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Examples: Continuous Probability Distributions
1. The magnitude of earthquakes recorded in a region of North America can be
modelled by an exponential distribution with mean 2.4 as measured on the
Richter scale. Find the probability that the next earthquake to strike this region
will (a) Exceed 3.0 on the Richter scale, and (b) Fall between 2.0 and 3.0 on the
Richter scale.
2. A pumping station operator observes that the demand for water at a certain hour
of the day can be modelled as an exponential random variable with a mean of
100 cubic feet per second. (a) Find the probability that the demand will exceed
200 cfs on a randomly selected day. (b) What is the maximum water-producing
capacity that the station should keep on line for this hour so that the demand will
exceed this production capacity with a probability of only 0.01?
3. Use Table 17 of the appendix to find the following probabilities for a standard
normal random variable Z. (a) P(0  Z  1.2), (b) P(-0.9  Z  0), (c) P(0.3  Z 
1.56), (d) P(-0.2  Z  0.2), and (e) P(-2.00  Z  -1.56).
4. For a standard normal random variable Z, use Table 17 of the Appendix to find a
number z0 such that (a) P(Z  z0) = 0.5, (b) P(Z  z0) = 0.8749, (c) P(Z  z0) =
0.117, (d) P(Z  z0) = 0.617, (e) P(-z0  Z  z0) = 0.90, and (f) P(-z0  Z  z0) =
0.95.
6.57/326) The weekly amount spent for maintenance and repairs in a certain company
has approximately a normal distribution with a mean of Php400 and a standard deviation
of Php20. If Php450 is budgeted to cover repairs for next week, what is the probability
that the actual costs will exceed the budgeted amount?
6.59/326) A machining operation produces steel shafts having diameters that are
normally distributed with a mean of 1.005 inches and a standard deviation of 0.01 inch.
Specifications call for diameters to fall within the interval 1.00  0.02 inches. What
percentage of the output of this operation will fail to meet specifications?
6.61/326) Wires manufactured for use in a certain computer system are specified to
have resistances between 0.12 and 0.14 ohm. The actual measured resistances of wires
produced by Company A have a normal probability distribution with a mean of 0.13 ohm
and a standard deviation of 0.005 ohm. (a) What is the probability that a randomly
selected wire from Company A’s production will meet the specifications? (b) If four such
wires are used in the system and all are selected form Company A, what is the
probability that all four will meet the specifications?
6.62/326) At a temperature of 26C, the resistances of a type of thermistor are normally
distributed with a mean of 10,000 ohms and a standard deviation of 4,000 ohms. The
thermistors are to be sorted, with those having resistances between 8,000 and 15,000
ohms being shipped to a vendor. What fraction of these thermistors will be shipped?
6.65/326) Sick leave time used by employees of a firm in one month has approximately
a normal distribution with a mean of 200 hours and a variance of 400. (a) Find the
probability that total sick leave for next month will be less than 150 hours. (b) In
planning schedules for next month, how much time should be budgeted for sick leave if
that amount is to be exceeded with a probability of only 0.10?
6.67/327) A machine for filling cereal boxes has a standard deviation of 1 ounce of fill
per box. What setting of the mean ounces of fill per box will allow 16-ounce boxes to
overflow only 1% of the time? Assume that the ounces of fill per box are normally
distributed?
6.114/353) The yield force of a steel-reinforcing bar of a certain type is found to be
normally distributed with a mean of 8,500 pounds and a standard deviation of 80
pounds. If three such bars are to be used on a certain project, find the probability that
all three will have yield forces in excess of 8,700 pounds.
Seatwork (20 pts)
(1/4 sheet)
The grade-point averages of a large population of college students are approximately
normally distributed with mean equal to 2.4 and standard deviation equal to 0.5. (a)
What fraction of the students will possess a grade-point average in excess of 3.0? (b) If
students possessing a grade-point average equal to or less than 1.9 are dropped from
college, what percentage of the students will be dropped?