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118 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

4-53. The reaction time of a driver to visual stimulus is nor- 4-57. The sick-leave time of employees in a firm in a month
mally distributed with a mean of 0.4 seconds and a standard is normally distributed with a mean of 100 hours and a stan-
deviation of 0.05 seconds. dard deviation of 20 hours.
(a) What is the probability that a reaction requires more than (a) What is the probability that the sick-leave time for next
0.5 seconds? month will be between 50 and 80 hours?
(b) What is the probability that a reaction requires between (b) How much time should be budgeted for sick leave if the
0.4 and 0.5 seconds? budgeted amount should be exceeded with a probability
(c) What is the reaction time that is exceeded 90% of the of only 10%?
time? 4-58. The life of a semiconductor laser at a constant power
4-54. The speed of a file transfer from a server on campus to is normally distributed with a mean of 7000 hours and a stan-
a personal computer at a student’s home on a weekday dard deviation of 600 hours.
evening is normally distributed with a mean of 60 kilobits per (a) What is the probability that a laser fails before 5000
second and a standard deviation of 4 kilobits per second. hours?
(a) What is the probability that the file will transfer at a speed (b) What is the life in hours that 95% of the lasers exceed?
of 70 kilobits per second or more? (c) If three lasers are used in a product and they are assumed
(b) What is the probability that the file will transfer at a speed to fail independently, what is the probability that all three
of less than 58 kilobits per second? are still operating after 7000 hours?
(c) If the file is 1 megabyte, what is the average time it will 4-59. The diameter of the dot produced by a printer is nor-
take to transfer the file? (Assume eight bits per byte.) mally distributed with a mean diameter of 0.002 inch and a
4-55. The length of an injection-molded plastic case that standard deviation of 0.0004 inch.
holds magnetic tape is normally distributed with a length of (a) What is the probability that the diameter of a dot exceeds
90.2 millimeters and a standard deviation of 0.1 millimeter. 0.0026 inch?
(a) What is the probability that a part is longer than 90.3 mil- (b) What is the probability that a diameter is between 0.0014
limeters or shorter than 89.7 millimeters? and 0.0026 inch?
(b) What should the process mean be set at to obtain the great- (c) What standard deviation of diameters is needed so that the
est number of parts between 89.7 and 90.3 millimeters? probability in part (b) is 0.995?
(c) If parts that are not between 89.7 and 90.3 millimeters are 4-60. The weight of a sophisticated running shoe is nor-
scrapped, what is the yield for the process mean that you mally distributed with a mean of 12 ounces and a standard de-
selected in part (b)? viation of 0.5 ounce.
4-56. In the previous exercise assume that the process is (a) What is the probability that a shoe weighs more than 13
centered so that the mean is 90 millimeters and the standard ounces?
deviation is 0.1 millimeter. Suppose that 10 cases are meas- (b) What must the standard deviation of weight be in order for
ured, and they are assumed to be independent. the company to state that 99.9% of its shoes are less than
(a) What is the probability that all 10 cases are between 89.7 13 ounces?
and 90.3 millimeters? (c) If the standard deviation remains at 0.5 ounce, what must
(b) What is the expected number of the 10 cases that are be- the mean weight be in order for the company to state that
tween 89.7 and 90.3 millimeters? 99.9% of its shoes are less than 13 ounces?

4-7 NORMAL APPROXIMATION TO THE BINOMIAL
AND POISSON DISTRIBUTIONS

We began our section on the normal distribution with the central limit theorem and the nor-
mal distribution as an approximation to a random variable with a large number of trials.
Consequently, it should not be a surprise to learn that the normal distribution can be used
to approximate binomial probabilities for cases in which n is large. The following example
illustrates that for many physical systems the binomial model is appropriate with an ex-
tremely large value for n. In these cases, it is difficult to calculate probabilities by using the
binomial distribution. Fortunately, the normal approximation is most effective in these
cases. An illustration is provided in Fig. 4-19. The area of each bar equals the binomial
probability of x. Notice that the area of bars can be approximated by areas under the nor-
mal density function.