33 Beverage Volume

EXAMPLE 5-33 Beverage Volume

  Soft-drink cans are filled by an automated filling machine. The

  E 1X 2 12.1 and V1X 2

  mean fill volume is 12.1 fluid ounces, and the standard devia-

  tion is 0.1 fluid ounce. Assume that the fill volumes of the cans are independent, normal random variables. What is the proba-

  Consequently,

  bility that the average volume of 10 cans selected from this process is less than 12 fluid ounces?

  X X 12 12.1

  1X 122 P c X 10.001 d

  Let X

  1 ,X 2 , p ,X 10 denote the fill volumes of the 10 cans.

  P

  The average fill volume (denoted as ) is a normal random X

  P 1Z 3.162 0.00079

  variable with

  EXERCISES FOR SECTION 5-4 5-54.

  X and Y are independent, normal random variables with

  (a) If a particular lamp is made up of these two inks only,

  E(X )

  0, V(X ) 4, E(Y ) 10, and V(Y ) 9.

  what is the probability that the total ink thickness is less

  Determine the following:

  than 0.2337 millimeter?

  (a) E 12X 3Y 2

  (b) V 12X 3Y 2

  (b) A lamp with a total ink thickness exceeding 0.2405 mil-

  (c) P 12X 3Y 302 (d) P

  12 X 3Y 402

  limeter lacks the uniformity of color demanded by the

  5-55.

  X and Y are independent, normal random variables

  customer. Find the probability that a randomly selected

  with E

  1X 2 2, V1X 2 5, E1Y 2 6, and V1Y 2 8. lamp fails to meet customer specifications.

  5-60. The width of a casing for a door is normally distrib-

  Determine the following:

  (a) E 13X 2Y 2

  (b) V 13X 2Y 2

  uted with a mean of 24 inches and a standard deviation of

  (c) P 13X 2Y 182 (d) P

  13X 2Y 282 兾8 inch. The width of a door is normally distributed with a

  mean of 23-7 兾8 inches and a standard deviation of 1兾16 inch.

  5-56. Suppose that the random variable X represents the

  Assume independence.

  length of a punched part in centimeters. Let Y be the length

  (a) Determine the mean and standard deviation of the differ-

  of the part in millimeters. If E(X )

  5 and V(X ) 0.25, what

  ence between the width of the casing and the width of the

  are the mean and variance of Y ?

  door.

  5-57.

  A plastic casing for a magnetic disk is composed of

  (b) What is the probability that the width of the casing minus

  two halves. The thickness of each half is normally distributed

  the width of the door exceeds 1 兾4 inch?

  with a mean of 2 millimeters and a standard deviation of

  (c) What is the probability that the door does not fit in the

  0.1 millimeter and the halves are independent.

  casing?

  (a) Determine the mean and standard deviation of the total

  5-61. An article in Knee Surgery Sports Traumatology,

  thickness of the two halves.

  Arthroscopy [“Effect of Provider Volume on Resource

  (b) What is the probability that the total thickness exceeds

  Utilization for Surgical Procedures” (2005, Vol. 13, pp.

  4.3 millimeters?

  273–279)] showed a mean time of 129 minutes and a standard

  5-58. Making handcrafted pottery generally takes two

  deviation of 14 minutes for ACL reconstruction surgery for

  major steps: wheel throwing and firing. The time of wheel

  high-volume hospitals (with more than 300 such surgeries per

  throwing and the time of firing are normally distributed

  year). If a high-volume hospital needs to schedule 10 surger-

  random variables with means of 40 min and 60 min and stan-

  ies, what are the mean and variance of the total time to com-

  dard deviations of 2 min and 3 min, respectively.

  plete these surgeries? Assume the times of the surgeries are in-

  (a) What is the probability that a piece of pottery will be fin-

  dependent and normally distributed.

  ished within 95 min?

  5-62. Soft-drink cans are filled by an automated filling

  (b) What is the probability that it will take longer than 110 min?

  machine and the standard deviation is 0.5 fluid ounce. Assume

  5-59. In the manufacture of electroluminescent lamps, sev-

  that the fill volumes of the cans are independent, normal

  eral different layers of ink are deposited onto a plastic sub-

  random variables.

  strate. The thickness of these layers is critical if specifications

  (a) What is the standard deviation of the average fill volume

  regarding the final color and intensity of light are to be met.

  of 100 cans?

  Let X and Y denote the thickness of two different layers of ink.

  (b) If the mean fill volume is 12.1 ounces, what is the proba-

  It is known that X is normally distributed with a mean of 0.1

  bility that the average fill volume of the 100 cans is below

  millimeter and a standard deviation of 0.00031 millimeter and

  12 fluid ounces?

  Y is also normally distributed with a mean of 0.23 millimeter

  (c) What should the mean fill volume equal so that the proba-

  and a standard deviation of 0.00017 millimeter. Assume that

  bility that the average of 100 cans is below 12 fluid ounces

  these variables are independent.

  is 0.005?

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  5-5 GENERAL FUNCTIONS OF RANDOM VARIABLES

  (d) If the mean fill volume is 12.1 fluid ounces, what should

  (a) Without measurement error, what is the probability that a

  the standard deviation of fill volume equal so that the

  part exceeds the specifications?

  probability that the average of 100 cans is below 12 fluid

  (b) With measurement error, what is the probability that a part

  ounces is 0.005?

  is measured as beyond specifications? Does this imply it is

  (e) Determine the number of cans that need to be measured

  truly beyond specifications?

  such that the probability that the average fill volume is

  (c) What is the probability that a part is measured beyond

  less than 12 fluid ounces is 0.01.

  specifications if the true weight of the part is one below

  5-63. The photoresist thickness in semiconductor manufac-

  the upper specification limit?

  turing has a mean of 10 micrometers and a standard deviation of

  5-66.

  A U-shaped component is to be formed from the three

  1 micrometer. Assume that the thickness is normally distributed

  parts A, B, and C. The picture is shown in Fig. 5-20. The length

  and that the thicknesses of different wafers are independent.

  of A is normally distributed with a mean of 10 millimeters and

  (a) Determine the probability that the average thickness of 10

  a standard deviation of 0.1 millimeter. The thickness of parts B

  wafers is either greater than 11 or less than 9 micrometers.

  and C is normally distributed with a mean of 2 millimeters and

  (b) Determine the number of wafers that need to be measured

  a standard deviation of 0.05 millimeter. Assume all dimensions

  such that the probability that the average thickness ex-

  are independent.

  ceeds 11 micrometers is 0.01. (c) If the mean thickness is 10 micrometers, what should the

  B C

  standard deviation of thickness equal so that the probability

  D

  that the average of 10 wafers is either greater than 11

  C

  or less than 9 micrometers is 0.001?

  5-64. Assume that the weights of individuals are indepen- dent and normally distributed with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 25 people

  A

  squeeze into an elevator that is designed to hold 4300 pounds. (a) What is the probability that the load (total weight) exceeds

  the design limit? (b) What design limit is exceeded by 25 occupants with prob-

  Figure 5-20 Figure for the

  ability 0.0001?

  U-shaped component.

  5-65. Weights of parts are normally distributed with variance

  2 . Measurement error is normally distributed with mean zero

  (a) Determine the mean and standard deviation of the length

  and variance 0.5 2 , independent of the part weights, and adds to

  of the gap D.

  the part weight. Upper and lower specifications are centered at

  (b) What is the probability that the gap D is less than 5.9 mil-

  3 about the process mean.

  limeters?