1 Normal Distributions Suppose that the fill-volume of soda bottles is Normally distributed with a mean of 10.0 oz. and a standard deviation of 0.25 oz. 1. What is the probability that the fill-volume is within one standard deviation of its mean? (Use the 68, 95, 99.7 percent rule.) (A) 34 (B) .50 (C) .68 (D) .95 (E) 997 2. What is the probability that the fill-volume is between 10.0 and 10.125 ca.? (A) 34 (B) .50 (C) .68 (D) .95 (E) .19 2 Random Sampling for Measured Characteristics In statistical quality control, samples are selected from production and various quality charac- teristies are measured in order to check that the process is "in control". Suppose that a bottling process is intended to fill bottles with, on average, 21 fluid ounces (fl. oz.) of beverage. Varia- tion around the mean follows a Normal distribution with a standard deviation of 0.5 fluid ounces. 3. If a technician samples n = 25 bottles (when the process is in control) and measures the volume of beverage in each, what is the standard deviation of the mean? (A) 0.5 (B) 0.02 (C) 0.1 (D) 1.0 (E) 5.0 f. ca. 4. What is the probability that the sample mean fill-level for the 25 bottles will exceed 21.2 il. oz. ? (A) 00135 (B) .0228 (C) 025 (D) .05 (E) .10 5. Find a such that the probability that the sample mean exceeds c when the process is in control is .05. (A) 21.0 (B) 21.1 (C) 21.1645 (D) 21.196 (E) 21.200 A. oz. 3 Exceeding a Specification Limit Suppose that the specification limits for diameters of bolts are 82 to 118 mm. Suppose that the diameters of individual bolts follow a Normal distribution with a mean of 100 mm. and a standard deviation of 6 mm. 6. What is the probabilty that the diameter of a bolt is within specs? (A) 34 (B) 50 (C) .68 (D) .95 (E) .997 7. Suppose that the manufacturer manages to improve the process so that the standard deviation is reduced to only 3 mm. But suppose that the mean of the process has slipped up to 109 mm. What proportion of diameters will now be expected to exceed the upper specification limit ? (A) 0013 (B) .0228 (C) .025 (D) 05 (E) .104 Normal Approximation to the Binomial Statistics released by Plum Computer Company claim that on average, one out of every 10 PCs is a Plum. Suppose that 400 PCs are randomly sampled. 8. The expected number of Plum Computers in the sample is ? (A) 10 (B) 20 (C) 30 (D) 40 (E) 50 9. The standard deviation of the number of Plum Computer is ? (A) 09 (B) 6 (C) 36 (D) 20 (E) 400 10. What is the probability that the number of Plum Computers will be more than 49 (Le.., at least 50)? In using the Normal approximation, use the number 49.5. (A) 1.583 (B) .943 (C) .057 (D) 500 (E) .999 11. What is the probability that the number of Plum Computers will be more than 50 (i.e.., at least 51)? In using the Normal approximation, use the number 50.5. (A) 1.75 (B) .943 (C) .057 (D) .500 (E) .0401 12. Approximate the probability that the number of Plum Computers will be exactly 40. Hint: Compute the area over the interval (39.5, 40.5) under the appropriate Normal curve. (A) (319 (B) .08 (C) .0638 (D) 0 (E) 1 5 Interval Estimation of a Binomial Probability In a random sample of n = 900 registered voters, 475 prefer Hardcastle for Governor. 13. What is the point estimate of the population proportion of registered voters preferring Hardcastle ? (A) .475 (B) .900 (C) .500 (D) .528 (E) .472 14. What is the estimate of the standard deviation of the sampling distribution of the sample proportion (p)? (A) 0.0166 (B) .472 (C) .528 (D) .90 (E) .95 15 The "margin of error" for a 95% confidence interval is 1.96 times this standard deviation. So here the margin of error is ? (A) 0327 (B) .05 (C) 10 (D) .125 (E) .127 16. The 95% confidence interval is the point estimate, plus or minus the margin of error. So what is this interval for the population proportion of individuals preferring Hardcastle? (A) (.495, -561) (B) (.500, .561) (C) (.495, .500) (D) (0, 495) (E) (.561, .999)6 Computation of the standard deviation Treat the nine digits 6, 5, 1, 3, 7, 4, 4, 8, 0 as a sample. 17. Compute the sum. (A) 37 (B) 38 (C) 39 (D) 40 (E) 41 18. Compute the sum of squares. (A) 216 (B) 217 (C) 218 (D) 219 (E) None of these 19. Compute the sum of squared deviations SD = )_, (2 -2) using the computational formula SSD = Sum of squares - Sum . (A) 38 (B) 216 (C) 55.56 (D) 1444 (E) 2444 20. Compute the sample variance. (A) 5.556 (B) 6.95 (C) 264 (D) 8 ], (E) 9