between 0 and z.005 is .5 − .005 = .4950. Using Table IV, Appendix B, z.005 = 2.575. c. For α = .05, α/2 = .05/2 = .025. zα/2 = z.025 is the z score with .025 of the area to the right of it. The area between 0 and z.025 is .5 − .025 = .4750. Using Table IV, Appendix B, z.025 = 1.96. d. For α = .20, α/2 = .20/2 = .10. zα/2 = z.10 is the z score with .10 of the area to the right of it. The area between 0 and z.10 is .5 − .10 = .4000. Using Table IV, Appendix B, z.10 = 1.28. a. zα/2 = 1.96, using Table IV, Appendix B, P(0 ≤ z ≤ 1.96) = .4750. Thus, α/2 = .5000 − .4750 = .025, α = 2(.025) = .05, and 1 − α = 1 - .05 = .95. The confidence level is 100% × .95 = 95%. b. zα/2 = 1.645, using Table IV, Appendix B, P(0 ≤ z ≤ 1.645) = .45. Thus, α/2 = .50 − .45 = .05, α = 2(.05) = .1, and 1 − α = 1 − .1 = .90. The confidence level is 100% × .90 = 90%. c. zα/2 = 2.575, using Table IV, Appendix B, P(0 ≤ z ≤ 2.575) = .495. Thus, α/2 = .500 − .495 = .005, α = 2(.005) = .01, and 1 − α = 1 − .01 = .99. The confidence level is 100% × .99 = 99%. d. zα/2 = 1.282, using Table IV, Appendix B, P(0 ≤ z ≤ 1.282) = .4. Thus, α/2 = .5 − .4 = .1, α = 2(.1) = .2, and 1 − α = 1 − .2 = .80. The confidence level is 100% × .80 = 80%. e. zα/2 = .99, using Table IV, Appendix B, P(0 ≤ z ≤ .99) = .3389. Thus, α/2 = .5000 − .3389 = .1611, α = 2(.1611) = .3222, and 1 − α = 1 − .3222 = .6778. The confidence level is 100% × .6778 = 67.78%. a. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: σ 28 ± 1.96 12 28 ± .784 (27.216, 28.784) x ± z.025 75 n b. x ± z.025 σ 102 ± 1.96 n c. x ± z.025 x ± z.025 e. 100 σ 4.05 ± 1.96 .83 n 102 ± .65 (101.35, 102.65) 200 σ 15 ± 1.96 .3 n d. 22 15 ± .0588 (14.9412, 15.0588) 100 4.05 ± .163 (3.887, 4.213) No. Since the sample size in each part was large (n ranged from 75 to 200), the Central Limit Theorem indicates that the sampling distribution of is approximately normal. 267 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 268 5.4 Chapter 5 a. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x ± z.025 b. s 25.9 ± 1.645 2.7 90 n s 25.9 ± 2.58 n a. 90 n 25.9 ± .56 (25.34, 26.46) 25.9 ± .47 (25.43, 26.37) For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: x ± z.005 5.5 2.7 25.9 ± 1.96 For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: x ± z.05 c. s 2.7 90 25.9 ± .73 (25.17, 26.63) For confidence coefficient .95, = .05 and /2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x zα / 2 s n 26.2 1.96 4.1 70 26.2 .96 (25.24, 27.16) b. The confidence coefficient of .95 means that in repeated sampling, 95% of all confidence intervals constructed will includeμ. c. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: x zα / 2 s n 26.2 2.58 4.1 70 26.2 1.26 (24.94, 27.46) d. As the confidence coefficient increases, the width of the confidence interval also increases. e. Yes. Since the sample size is 70, the Central Limit Theorem applies. This ensures the distribution of is normal, regardless of the original distribution. 5.6 If we were to repeatedly draw samples from the population and form the interval x ± 1.96 σ x each time, approximately 95% of the intervals would contain μ. We have no way of knowing whether our interval estimate is one of the 95% that contain μ or one of the 5% that do not. 5.7 A point estimator is a single value used to estimate the parameter, μ. An interval estimator is two values, an upper and lower bound, which define an interval with which we attempt to enclose the parameter, . An interval estimate also has a measure of confidence associated with it. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.8 a. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x ± z.025 s 33.9 ± 1.96 3.3 33.9 ± 1.96 3.3 100 n b. x ± z.025 s 400 n c. 269 33.9 ± .647 (33.253, 34.547) 33.9 ± .323 (33.577, 34.223) For part a, the width of the interval is 2(.647) = 1.294. For part b, the width of the interval is 2(.323) = .646. When the sample size is quadrupled, the width of the confidence interval is halved. 5.9 Yes. As long as the sample size is sufficiently large, the Central Limit Theorem says the distribution of is approximately normal regardless of the original distribution. 5.10 a. A point estimate for the average number of latex gloves used per week by all healthcare workers with latex allergy is x = 19.3 . b. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x zα / 2 n 19.3 1.96 11.9 46 19.3 3.44 (15.86, 22.74) c. We are 95% confident that the true average number of latex gloves used per week by all healthcare workers with a latex allergy is between 15.86 and 22.74. d. The conditions required for the interval to be valid are: a. b. 5.11 s The sample selected was randomly selected from the target population. The sample size is sufficiently large, i.e. n > 30. For confidence coefficient .90, α = 1 - .90 = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The 90% confidence interval is: x zα / 2 s 2, 484 6,563 97.65 (6, 465.35, 6,660.65) 1, 751 We are 90% confident that the true mean expenses per full-time equivalent employees of all U.S. Army hospitals is between $6,465.35 and $6,660.65. 5.12 n 6,563 1.645 a. The point estimate for the mean charitable commitment of tax-exempt organizations is x = 79.67. b. From the printout, the 95% confidence interval is (75.84, 83.49). c. The probability of estimating the true mean charitable commitment with a single number is 0. By estimating the true mean charitable commitment with an interval, we can be pretty confident that the true mean is in the interval. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 270 5.13 Chapter 5 a. For confidence coefficient .99, α = 1 - .99 = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The 99% confidence interval is: s x zα / 2 5.14 5.15 n 4.25 2.58 12.02 56 4.25 4.14 (0.11, 8.39) b. We are 99% confident that the true mean number of blogs/forums per site of all Fortune 500 firms that provide blogs and forums for marketing tools is between 0.11 and 8.39. c. No. Since our sample size is 56, the sampling distribution of x is approximately normal by the Central Limit Theorem. a. From the printout, the 95% confidence interval is (1.6711, 2.1989). b. We are 95% confident that the true mean failure time of used colored display panels is between 1.6711 and 2.1989. c. If 95% confidence intervals are formed, then approximately .95 of the intervals will contain the true mean failure time. a. The target parameter is the population mean 2008 salary of these 500 CEOs who participated in the Forbes’ survey. b. Using MINITAB, a sample of 50 CEOs was selected. The ranks of the 50 selected are: 9, 10, 14, 18, 19, 22, 25, 32, 38, 39, 45, 49, 50, 55, 60, 66, 69, 77, 96, 104, 106, 115, 147, 152, 192, 197, 209, 213, 229, 241, 245, 261, 268, 278, 283, 292, 305, 309, 325, 337, 342, 358, 364, 370, 376, 384, 405, 417, 433, 470. c. Using MINITAB, the descriptive statistics are: Descriptive Statistics: PAY2005 ($mil) Variable PAY ($mil) N 50 Mean 19.56 StDev 19.95 Minimum 1.23 Q1 4.70 Median 8.99 Q3 32.78 Maximum 73.17 The sample mean is x = 19.56 and the sample standard deviation is s = 19.95. d. Using MINITAB, the descriptive statistics for the entire data set is: Descriptive Statistics: Pay ($mil) Variable Pay ($mil) N 497 Mean 12.874 StDev 18.502 Minimum 0.000000000 Q1 3.395 Median 6.470 Q3 14.250 Maximum 192.920 From the above, the standard deviation of the population is $18.5 million. e. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: x zα / 2 s n 19.56 2.58 19.95 50 19.56 7.28 (12.28, 26.84) Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.16 271 f. We are 99% confident that the true mean salary of all 500 CEOs in the Forbes’ survey is between $12.28 million and $26.84 million. g. From part d, the true mean salary of all 500 CEOs is $12.87 million. This value does fall within the 99% confidence interval that we found in part e. a. Using MINITAB, the descriptive statistics are: Descriptive Statistics: Rate Variable Rate N 30 Mean 79.73 Median 80.00 TrMean 80.15 Variable Rate Minimum 60.00 Maximum 90.00 Q1 76.75 Q3 84.00 StDev 5.96 SE Mean 1.09 For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: x zα / 2 s n 79.73 1.645 5.96 30 79.73 1.79 (77.94, 81.52) 5.17 b. We are 90% confident that the mean participation rate for all companies that have 401(k) plans is between 77.94% and 81.52%. c. We must assume that the sample size (n = 30) is sufficiently large so that the Central Limit Theorem applies. d. Yes. Since 71% is not included in the 90% confidence interval, it can be concluded that this company's participation rate is lower than the population mean. e. The center of the confidence interval is . If 60% is changed to 80%, the value of will increase, thus indicating that the center point will be larger. The value of s2 will decrease if 60% is replaced by 80%, thus causing the width of the interval to decrease. a. An estimate of the true mean Mach rating score of all purchasing managers is x =99.6. b. For confidence coefficient .95, α = 1 - .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The 95% confidence interval is: x zα / 2 s n 99.6 1.96 12.6 122 99.6 2.24 (97.36, 101.84) c. We are 95% confident that the true Mach rating score of all purchasing managers is between 97.36 and 101.84. d. Yes, there is evidence to dispute this claim. We are 95% confident that the true mean Mach rating score is between 97.36 and 101.84. It would be very unlikely that the true means Mach scores is as low as 85. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 272 5.18 Chapter 5 a. Using MINITAB, I generated 30 random numbers using the uniform distribution from 1 to 308. The random numbers were: 9, 15, 19, 36, 46, 47, 63, 73, 90, 92, 108, 112, 117, 127, 144, 145, 150, 151, 172, 178, 218, 229, 230, 241, 242, 246, 252, 267, 274, 282 I numbered the 308 observations in the order that they appear in the file. Using the random numbers generated above, I selected the 9th, 15th, 19th, etc. observations for the sample. The selected sample is: .31, .34, .34, .50, .52, .53, .64, .72, .70, .70, .75, .78, 1.00, 1.00, 1.03, 1.04, 1.07, 1.10, .21, .24, .58, 1.01, .50, .57, .58, .61, .70, .81, .85, 1.00 b. Using MINITAB, the descriptive statistics for the sample of 30 observations are: Descriptive Statistics: carats-samp Variable carats-s N 30 Mean 0.6910 Median 0.7000 TrMean 0.6965 Variable carats-s Minimum 0.2100 Maximum 1.1000 Q1 0.5150 Q3 1.0000 StDev 0.2620 SE Mean 0.0478 From above, x =.6910 and s = .2620. c. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x zα / 2 d. e. 5.19 s n .691 1.96 .262 30 .691 .094 (.597, .785) We are 95% confident that the mean number of carats is between .597 and .785. From Exercise 2.49, we computed the “population” mean to be .631. This mean does fall 95% confidence interval we computed in part d. in the To answer the question, we will first form 90% confidence intervals for each of the 2 SAT scores. For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: x zα / 2 s n 19 1.645 65 265 19 6.57 (12.43, 25.57) We are 90% confident that the mean change in SAT-Mathematics score is between 12.43 and 25.57 points. For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: x zα / 2 s n 7 1.645 49 265 7 4.95 (2.05, 11.95) We are 90% confident that the mean change in SAT-Verbal score is between 2.05 and 11.95 points. The SAT-Mathematics test would be the most likely of the two to have 15 as the mean change in score. This value of 15 is in the 90% confidence interval for the mean change in SAT-Mathematics score. However, 15 does not fall in the 90% confidence interval for the mean SAT-Verbal test. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.20 x 273 11, 298 = 2.26 5, 000 For confidence coefficient, .95, α = .05 and α/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x ± zα/2 s n 2.26 ± 1.96 1.5 5000 2.26 ± .04 (2.22, 2.30) We are 95% confident the mean number of roaches produced per roach per week is between 2.22 and 2.30. 5.21 5.22 5.23 a. For confidence coefficient .80, α = 1 .80 = .20 and α/2 = .20/2 = .10. From Table IV, Appendix B, z.10 = 1.28. From Table V, with df = n 1 = 5 1 = 4, t.10 = 1.533. b. For confidence coefficient .90, α = 1 .90 = .05 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. From Table V, with df = n 1 = 5 1 = 4, t.05 = 2.132. c. For confidence coefficient .95, α = 1 .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. From Table V, with df = n 1 = 5 1 = 4, t.025 = 2.776. d. For confidence coefficient .98, α = 1 .98 = .02 and α/2 = .02/2 = .01. From Table IV, Appendix B, z.01 = 2.33. From Table V, with df = n 1 = 5 1 = 4, t.01 = 3.747. e. For confidence coefficient .99, α = 1 .99 = .02 and α/2 = .02/2 = .005. From Table IV, Appendix B, z.005 = 2.575. From Table V, with df = n 1 = 5 1 = 4, t.005 = 4.604. f. Both the t- and z-distributions are symmetric around 0 and mound-shaped. The t-distribution is more spread out than the z-distribution. a. If x is normally distributed, the sampling distribution of x is normal, regardless of the sample size. b. If nothing is known about the distribution of x, the sampling distribution of x is approximately normal if n is sufficiently large. If n is not large, the distribution of x is unknown if the distribution of x is not known. a. P(−t0 < t < t0) = .95 where df = 10 Because of symmetry, the statement can be written P(0 < t < t0) = .475 where df = 10 P(t ≥ t0) = .025 t0 = 2.228 P(t ≤ (−t0 or t ≥ t0) = .05 where df = 10 2P(t ≥ t0) = .05 P(t ≥ t0) = .025 where df = 10 t0 = 2.228 b. c. P(t ≤ t0) = .05 where df = 10 Because of symmetry, the statement can be written P(t ≥ −t0) = .05 where df = 10 t0 = −1.812 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 274 5.24 Chapter 5 d. P(t < −t0 or t > t0) = .10 where df = 20 2P(t > t0) = .10 P(t > t0) = .05 where df = 20 t0 = 1.725 e. P(t ≤ −t0 or t ≥ t0) = .01 where df = 5 2P(t ≥ t0) = .01 P(t ≥ t0) = .005 where df = 5 t0 = 4.032 a. P(t ≥ t0) = .025 where df = 11 t0 = 2.201 b. P(t ≥ t0) = .01 where df = 9 t0 = 2.821 c. P(t ≤ t0) = .005 where df = 6 Because of symmetry, the statement can be rewritten P(t ≥ −t0) = .005 where df = 6 t0 = −3.707 d. 5.25 P(t ≤ t0) = .05 where df = 18 t0 = −1.734 First, we must compute x and s. x s2 x 2 x 30 5 n x n n 1 6 2 (30) 2 6 26 5.2 6 1 5 176 s 5.2 2.2804 a. For confidence coefficient .90, α = 1 .90 = .10 and α/2 = .10/2 = .05. From Table V, Appendix B, with df = n 1 = 6 1 = 5, t.05 = 2.015. The 90% confidence interval is: s x t0.5 b. n 5 ± 2.015 2.2804 6 5 ± 1.88 (3.12, 6.88) For confidence coefficient .95, α = 1 .95 = .05 and α/2 = .05/2 = .025. From Table V, Appendix B, with df = n − 1 = 6 − 1 = 5, t.025 = 2.571. The 95% confidence interval is: x t.025 s n 5 ± 2.571 2.2804 6 5 ± 2.39 (2.61, 7.39) Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals c. For confidence coefficient .99, α = 1 .99 = .01 and α/2 = .01/2 = .005. From Table V, Appendix B, with df = n 1 = 6 1 = 5, t.005 = 4.032. The 99% confidence interval is: x t.005 d. 275 s 5 ± 4.032 n 2.2804 6 5 ± 3.75 (1.25, 8.75) For confidence coefficient .90, α = 1 .90 = .10 and α/2 = .10/2 = .05. From Table V, Appendix B, with df = n 1 = 25 1 = 24, t.05 = 1.711. The 90% confidence interval is: a) s x t.05 2.2804 5 ± .78 (4.22, 5.78) n 25 For confidence coefficient .95, α = 1 .95 = .05 and α/2 = .05/2 = .025. From Table V, Appendix B, with df = n 1 = 25 1 = 24, t.025 = 2.064. The 95% confidence interval is: b) 5 ± 1.711 s x t.025 n 5 ± 2.064 2.2804 25 5 ± .94 (4.06, 5.94) For confidence coefficient .99, α = 1 .99 = .01 and α/2 = .01/2 = .005. From Table V, Appendix B, with df = n 1 = 25 1 = 24, t.005 = 2.797. The 99% confidence interval is: c) s x t.005 n 5 ± 2.797 2.2804 25 5 ± 1.28 (3.72, 6.28) Increasing the sample size decreases the width of the confidence interval. 5.26 For this sample, x s2 = x 1567 n 16 x2 = 97.9375 x n 1 n 2 1567 2 16 = 159.9292 16 1 155,867 s= s 2 = 12.6463 a. For confidence coefficient, .80, α = 1 − .80 = .20 and α/2 = .20/2 = .10. From Table V, Appendix B, with df = n − 1 = 16 − 1 = 15, t.10 = 1.341. The 80% confidence interval for μ is: 12.6463 s x ± t.10 97.94 ± 1.341 97.94 ± 4.240 (93.700, 102.180) 16 n b. For confidence coefficient, .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table V, Appendix B, with df = n − 1 = 24 − 1 = 23, t.025 = 2.131. The 95% confidence interval for μ is: x ± t.025 s n 97.94 ± 2.131 12.6463 16 97.94 ± 6.737 (91.203, 104.677) The 95% confidence interval for μ is wider than the 80% confidence interval for μ found in part a. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 276 Chapter 5 c. For part a: We are 80% confident that the true population mean lies in the interval 93.700 to 102.180. For part b: We are 95% confident that the true population mean lies in the interval 91.203 to 104.677. The 95% confidence interval is wider than the 80% confidence interval because the more confident you want to be that μ lies in an interval, the wider the range of possible values. 5.27 For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table V, Appendix B, with df = n – 1 = 25 – 1 = 24, t.05 = 1.711. The 90% confidence interval is: x t.05 s n 75.4 1.711 10.9 25 75.4 3.73 (71.67, 79.13) We are 90% confident that the mean breaking strength of the white wood is between 71.67 and 79.13. 5.28 We must assume that the distribution of the LOS's for all patients is normal. a. For confidence coefficient .90, α = 1 − .90 = .10 and α/2 = .10/2 = .05. From Table V, Appendix B, with df = n − 1 = 20 − 1 = 19, t.05 = 1.729. The 90% confidence interval is: x t.05 5.29 5.30 s n 3.8 ± 1.729 1.2 20 3.8 ± .464 (3.336, 4.264) b. We are 90% confident that the mean LOS is between 3.336 and 4.264 days. c. “90% confidence” means that if repeated samples of size n are selected from a population and 90% confidence intervals are constructed, 90% of all intervals thus constructed will contain the population mean. a. From the printout, the 95% confidence interval is (−320%, 5,922%). We are 95% confident that the true mean 5-year revenue growth rate for the 12 companies is between −320% and 5,922%. b. The population being sampled from must be normally distributed. c. From the stem-and-leaf display in the printout, the data appear to be skewed to the right. The data do not appear to have come from a normal distribution. Therefore, the 95% confidence interval in part a may not be valid. a. The 95% confidence interval for the mean surface roughness of coated interior pipe is (1.63580, 2.12620). b. No. Since 2.5 does not fall in the 95% confidence interval, it would be very unlikely that the average surface roughness would be as high as 2.5 micrometers. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals a. Using MINITAB, the descriptive statistics are: Descriptive Statistics: Skid Variable Skid N 20 N* 0 Mean 358.5 SE Mean 26.3 StDev 117.8 Minimum 141.0 Q1 276.0 Median 367.5 Q3 438.0 Maximum 574.0 For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table V, Appendix B, with df = n – 1 = 20 – 1 = 19, t.025 = 2.093. The 95% confidence interval is: x t.05 s n 358.5 2.093 117.8 20 358.5 55.13 (303.37, 413.63) b. We are 95% confident that the mean skidding distance is between 303.37 and 413.63 meters. c. In order for the inference to be valid, the skidding distances must be from a normal distribution. We will use the four methods to check for normality. First, we will look at a histogram of the data. Using MINITAB, the histogram of the data is: Histogram of Skid 4 3 Fr equency 5.31 277 2 1 0 200 300 400 500 Skid From the histogram, the data appear to be fairly mound-shaped. This indicates that the data may be normal. Next, we look at the intervals x s, x 2s, x 3s . If the proportions of observations falling in each interval are approximately .68, .95, and 1.00, then the data are approximately normal. Using MINITAB, the summary statistics are: x s 358.5 117.8 (240.7, 476.3) 14 of the 20 values fall in this interval. The proportion is .70. This is very close to the .68 we would expect if the data were normal. x 2s 358.5 2(117.8) 358.5 235.6 (122.9, 594.1) 20 of the 20 values fall in this interval. The proportion is 1.00. This is a larger than the .95 we would expect if the data were normal. x 3s 358.5 3(117.8) 358.5 353.4 (5.1, 711.9) 20 of the 20 values fall in this interval. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 278 Chapter 5 The proportion is 1.00. This is exactly the 1.00 we would expect if the data were normal. From this method, it appears that the data may be normal. Next, we look at the ratio of the IQR to s. IQR = QU – QL = 438 – 276 = 162. IQR 162 1.37 This is fairly close to the 1.3 we would expect if the data were normal. This s 117.8 method indicates the data may be normal. Finally, using MINITAB, the normal probability plot is: Probability Plot of Skid N ormal - 95% C I 99 95 90 Mean StDev 358.5 117.8 N AD P-Value 20 0.170 0.921 P er cent 80 70 60 50 40 30 20 10 5 1 0 100 200 300 400 Skid 500 600 700 800 Since the data form a fairly straight line, the data may be normal. From above, all the methods indicate the data may be normal. It appears that the assumption that the data come from a normal distribution is probably valid. 5.32 d. No. A distance of 425 meters falls above the 95% confidence interval that was computed in part a. It would be very unlikely to observe a mean skidding distance of at least 425 meters. a. Using MINITAB, the descriptive statistics are: Descriptive Statistics: MTBE Variable MTBE N 12 N* 0 Mean 97.2 SE Mean 32.8 StDev 113.8 Minimum 8.00 Q1 12.0 Median 50.5 Q3 146.0 Maximum 367.0 A point estimate for the true mean MTBE level for all well sites located near the New Jersey gasoline service station is x 97.2 . Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals b. 279 For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table V, Appendix B, with df = n – 1 = 12 – 1 = 11, t.005 = 3.106. The 99% confidence interval is: x t.005 s n 97.2 3.106 113.8 12 97.2 102.04 (4.84, 199.24) We are 99% confident that the true mean MTBE level for all well sites located near the New Jersey gasoline service station is between −4.84 and 199.24. We must assume that the data were sampled from a normal distribution. We will use the four methods to check for normality. First, we will look at a histogram of the data. Using MINITAB, the histogram of the data is: Histogram of MTBE 5 4 Fr equency c. 3 2 1 0 0 50 100 150 200 M T BE 250 300 350 From the histogram, the data do not appear to be mound-shaped. This indicates that the data may not be normal. Next, we look at the intervals x s, x 2s, x 3s . If the proportions of observations falling in each interval are approximately .68, .95, and 1.00, then the data are approximately normal. Using MINITAB, the summary statistics are: x s 97.2 113.8 (16.6, 211.0) 10 of the 12 values fall in this interval. The proportion is .83. This is not very close to the .68 we would expect if the data were normal. x 2s 97.2 2(113.8) 97.2 227.6 (130.4, 324.8) 11 of the 12 values fall in this interval. The proportion is .92. This is a somewhat smaller than the .95 we would expect if the data were normal. x 3s 97.2 3(113.8) 97.2 341.4 (244.2, 438.6) 12 of the 12 values fall in this interval. The proportion is 1.00. This is exactly the 1.00 we would expect if the data were normal. From this method, it appears that the data may not be normal. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 280 Chapter 5 Next, we look at the ratio of the IQR to s. IQR = QU – QL = 146.0 – 12.0 = 134.0. IQR 134.0 1.18 This is somewhat smaller than the 1.3 we would expect if the data were normal. s 113.8 This method indicates the data may not be normal. Finally, using MINITAB, the normal probability plot is: Probability Plot of MTBE N ormal - 95% C I 99 95 90 Mean StDev 97.17 113.8 N AD P-Value 12 0.929 0.012 P er cent 80 70 60 50 40 30 20 10 5 1 -300 -200 -100 0 100 200 M T BE 300 400 500 Since the data do not form a fairly straight line, the data may not be normal. From above, the all methods indicate the data may not be normal. It appears that the data probably are not normal. 5.33 a. Using MINITAB, the descriptive statistics are: Descriptive Statistics: Alqaeda Variable Alqaeda N 21 Mean 1.857 StDev 1.195 Minimum 1.000 Q1 1.000 Median 1.000 Q3 2.000 The sample mean is x = 1.857 and the sample standard deviation is s = 1.195. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Maximum 5.000 Inferences Based on a Single Sample: Estimation with Confidence Intervals b. 281 Using MINITAB, a histogram of the data is: Histogram of Alqaeda 12 10 Frequency 8 6 4 2 0 1 2 3 Alqaeda 4 5 The data are skewed to the right. c. Since the sample size is small (n = 21), the Central Limit Theorem does not apply. In order to use a small sample confidence interval, the population being sampled from must be normal. In this case, it does not look like the population being sampled from is normal. Thus, the confidence interval may not be valid. For confidence coefficient .90, α = 1 - .90 = .10 and α/2 = .10/2 = .05. From Table V, Appendix B, with df = n – 1 = 21 – 1 = 20, t.05 = 1.725. The 90% confidence interval is: x tα / 2 5.34 s n 1.857 1.725 1.195 21 1.857 .450 (1.407, 2.307) d. We are 90% confident that the mean number of individual suicide bombings and/or attacks per incident is between 1.407 and 2.307. e. If the distribution being sampled from is normal, then .90 of all intervals formed will contain the true mean. Using MINITAB, the descriptive statistics are: Descriptive Statistics: Comp Variable Comp N 10 Mean 1368 StDev 463 Minimum 720 Q1 1016 Median 1352 Q3 1652 Maximum 2112 For confidence coefficient .90, α = 1 - .90 = .10 and α/2 = .10/2 = .05. From Table V, Appendix B, with df = n – 1 = 10 – 1 = 9, t.05 = 1.833. The 90% confidence interval is: x tα / 2 s n 1368 1.833 463 10 1368 268.38 (1, 099.62, 1, 636.38) We are 90% confident that the true mean threshold compensation level for all major airlines is between $1,099.62 and $1,636.38. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 282 5.35 Chapter 5 a. The population from which the sample was drawn is the Forbes 441 Biggest Private companies. b. Using MINITAB, the descriptive statistics are: Descriptive Statistics: Revenue Variable Revenue N 15 Mean 4.61 StDev 4.51 Minimum 1.10 Q1 1.75 Median 2.20 Q3 5.05 Maximum 14.32 For confidence coefficient .98, α = .02 and α/2 = .02/2 = .01. From Table V, Appendix B, with df = n – 1 = 15 – 1 = 14, t.01 = 2.624. The 98% confidence interval is: x t.025 s n 4.61 2.624 4.51 15 4.61 3.06 (1.55, 7.67) c. We are 98% confident that the mean revenue is between $1.554 and $7.666 billion . d. The population must be normally distributed in order for the procedure used in part b to be valid. e. Yes. The value of $5.0 billion dollars falls in the 98% confidence interval computed in part b. Therefore, we should believe the claim. 5.36 By the Central Limit Theorem, the sampling distribution of is approximately normal with mean 5.37 pq . n The sample size is large enough if both npˆ 15 and nqˆ 15 . μ pˆ = p and standard deviation σ pˆ a. When n = 400, pˆ = .10: npˆ = 400(.10) = 40 and nqˆ = 400(.90) = 360 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. b. When n = 50, pˆ = .10: npˆ = 50(.10) = 5 and nqˆ = 50(.90) = 45 Since npˆ is less than 15, the sample size is not large enough to conclude the normal approximation is reasonable. c. When n = 20, pˆ = .5: npˆ = 20(.5) = 10 and nqˆ = 20(.5) = 10 Since both numbers are less than 15, the sample size is not large enough to conclude the normal approximation is reasonable. d. When n = 20, pˆ = .3: npˆ = 20(.3) = 6 and nqˆ = 20(.7) = 14 Since both numbers are less than 15, the sample size is not large enough to conclude the normal approximation is reasonable. 5.38 a. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 121(.88) = 106.48 and nqˆ = 121(.12) = 14.52 Since nqˆ is less than 15, the sample size is not large enough to conclude the normal approximation is reasonable. However, 14.52 is very close to 15, so the normal approximation may work fairly well. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals b. For confidence coefficient .90, α = .10 and α/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The 90% confidence interval is: pˆ z .05 5.39 283 pq pˆ ± 1.645 n ˆˆ pq .88(.12) .88 ± 1.645 1.645 .88 ± .049 n 121 (.831, .929) c. We must assume that the sample is a random sample from the population of interest. a. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 225(.46) = 103.5 and nqˆ = 225(.54) = 121.5 b. Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = .05 and α/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The 95% confidence interval is: pˆ ± z.025 5.40 pq pˆ ± 1.96 n ˆˆ pq .46(1 .46) .46 ± 1.96 .46 ± .065 (.395, .525) n 225 c. We are 95% confident the true value of p will fall between .395 and .525. d. "95% confidence interval" means that if repeated samples of size 225 were selected from the population and 95% confidence intervals formed, 95% of all confidence intervals will contain the true value of p. a. Of the 50 observations, 15 like the product pˆ 15 = .30. 50 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 50(.3) = 15 and nqˆ = 50(.7) = 35 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For the confidence coefficient .80, α = .20 and α/2 = .10. From Table IV, Appendix B, z.10 = 1.28. The confidence interval is: pˆ ± z.10 5.41 .3(.7) ˆˆ pq .3 ± 1.28 .3 ± .083 (.217, .383) 50 n b. We are 80% confident the proportion of all consumers who like the new snack food is between .217 and .383. a. The population of interest is all American adults. b. The sample is the 1,000 adults surveyed. c. The parameter of interest is the proportion of all American adults who think Starbucks coffee is overpriced. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 284 Chapter 5 d. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 1,000(.73) = 730 and nqˆ = 1,000(.27) = 270 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = 1 - .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The 95% confidence interval is: pˆ zα / 2 pq pˆ zα / 2 n ˆˆ .73(.27) pq .73 1.96 .73 .028 (.702, .758) n 1000 We are 95% confident that the true proportion of all American adults who say Starbucks coffee is overpriced is between .702 and .758. 5.42 a. An estimate of the true proportion of satellite radio subscribers who have a satellite radio receiver x 396 in their car is pˆ .79 . n 501 b. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 501(.79) = 395.79 and nqˆ = 501(.21) = 105.21 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .90, α = 1 - .90 = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The 90% confidence interval is: pˆ zα / 2 5.43 pq pˆ zα / 2 n ˆˆ .79(.21) pq .79 1.645 .79 .036 (.754, .826) n 501 c. We are 90% confident that the true proportion of satellite radio subscribers who have a satellite receiver in their car is between .754 and .826. a. The point estimate of p is pˆ b. x 414 .46 . n 900 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 900(.46) = 414 and nqˆ = 900(.54) = 486 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: pˆ z.05 ˆˆ .46(.54) pq .46 1.645 .46 .027 (.433, .487) n 900 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.44 c. We are 90% confident that the true proportion of contractors in the U.S. who have a company website or will have one by the end of the year is between .433 and .487. d. The meaning of “90% confident” is that in repeated sampling, 90% of all confidence intervals constructed will contain the true proportion and 10% will not. a. The point estimate of p is pˆ .11 . b. The sample size is large enough if both npˆ 15 and nqˆ 15 . 285 npˆ = 150(.11) = 16.5 and nqˆ = 150(.89) = 133.5 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ z.025 5.45 ˆˆ .11(.89) pq .11 1.645 .11 .05 (.06, .16) n 150 c. We are 95% confident that the true proportion of MSDS that are satisfactorily completed is between .06 and .16. a. The point estimate of p is pˆ b. The sample size is large enough if both npˆ 15 and nqˆ 15 . x 35 .03 . n 1,165 npˆ = 1,165(.03) = 34.95 and nqˆ = 1,165(.97) = 1130.05 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ z.025 5.46 ˆˆ pq .03(.97) .03 1.96 .03 .01 (.02, .04) n 1,165 c. We are 95% confident that the true proportion of drivers that use their cell phone while driving is between .02 and .04. a. Since all the people surveyed were from Muncie, Indiana, the population of interest is all consumers in Muncie, Indiana. b. The characteristic of interest in the population is the proportion of shoppers who believe that “Made in the USA” means that 100% of labor and materials are from the USA. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 286 Chapter 5 c. The point estimate of p is pˆ x 64 .604 . n 106 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 106(.604) = 64.024 and nqˆ = 106 (.396) = 41.976 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: pˆ z.05 5.47 ˆˆ pq .604(.396) .604 1.645 .604 .078 (.526, .682) n 106 d. We are 90% confident that the true proportion of shoppers who believe that “Made in the USA” means that 100% of labor and materials are from the USA is between .526 and .682. e. 90% confidence means that if we took repeated samples of size 106 and computed 90% confidence intervals for the true proportion shoppers who believe that “Made in the USA” means that 100% of labor and materials are from the USA, 90% of the intervals computed will contain the true proportion. a. A point estimate for the true percentage of all American adults who have access to a high-speed internet connection is 42%. Changing this to a proportion, the point estimate is pˆ .42 . b. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 4,000(.42) = 1,680 and nqˆ = 4,000(.58) = 2,320 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = 1 - .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The 95% confidence interval is: pˆ zα / 2 pq pˆ zα / 2 n ˆˆ pq .42(.58) .42 1.96 .42 .015 (.405, .435) n 4, 000 We are 95% confident that the true proportion of all American adults who have access to a high-speed internet connection at home is between .405 and .435. c. Yes, since the interval constructed in part b contains values greater than .30, we could conclude that the percentage has increased since 2005. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.48 a. The population is all senior human resource executives at U.S. companies. b. The population parameter of interest is p, the proportion of all senior human resource executives at U.S. companies who believe that their hiring managers are interviewing too many people to find qualified candidates for the job. The point estimate of p is pˆ 287 x 211 .42 . n 502 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 502(.42) = 241.84 and nqˆ = 502(.58) = 291.16 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. d. For confidence coefficient .98, α = .02 and α/2 = .02/2 = .01. From Table IV, Appendix B, z.01 = 2.33. The confidence interval is: pˆ z.01 ˆˆ pq .42(.58) .42 2.33 .42 .051 (.369, .471) n 502 We are 98% confident that the true proportion of all senior human resource executives at U.S. companies who believe that their hiring managers are interviewing too many people to find qualified candidates for the job is between .369 and .471. e. 5.49 A 90% confidence interval would be narrower. If the interval was narrower, it would contain fewer values, thus, we would be less confident. Of the 2,778 sampled firms, 748 announced one or more acquisitions during the year 2000. Thus, pˆ x 748 .269 n 2, 778 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 2,778(.269) = 747 and nqˆ = 2,778(.731) = 2031 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .90, α = 1 - .90 = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The 90% confidence interval is: pˆ zα / 2 pq pˆ zα / 2 n ˆˆ pq .269(.731) .269 1.645 .269 .014 (.255, .283) n 2, 778 We are 90% confident that the true proportion of all firms that announced one or more acquisitions during the year 2000 is between .255 and .283. Changing these to percentages, the results would be 25.5% and 28.3%. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 288 5.50 Chapter 5 a. x 16 .052 . n 308 The sample size is large enough if both npˆ 15 and nqˆ 15 . The point estimate of p is pˆ npˆ = 308(.052) = 16 and nqˆ = 502(.948) = 476 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: pˆ z.005 b. ˆˆ .052(.948) pq .052 2.58 .052 .033 (.019, .085) 308 n We are 99% confident that the true proportion of diamonds for sale that are classified as “D” color is between .019 and .085. x 81 .263 . The point estimate of p is pˆ n 308 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 308(.263) = 81 and nqˆ = 308(.737) = 227 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: pˆ z.005 ˆˆ .263(.737) pq .263 2.58 .263 .065 (.198, .328) 308 n We are 99% confident that the true proportion of diamonds for sale that are classified as “VS1” clarity, is between .198 and .328. 5.51 x 52 .867 . n 60 a. The point estimate of p is pˆ b. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ z.025 c. ˆˆ pq .867(.133) .867 1.96 .867 .085 (.781, .953) n 60 We are 95% confident that the true proportion of Wal-Mart stores in California that have more than 2 inaccurately priced items per 100 scanned is between .781 and .953. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals d. e. 289 If 99% of the California Wal-Mart stores are in compliance, then only 1% or .01 would not be. However, we found the 95% confidence interval for the proportion that are not in compliance is between .781 and .953. The value of .01 is not in this interval. Thus, it is not a likely value. This claim is not believable. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 60(.867) = 52 and nqˆ = 60(.133) = 8 Since nqˆ is less than 15, the sample size is not large enough to conclude the normal approximation is reasonable. Thus, the confidence interval constructed in part b may not be valid. Any inference based on this interval is questionable. 5.52 Using Wilson’s adjustment, the point estimate of the true proportion of all working nannies who are men is x2 24 2 26 p .0062 n 4 4,176 4 4,180 For confidence coefficient .95, α = 1 - .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Wilson’s adjusted 95% confidence interval is: p zα / 2 pq .0062(.9938) .0062 1.96 .0062 .0024 (.0038, .0086) n 4,176 4 We are 95% confident that the true proportion of all working nannies who are men is between .0038 and .0086. 5.53 a. The parameter of interest is p, the proportion of all fillets that are red snapper. b. The estimate of p is pˆ x 22 17 .23 n 22 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 22(.23) = 5 and nqˆ = 22(.77) = 17 Since npˆ is less than 15, the sample size is not large enough to conclude the normal approximation is reasonable. c. We will use Wilson’s adjustment to form the confidence interval. Using Wilson’s adjustment, the point estimate of the true proportion of all fillets that are not red snapper is p x2 5 2 7 .27 n 4 22 4 26 For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Wilson’s adjusted 95% confidence interval is: p zα / 2 .27(.73) pq .27 1.96 .27 .170 (.10, .44) 22 4 n Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 290 Chapter 5 d. 5.54 We are 95% confident that the true proportion of all fillets that are red snapper is between .10 and .44. The point estimate of p is pˆ x 36 .434 . n 83 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 83(.434) = 36 and nqˆ = 83(.566) = 47 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ z.025 ˆˆ pq .434(.566) .434 1.96 .434 .107 (.327, .541) n 83 We are 95% confident that the true proportion of healthcare workers with latex allergies actually suspects the he or she actually has the allergy is between .327 and .541. 5.55 We will use a 99% confidence interval to estimate the true proportion of mailed items that are delivered on time. First, we must compute pˆ : pˆ = x 282, 200 = = .85 n 332, 000 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 332,000(.85) = 282,200 and nqˆ = 332,000(.15) = 49,800 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: pˆ ± z.005 pq ≈ pˆ ± 2.58 n ˆˆ .85(.15) pq .85 ± 2.58 .85 ± .002 (.848, .852) 332,000 n We are 99% confident that the true percentage of items delivered on time by the U.S. Postal Service is between 84.8% and 85.2%. 5.56 To compute the necessary sample size, use n= zα / 2 2 σ 2 SE 2 where α = 1 − .95 = .05 and α/2 = .05/2 = .025. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals From Table IV, Appendix B, z.025 = 1.96. Thus, n= (1.96) 2 (7.2) = 307.328 ≈ 308 .32 You would need to take 308 samples. 5.57 a. An estimate of σ is obtained from: range ≈ 4s range 34 30 =1 s≈ 4 4 To compute the necessary sample size, use n= zα / 2 2σ 2 ( SE ) 2 where α = 1 − .90 = .10 and α/2 = .05. From Table IV, Appendix B, z.05 = 1.645. Thus, n= b. (1.645) 2(1) 2 = 67.65 ≈ 68 .2 2 A less conservative estimate of σ is obtained from: range ≈ 6s range 34 30 = .6667 s≈ 6 6 Thus, n = 5.58 a. zα / 2 2σ 2 ( SE ) 2 = (1.645) 2(.6667) 2 = 30.07 ≈ 31 .2 2 To compute the needed sample size, use: n= z α / 22 pq SE 2 Thus, n = where z.025 = 1.96 from Table IV, Appendix B. (1.96) 2 (.2)(.8) = 96.04 ≈ 97 .08 2 You would need to take a sample of size 97. b. To compute the needed sample size, use: n= z α / 22 pq SE 2 = (1.96) 2(.5)(.5) = 150.0625 ≈ 151 .08 2 You would need to take a sample of size 151. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 291 292 5.59 Chapter 5 For confidence coefficient .90, α = .10 and α/2 = .05. From Table IV, Appendix B, z.05 = 1.645. We know pˆ is in the middle of the interval, so pˆ The confidence interval is pˆ z.05 We know .4 1.645 .4 − .8059 .4(.6) = .26 n .8059 n n = 5.756² = 33.1 ≈ 34 5.60 a. ˆˆ pq .4(.6) .4 1.645 n n = .26 n .4 − .26 = .54 .26 = .4 2 n .8059 = 5.756 .14 For a width of 5 units, SE = 5/2 = 2.5. To compute the needed sample size, use n= z α / 22σ 2 SE 2 where α = 1 − .95 = .05 and α/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Thus, n= (1.96)2 (14)2 2.52 = 120.47 ≈ 121 You would need to take 121 samples at a cost of 121($10) = $1210. Yes, you do have sufficient funds. b. For confidence coefficient .90, α = 1 − .90 = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. n= (1.645) 2 (14)2 2.52 = 84.86 ≈ 85 You would need to take 85 samples at a cost of 85($10) = $850. You still have sufficient funds but have an increased risk of error. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.61 a. The width of a confidence interval is 2(SE) = 2zα/2 293 σ n For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. For n = 16, W = 2zα/2 σ = 2(1.96) 1 16 n = 0.98 For n = 25, W = 2zα/2 σ = 2(1.96) 1 25 n = 0.784 For n = 49, W = 2zα/2 σ = 2(1.96) 1 49 n = 0.56 For n = 100, W = 2zα/2 σ = 2(1.96) n 1 100 = 0.392 For n = 400, W = 2zα/2 σ = 2(1.96) n 1 400 = 0.196 b. 5.62 The sample size will be larger than necessary for any p other than .5. 5.63 For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. Thus, n ( zα / 2 )2 σ 2 ( SE ) 2 1.6452 (10.9) 2 42 20.09 21 Thus, we would need a sample of size 21. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 294 5.64 Chapter 5 To compute the necessary sample size, use: n zα / 2 2 σ 2 SE 2 where α = 1 - .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Thus, n 5.65 1.52 245.86 246 For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. Since we have no estimate given for the value of p, we will use .5. The sample size is: n= 5.66 1.962 122 zα2 / 2 pq ( SE ) 2 1.6452 (.5)(.5) .022 = 1,691.3 ≈ 1,692 a. The confidence level desired by the researchers is 90%. b. The sampling error desired by the researchers is SE = .05. c. For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, x 64 z.05 = 1.645. From the problem, we will use pˆ .604 to estimate p. n 106 Thus, n ( zα / 2 )2 pq ( SE )2 1.6452.604(.396) .052 258.9 259 Thus, we would need a sample of size 259. 5.67 For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.575. From the previous estimate, we will use pˆ = 1/3 to estimate p. n 5.68 zα2 / 2 pq ( SE )2 2.5752 (1 / 3)(2 / 3) .012 14, 734.7 14, 735 To compute the necessary sample size, use: n zα / 2 2 pq SE 2 where α = 1 - .90 = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. We will use pˆ = .867 from Exercise 5.51 to estimate p. Thus, n 1.962 .867(.133) .052 177.2 178 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.69 295 To compute the needed sample size, use n= zα / 2 2σ 2 ( SE ) 2 where α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Thus, for s = 10, n = For s = 20, n = For s = 30, n = 5.70 (1.96) 2(10) 2 32 (1.96) 2(20) 2 32 (1.96) 2(30) 2 32 = 42.68 ≈ 43 = 170.74 ≈ 171 = 384.16 ≈ 385 To compute the necessary sample size, use n= ( zα / 2 ) 2 σ 2 SE 2 where α = 1 − .90 = .10 and α/2 = .05. From Table IV, Appendix B, z.05 = 1.645. Thus, n= 5.71 (1.645) 2 (10)2 12 = 270.6 ≈ 271 The bound is SE = .05. For confidence coefficient .99, α = 1 − .99 = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.575. We estimate p with = 11/27 = .407. Thus, n= zα / 2 2 pq ( SE )2 2.5752 (.407)(.593) .052 ≈ 640.1 641 The necessary sample size would be 641. The sample was not large enough. 5.72 a. To compute the needed sample size, use n= ( zα / 2 ) 2 σ 2 SE 2 where α = 1 − .90 = .10 and α/2 = .05. From Table IV, Appendix B, z.10 = 1.645. Thus, n= b. (1.645)2 (2) 2 .12 = 1,082.41 ≈ 1,083 As the sample size decreases, the width of the confidence interval increases. Therefore, if we sample 100 parts instead of 1,083, the confidence interval would be wider. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 296 Chapter 5 c. To compute the maximum confidence level that could be attained meeting the management's specifications, n= ( zα / 2 ) 2 σ 2 SE 2 100 = ( zα / 2 )(2)2 2 .1 ( zα / 2 )2 100(.01) = .25 zα/2 = .5 4 Using Table IV, Appendix B, P(0 ≤ z ≤ .5) = .1915. Thus, α/2 = .5000 − .1915 = .3085, α = 2(.3085) = .617, and 1 − α = 1 − .617 = .383. The maximum confidence level would be 38.3%. 5.73 a. Percentage sampled = n 1000 (100%) (100%) = 40% N 2500 Finite population correction factor: N n 2500 1000 .6 = .7746 N 2500 b. Percentage sampled = n 1000 (100%) (100%) = 20% N 5000 Finite population correction factor: N n 5000 1000 .8 = .8944 N 5000 c. Percentage sampled = n 1000 (100%) (100%) = 10% N 10, 000 Finite population correction factor: N n 10, 000 1000 .9 = .9487 N 10, 000 d. Percentage sampled = n 1000 (100%) (100%) = 1% N 100, 000 Finite population correction factor: N n 100, 000 1000 .99 = .995 N 100, 000 5.74 σx = σ n a. σx b. σx N n N 200 1000 200 1000 2500 1000 = 4.90 2500 5000 1000 = 5.66 5000 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.75 5.76 200 10, 000 1000 = 6.00 10, 000 c. σx = d. σx a. σˆ x b. σˆ x c. σˆ x d. As n increases, σ x decreases. e. We are computing the standard error of x . If the entire population is sampled, then x = μ. There is no sampling error, so σ x = 0. a. For n = 36, with the finite population correction factor: N n 24 5000 64 σˆ x s / n 3 .9872 = 2.9807 N 5000 64 1000 200 1000 100, 000 1000 = 6.293 100, 000 N n N s n 50 4000 50 10,000 50 2000 10, 000 2000 = 1.00 10, 000 10, 000 4000 = .6124 10, 000 10, 000 10, 000 =0 10, 000 without the finite population correction factor: 24 σˆ x s / n =3 64 σˆ x without the finite population correction factor is slightly larger. b. For n = 400, with the finite population correction factor: N n 24 5000 400 σˆ x s / n 1.2 .92 = 1.1510 5000 N 400 without the finite population correction factor: 24 σˆ x s / n = 1.2 400 c. In part a, n is smaller relative to N than in part b. Therefore, the finite population correction factor did not make as much difference in the answer in part a as in part b. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 297 298 5.77 Chapter 5 The approximate 95% confidence interval for p is pˆ ± 2σˆ pˆ pˆ ± 2 .42 ± 2 5.78 5.79 pˆ (1 pˆ ) n N n N .42(.58) 6000 1600 .42 ± .011 (.409, .431) 1600 6000 An approximate 95% confidence interval for μ is: s N n 14 x ± 2σˆ x x ± 2 422 ± 2 N 40 n a. x= 2 s = 375 40 422 ± 4.184 (417.816, 426.184) 375 x 1081 = 36.03 30 n x 2 ( x) − 2 n = 41, 747 − n −1 μˆ = x = 36.03 x ± 2σˆ x s x ±2 n 36.03 ± 2 2 1081 = 96.3782 30 N n N 96.3782 30 300 30 36.03 ± 3.40 300 (32.63, 39.43) b. pˆ = x 21 = .7 n 30 pˆ (1 pˆ ) n x ± 2σˆ pˆ pˆ ± 2 .7(.3) 300 30 .7 ± .159 (.541, .859) 30 300 .7 ± 2 5.80 a. N n N For N = 2,193, n = 223, x =116,754, and s = 39,185, the 95% confidence interval is: N n 39,185 2,193 223 116, 754 2 N 2,193 n 223 116, 754 4,974.06 (111, 779.94, 121, 728.06) x 2σˆ x x 2 b. s We are 95% confident that the mean salary of all vice presidents who subscribe to Quality Progress is between $111,777.94 and $121,728.06. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.81 a. First, we must estimate p: pˆ 299 x 759 .560 n 1,355 For confidence coefficient .95, α = 1 - .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Since n/N = 1,355/1,696 = .799 > .05, we must use the finite population correction factor. The 95% confidence interval is: pˆ zα / 2 ˆˆ N n pq .560(.440) 1, 696 1,355 .560 1.96 n N 1,355 1, 696 .560 .012 (.548, .572) 5.82 b. We used the finite correction factor because the sample size was very large compared to the population size. c. We are 95% confident that the true proportion of active NFL players who select a professional coach as the most influential in their career is between .548 and .572. a. The population of interest is the set of all households headed by women that have incomes of $25,000 or more in the database. b. Yes. Since n/N = 1,333/25,000 = .053 exceeds .05, we need to apply the finite population correction. c. The standard error for pˆ should be: pˆ (1 pˆ ) N n .708(1 .708) 25, 000 1,333 = .012 N n 1333 25, 000 σˆ pˆ d. For confidence coefficient .90, α = 1 − .90 = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The approximate 90% confidence interval is: pˆ ± 1.645σˆ pˆ .708 ± 1.645(.012) (.688, .728) 5.83 a. First, we must calculate the sample mean: 15 fx i i x i 1 n 3(108) 2(55) 1(500) 19(100) 15,646 156.46 100 100 The point estimate of the mean value of the parts inventory is x = 156.46. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 300 Chapter 5 b. The sample variance and standard deviation are: s2 f x 2 15 f i xi2 i 1 i i n 3(108)2 2(55) 2 19(100)2 n 1 15, 6462 6, 776,336 100 43, 720.83677 99 100 1 15, 6462 100 s s 2 43,720.83677 209.10 The estimated standard error is: σˆ x = c. s n N n 209.10 500 100 = = 18.7025 N 500 100 The approximate 95% confidence interval is: s N n x ± 2σˆ x x ± 2 156.46 ± 2(18.7025) 156.46 ± 37.405 N n (119.055, 193.865) We are 95% confident that the mean value of the parts inventory is between $119.06 and $193.87. d. 5.84 Since the interval in part c does not include $300, the value of $300 is not a reasonable value for the mean value of the parts inventory. For N = 1,500, n = 35, x = 1, and s = 124, the 95% confidence interval is: s N n 124 1,500 35 x ± 2σˆ x x ± 2 1 ± 2 1 ± 41.43 N 1,500 n 35 (−40.43, 42.43) We are 95% confident that the mean error of the new system is between -$40.43 and $42.43. 5.85 pˆ x 15 .086 n 175 The standard error of pˆ is: σˆ pˆ = pˆ (1 pˆ ) N n = n N .086(1 .086) 3000 175 = .0206 175 3000 An approximate 95% confidence interval for p is: pˆ ± 2 σˆ pˆ .086 ± 2(.0206) .086 ± .041 (.045, .127) Since .07 falls in the 95% confidence interval, it is not an uncommon value. Thus, there is no evidence that more than 7% of the corn-related products in this state have to be removed from shelves and warehouses. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.86 5.87 5.88 301 a. For a small sample from a normal distribution with unknown standard deviation, we use the t statistic. For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table V, Appendix B, with df = n − 1 = 23 − 1 = 22, t.025 = 2.074. b. For a large sample from a distribution with an unknown standard deviation, we can estimate the population standard deviation with s and use the z statistic. For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. c. For a small sample from a normal distribution with known standard deviation, we use the z statistic. For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. d. For a large sample from a distribution about which nothing is known, we can estimate the population standard deviation with s and use the z statistic. For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. e. For a small sample from a distribution about which nothing is known, we can use neither z nor t. a. P(t ≤ t0) = .05 where df = 20 t0 = −1.725 b. P(t ≥ t0) = .005 where df = 9 t0 = 3.250 c. P(t ≤ −t0 or t ≥ t0) = .10 where df = 8 is equivalent to P(t ≥ t0) = .10/2 = .05 where df = 8 t0 = 1.860 d. P(t ≤ −t0 or t ≥ t0) = .01 where df = 17 is equivalent to P(t ≥ t0) = .01/2 = .005 where df = 17 t0 = 2.898 a. Of the 400 observations, 227 had the characteristic pˆ = 227/400 = .5675. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 400(.5675) = 227 and nqˆ = 400(.4325) = 173 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ ± z.025 pq ± 1.96 n ˆˆ .5675(.4325) pq .5675 ± 1.96 .5675 ± .0486 400 n (.5189, .6161) Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 302 Chapter 5 b. For this problem, SE = .02. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Thus, n= ( zα / 2 ) 2 pq SE 2 (1.96)2 (.5675)(.4325) .022 = 2,357.2 ≈ 2,358 Thus, the sample size was 2,358. 5.89 a. For confidence coefficient .99, α = .01 and α/2 = .005. From Table IV, Appendix B, z.005 = 2.575. The confidence interval is: x ± zα/2 5.90 s n 30 225 zα / 2 σ 2 32.5 ± 5.15 (27.35, 37.65) 2.5752 (30) 2 = 27,870.25 ≈ 23,871. .5 2 b. The sample size is n = c. "99% confidence" means that if repeated samples of size 225 were selected from the population and 99% confidence intervals constructed for the population mean, then 99% of all the intervals constructed will contain the population mean. a. The finite population correction factor is: ( N n) = N b. c. ( SE ) 2 (2, 000 50) = .9874 2, 000 The finite population correction factor is: ( N n) = N (100 20) = .8944 100 The finite population correction factor is: ( N n) = N 5.91 32.5 ± 2.575 (1,500 300) = .8944 1,500 The parameters of interest for the problems are: (1) (2) (3) (4) The question requires a categorical response. One parameter of interest might be the proportion, p, of all Americans over 18 years of age who think their health is generally very good or excellent. A parameter of interest might be the mean number of days, μ, in the previous 30 days that all Americans over 18 years of age felt that their physical health was not good because of injury or illness. A parameter of interest might be the mean number of days, μ, in the previous 30 days that all Americans over 18 years of age felt that their mental health was not good because of stress, depression, or problems with emotions. A parameter of interest might be the mean number of days, μ, in the previous 30 days that all Americans over 18 years of age felt that their physical or mental health prevented them from performing their usual activities. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.92 a. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The 99% confidence interval is: x zα / 2 5.93 303 17.77 141.31 2.58 141.31 .145 (141.165, 141.455) n 100, 000 s b. We are 99% confident that the mean number of semester hours taken by all first-time candidates for the CPA exam is between 141.165 and 145.455. c. Since the sample size was so large, no conditions must hold for the interpretation in part b to be valid. a. Of the 1000 observations, 29% said they would never give personal information to a company pˆ = .29 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 1000(.29) = 290 and nqˆ = 1000(.71) = 710 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. b. For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The 95% confidence interval is: pˆ ± z.025 ˆˆ pq .29 ± 1.96 n .29(.71) .29 ± .028 (.262, .318) 1000 We are 95% confident that the proportion of Internet users who would never give personal information to a company is between .262 and .318. 5.94 x 67 .638 . n 105 a. The point estimate of p is pˆ b. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 105(.638) = 67 and nqˆ = 105(.362) = 38 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ z.025 c. ˆˆ pq .638(.362) .638 1.96 .638 .092 (.546, .730) n 105 We are 95% confident that the true proportion of on-the-job homicide cases that occurred at night is between .546 and .730. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 304 5.95 Chapter 5 For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. For this study, n= ( zα / 2 ) 2 σ 2 SE 2 1.962 (5) 2 12 = 96.04 ≈ 97 The sample size needed is 97. 5.96 a. From the printout, the 90% confidence interval for the mean lead level is (0.61, 5.16). b. From the printout, the 90% confidence interval for the mean copper level is (0.2637, 0.5529). c. We are 95% confident that the mean lead level in water specimens from Crystal Lakes Manors is between .61 and 5.16. We are 95% confident that the mean copper level in water specimens from Crystal Lakes Manors is between .2637 and .5529. d. 5.97 99% confidence means that if repeated samples of size n are selected and 99% confidence intervals formed, 99% of all confidence intervals will contain the true mean. First, we must estimate p: pˆ pˆ 2 x 50 .694 . The 95% confidence interval is: n 72 ˆˆ N n pq .694(.306) 251 72 .694 2 .694 .092 (.602, .786) n N 72 251 We are 95% confident that the proportion of all New Jersey Governor’s Council business members that have employees with substance abuse problems is between .602 and .786. 5.98 a. For confidence coefficient .90, α = .10 and α/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The 90% confidence interval is: x ± z.05 σ x ± 1.645 s 12.2 ± 1.645 10 n n 100 12.2 ± 1.645 (10.555, 13.845) We are 90% confident that the mean number of days of sick leave taken by all its employees is between 10.555 and 13.845. b. For confidence coefficient .99, α = .01 and α/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The sample size is n = zα / 2 2 σ 2 SE 2 = (2.58) 2 (10) 2 22 = 166.4 ≈ 167 You would need to take n = 167 samples. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.99 305 There are a total of 96 channel catfish in the sample. The point estimate of p is x 96 pˆ .667 . n 144 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 144(.667) = 96 and nqˆ = 144(.333) = 48 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: pˆ z.05 ˆˆ pq .667(.333) .667 1.645 .667 .065 (.602, .732) n 144 We are 90% confident that the true proportion of channel catfish in the population is between .602 and .732. 5.100 a. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: x zα / 2 s n 1.13 2.58 2.21 72 1.13 .67 (.46, 1.80) We are 99% confident that the mean number of pecks at the blue string is between .46 and 1.80. 5.101 b. Yes. The mean number of pecks at the white string is 7.5. This value does not fall in the 99% confident interval for the blue string found in part a. Thus, the chickens are more apt to peck at white string. a. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table V, Appendix B, with df = n − 1 = 3 − 1 = 2, t.005 = 9.925. The confidence interval is: x ± t.005 s n 49.3 ± 9.925 1.5 3 49.3 ± 8.60 (40.70, 57.90) b. We are 99% confident that the mean percentage of B(a)p removed from all soil specimens using the poison is between 40.70% and 57.90%. c. We must assume that the distribution of the percentages of B(a)p removed from all soil specimens using the poison is normal. d. Since the 99% confidence interval for the mean percent removed contains 50%, this would be a very possible value. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 306 5.102 Chapter 5 For confidence coefficient .90, α = .10 and α/2 = .05. From Table IV, Appendix B, z.05 = 1.645. For a width of .06, SE = .06/2 = .03 The sample size is n = ( zα / 2 ) 2 pq = SE 2 (.1645)2 (.17)(.83) .032 = 424.2 ≈ 425 You would need to take n = 425 samples. 5.103 a. Using MINITAB, the descriptive statistics are: Descriptive Statistics: IQ25, IQ60 Variable IQ25 IQ60 N 36 36 Mean 66.83 45.39 Median 66.50 45.00 TrMean 66.69 45.22 Variable IQ25 IQ60 Minimum 41.00 22.00 Maximum 94.00 73.00 Q1 54.25 36.25 Q3 80.00 58.00 StDev 14.36 12.67 SE Mean 2.39 2.11 For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.58. The confidence interval is: x zα / 2 s n 66.83 2.58 14.36 36 66.83 6.17 (60.66, 73.00) We are 99% confident that the mean raw IQ score for all 25 year olds is between 60.66 and 73.00. b. We must assume that the sample is random and that the observations are independent. c. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x zα / 2 s n 45.39 1.96 12.67 36 45.39 4.14 (41.25, 49.53) We are 95% confident that the mean raw IQ score for all 60 year olds is between 41.25 and 49.53. 5.104 pˆ = x/n = 35/55 = .636. a. The point estimate of p is b. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 55(.636) = 35 and nqˆ = 55(.364) = 20 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. For confidence coefficient, .99, α = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.575. The confidence interval is: pˆ ± z.005 .636(.364) ˆˆ pq .636 ± 2.575 .636 ± .167 (.469, .803) 55 n Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.105 307 c. We are 99% confident that the true proportion of fatal accidents involving children is between .469 and .803. d. The sample proportion of children killed by air bags who were not wearing seat belts or were improperly restrained is 24/35 = .686. This is rather large proportion. Whether a child is killed by an airbag could be related to whether or not he/she was properly restrained. Thus, the number of children killed by air bags could possibly be reduced if the child were properly restrained. The bound is SE = .1. For confidence coefficient .99, α = 1 − .99 = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.575. We estimate p with from Exercise 5.104 which is pˆ = .636. Thus, n= zα / 2 2 pq ( SE ) 2 2.5752 (.636)(.364) .12 1 = 153.5 154 The necessary sample size would be 154. 5.106 a. Of the 100 cancer patients, 7 were fired or laid off pˆ = 7/100 = .07. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 100(.07) = 7 and nqˆ = 100(.93) = 93 Since npˆ is less than 15, the sample size is not sufficiently large to conclude the normal approximation is reasonable. We will iuse Wilson’s adjustment to form the confidence interval. p x2 72 9 .087 n 4 100 4 104 For confidence coefficient .90, α = .10 and α/2 = .10/2 = .05. From Table IV, Appendix B, z.05 = 1.645. The confidence interval is: p ± z.05 pq p ± 1.645 n pq .087(.913) .087 ± 1.645 .087 ± .045 (.042, .132) n 100 4 Converting these to percentages, we get (4.2%, 13.2%). 5.107 b. We are 90% confident that the percentage of all cancer patients who are fired or laid off due to their illness is between 4.2% and 13.2%. c. Since the rate of being fired or laid off for all Americans is 1.3% and this value falls outside the confidence interval in part b, there is evidence to indicate that employees with cancer are fired or laid off at a rate that is greater than that of all Americans. a. Of the 24 observations, 20 were 2 weeks of vacation pˆ = 20/24 = .833. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: pˆ z.025 ˆˆ pq .833(.167) .833 1.96 .833 .149 (.683, .982) n 24 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 308 Chapter 5 b. The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 24(.833) = 20 and nqˆ = 24(.167) = 4 Since nqˆ is less than 15, the sample size is not sufficiently large to conclude the normal approximation is reasonable. The validity of the confidence interval is in question. c. The bound is SE = .02. For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. Thus, n= zα / 2 2 pq ( SE )2 = 1.962 (.833)(.167) .022 = 1,336.02 ≈ 1,337. Thus, we would need a sample size of 1,337. 5.108 Using MINITAB, the descriptive statistics are: Descriptive Statistics: r Variable r N 34 Mean 0.4224 Median 0.4300 TrMean 0.4310 Variable r Minimum -0.0800 Maximum 0.7400 Q1 0.2925 Q3 0.6000 StDev 0.1998 SE Mean 0.0343 For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: x zα / 2 s n .4224 1.96 .1998 34 .4224 .0672 (.3552, .4895) We are 95% confident that the mean value of r is between .3552 and .4895. 5.109 Sampling error has to do with chance. In a population, there is variation – not all observations are the same. The sampling error has to do with the variation within a sample. By chance, one might get a sample that overestimates the mean just because all the observations in the sample happen to be high. Nonsampling error has to do with errors that have nothing to do with the sampling. These errors could be due to misunderstanding the question being asked, asking a question that the respondent does not know how to answer, etc. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Inferences Based on a Single Sample: Estimation with Confidence Intervals 5.110 a. The point estimate for the fraction of the entire market who refuse to purchase bars is: pˆ b. 309 x 23 .094 n 244 The sample size is large enough if both npˆ 15 and nqˆ 15 . npˆ = 244(.094) = 22.9 and nqˆ = 244(.906) = 221.1 Since both numbers are greater than or equal to 15, the sample size is sufficiently large to conclude the normal approximation is reasonable. c. For confidence coefficient .95, α = 1 − .95 = .05 and α/2 = .05/2 = .025. From Table IV, Appendix B, z.025 = 1.96. The confidence interval is: ˆˆ pq (.094)(.906) .094 1.96 .094 .037 (.057, .131) n 244 pˆ z.025 d. 5.111 The best estimate of the true fraction of the entire market who refuse to purchase bars six months after the poisoning is .094. We are 95% confident the true fraction of the entire market who refuse to purchase bars six months after the poisoning is between .057 and .131. For confidence coefficient .95, α = .05 and α/2 = .025. From Table IV, Appendix B, z.025 = 1.96. From Exercise 5.110, a good approximation for p is .094. Also, SE = .02. The sample size is n = zα / 2 2 pq ( SE ) 2 = (1.96) 2(.094)(.906) .022 = 817.9 ≈ 818 You would need to take n = 818 samples. 5.112 The bound is SE = .1. For confidence coefficient .99, α = 1 − .99 = .01 and α/2 = .01/2 = .005. From Table IV, Appendix B, z.005 = 2.575. We estimate p with from Exercise 7.48 which is = .636. Thus, n= ( zα / 2 ) 2 pq SE 2 2.5752 (.636)(.364) = 153.5 154 .12 The necessary sample size would be 154. 5.113 Solution will vary. See page 1037 for Guided Solutions. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 310 5.114 Chapter 5 Since the manufacturer wants to be reasonably certain the process is really out of control before shutting down the process, we would want to use a high level of confidence for our inference. We will form a 99% confidence interval for the mean breaking strength. For confidence coefficient .99, α = .01 and α/2 = .01/2 = .005. From Table V, Appendix B, with df = n – 1 = 9 – 1 = 8, t.005 = 3.355. The 99% confidence interval is: x t.005 s n 985.6 3.355 22.9 9 985.6 25.61 (959.99, 1,011.21) We are 99% confident that the true mean breaking strength is between 959.99 and 1,011.21. Since 1,000 is contained in this interval, it is not an unusual value for the true mean breaking strength. Thus, we would recommend that the process is not out of control. 5.115 a. As long as the sample is random (and thus representative), a reliable estimate of the mean weight of all the scallops can be obtained. b. The government is using only the sample mean to make a decision. Rather than using a point estimate, they should probably use a confidence interval to estimate the true mean weight of the scallops so they can include a measure of reliability. c. We will form a 95% confidence interval for the mean weight of the scallops. Using MINITAB, the descriptive statistics are: Descriptive Statistics: Weight Variable Weight N 18 N* 0 Variable Weight Maximum 1.1400 Mean 0.9317 SE Mean 0.0178 StDev 0.0753 Minimum 0.8400 Q1 0.8800 Median 0.9100 Q3 9800 For confidence coefficient .95, α = .05 and α/2 = .05/2 = .025. From Table V, Appendix A, with df = n – 1 = 18 – 1 = 17, t.025 = 2.110. The 95% confidence interval is: x t.025 s n .932 2.110 .0753 18 .932 .037 (.895, .969) We are 95% confident that the true mean weight of the scallops is between .8943 and .9691. Recall that the weights have been scaled so that a mean weight of 1 corresponds to 1/36 of a pound. Since the above confidence interval does not include 1, we have sufficient evidence to indicate that the minimum weight restriction was violated. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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