Homework Answers for Chapter 6 Exercises 21-30
21. a. 46.41%, found by (20.27 – 20.00)/0.15 = 1.8
b. 3.59%, found by 0.5000 – 0.4641
c. 81.85%, found by 0.3413 + 0.4772
d. 27.43%, found by 0.5000 – 0.2257
22. a. About 34.13%, found by (34 – 32)/2 = 1.00, the area for 1.00 is 0.3413
b. About 4.95%, found by (28.7 – 32)2 = – 1.65. The area for – 1.65 is 0.4505, then 0.5000 – 0.4505 = 0.0495
c. About 77.45%, found by (29 – 32)/2 = – 1.5. The area for – 1.5 is 0.4332. Then 0.4332 + 0.3413(from part a) = 0.7745
d. About 35.3 hours, found by 1.65 = (X – 32)/2
23. a. – 0.4 for net sales, found by (170 – 180)/25 and 2.92 for employees, found by (1850 – 1500)/120
b. Net sales are 0.4 standard deviations below the mean. Employees is 2.92 standard deviations above the mean. Draw a normal curve and locate the approximate position of these values.
c. 65.54% of the aluminum fabricators have greater net sales compared with Clarion, found by 0.1554 + 0.5000. Only 0.18% have more employees than Clarion, found by 0.5000 – 0.4982
24. a. 15.87%, found by (15 – 20)/5 = – 1.0. The area for –1.0 is 0.3413. Then 0.5000 – 0.3413 = 0.1587
b. 0.5403, first, the area between 18 and 20 is 0.1554. The area between 20 and 26 is 0.3849. 0.1554 + 0.3849 = 0.5403
c. About 1 person, found by z = (7 – 20)/5 = – 2.6, for which the area is 0.4953. Then 0.5000 – 0.4953 = 0.0047. Finally, 200(0.0047) = 0.94, which is about one customer.
25. 60.06%, found by (42,000 – 40,000)/5000 = 0.40. The area under the curve for 0.40 is 0.1554. similarly, the area between 32,000 and 40,000 is 0.4452. 0.1554 + 0.4452 = 0.6006
26. a. (860 – 1000)/50 = – 2.8. The area below 860 is 0.0026, found by 0.5000 – 0.4974, or 0.26 percent
b. (1055 – 1000)/50 = 1.1. The area between 1000 and 1055 is 0.3643. The area between 1000 and 1100 is 0.4772. Subtracting: 0.4772 – 0.3643 = 0.1129 or 11.29%.
27. About 4099 units found by solving for X. 1.65 = (X – 4000)/60
28. a. About 0.47% (65,200 – 60,000)/2000 = 2.6. Then 0.5000 – 0.4953 = 0.0047
b. About 22 trucks, (55,000 – 60,000)/2000 = – 2.5. Then 0.5000 – 0.4938 = 0.0062. Multiplying, 0.0062 x 3500 = 21.7
c. About 2945, (62,000 – 60,000)/2000 = 1.00. then 0.5000 + 0.3413 = 0.8413. Multiplying 0.8413 x 3500 = 2944.55
29. a. Only 2.28% earn more than John: (30,400 – 28,000)/1200 = 2.00. Then 0.5000 – 0.4772 = 0.0228
b. Of the other supervisors, 97.72% have more service. (10 – 20)/5 = – 2.00. Then 0.4772 + 0.5000 = 0.9772
30. a. 26.43%, found by (30 – 35)/8 = – 0.63, Then 0.500 – 0.2357 = 0.2643
b. 26.43%, found by (40 – 35)/8 = 0.63. Then 0.5000 – 0.2357 = 0.2643
c. The normal distribution is continuous. Thus, the probability of an exact value is very small.
d. About 4.26%. You could find the probability of 39.5 and 40.5
z = (39.5 – 35)/8 = 0.56. Area is 0.2123
z = (40.5 – 35)/8 = 0.69.
Area is 0.2549
Subtracting: 0.2549 – 0.2123 =
0.0426
e. 45.24 minutes or longer, found by solving for X. 1.28 = (X – 35)/8