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186 Chapter 6 Some Continuous Probability Distributions

(a) What is the probability that the individual waits 6.11 A soft-drink machine is regulated so that it dis-
more than 7 minutes? charges an average of 200 milliliters per cup. If the
(b) What is the probability that the individual waits amount of drink is normally distributed with a stan-
between 2 and 7 minutes? dard deviation equal to 15 milliliters,
(a) what fraction of the cups will contain more than
6.5 Given a standard normal distribution, find the 224 milliliters?
area under the curve that lies (b) what is the probability that a cup contains between
(a) to the left of z = −1.39; 191 and 209 milliliters?
(b) to the right of z = 1.96; (c) how many cups will probably overflow if 230-
(c) between z = −2.16 and z = −0.65; milliliter cups are used for the next 1000 drinks?
(d) to the left of z = 1.43; (d) below what value do we get the smallest 25% of the
drinks?
(e) to the right of z = −0.89;
(f) between z = −0.48 and z = 1.74. 6.12 The loaves of rye bread distributed to local
stores by a certain bakery have an average length of 30
6.6 Find the value of z if the area under a standard centimeters and a standard deviation of 2 centimeters.
normal curve Assuming that the lengths are normally distributed,
(a) to the right of z is 0.3622; what percentage of the loaves are
(b) to the left of z is 0.1131; (a) longer than 31.7 centimeters?
(c) between 0 and z, with z > 0, is 0.4838; (b) between 29.3 and 33.5 centimeters in length?
(d) between −z and z, with z > 0, is 0.9500. (c) shorter than 25.5 centimeters?

6.7 Given a standard normal distribution, find the 6.13 A research scientist reports that mice will live an
value of k such that average of 40 months when their diets are sharply re-
stricted and then enriched with vitamins and proteins.
(a) P (Z > k) = 0.2946; Assuming that the lifetimes of such mice are normally
(b) P (Z < k) = 0.0427; distributed with a standard deviation of 6.3 months,
(c) P (−0.93 < Z < k) = 0.7235. find the probability that a given mouse will live
(a) more than 32 months;
6.8 Given a normal distribution with μ = 30 and (b) less than 28 months;
σ = 6, find (c) between 37 and 49 months.
(a) the normal curve area to the right of x = 17;
(b) the normal curve area to the left of x = 22; 6.14 The finished inside diameter of a piston ring is
(c) the normal curve area between x = 32 and x = 41; normally distributed with a mean of 10 centimeters and
(d) the value of x that has 80% of the normal curve a standard deviation of 0.03 centimeter.
area to the left; (a) What proportion of rings will have inside diameters
(e) the two values of x that contain the middle 75% of exceeding 10.075 centimeters?
the normal curve area. (b) What is the probability that a piston ring will have
an inside diameter between 9.97 and 10.03 centime-
6.9 Given the normally distributed variable X with ters?
mean 18 and standard deviation 2.5, find (c) Below what value of inside diameter will 15% of the
(a) P (X < 15); piston rings fall?
(b) the value of k such that P (X < k) = 0.2236;
(c) the value of k such that P (X > k) = 0.1814; 6.15 A lawyer commutes daily from his suburban
home to his midtown office. The average time for a
(d) P (17 < X < 21). one-way trip is 24 minutes, with a standard deviation
of 3.8 minutes. Assume the distribution of trip times
6.10 According to Chebyshev’s theorem, the proba- to be normally distributed.
bility that any random variable assumes a value within (a) What is the probability that a trip will take at least
3 standard deviations of the mean is at least 8/9. If it 1/2 hour?
is known that the probability distribution of a random
(b) If the office opens at 9:00 A.M. and the lawyer leaves
variable X is normal with mean μ and variance σ 2 ,
his house at 8:45 A.M. daily, what percentage of the
what is the exact value of P (μ − 3σ < X < μ + 3σ)?
time is he late for work?
6.5 Normal Approximation to the Binomial 187

(c) If he leaves the house at 8:35 A.M. and coffee is (a) what percentage of the workers receive wages be-
served at the office from 8:50 A.M. until 9:00 A.M., tween $13.75 and $16.22 an hour inclusive?
what is the probability that he misses coffee? (b) the highest 5% of the employee hourly wages is
(d) Find the length of time above which we find the greater than what amount?
slowest 15% of the trips.
(e) Find the probability that 2 of the next 3 trips will 6.20 The weights of a large number of miniature poo-
take at least 1/2 hour. dles are approximately normally distributed with a
mean of 8 kilograms and a standard deviation of 0.9
6.16 In the November 1990 issue of Chemical Engi- kilogram. If measurements are recorded to the nearest
neering Progress, a study discussed the percent purity tenth of a kilogram, find the fraction of these poodles
of oxygen from a certain supplier. Assume that the with weights
mean was 99.61 with a standard deviation of 0.08. As- (a) over 9.5 kilograms;
sume that the distribution of percent purity was ap- (b) of at most 8.6 kilograms;
proximately normal. (c) between 7.3 and 9.1 kilograms inclusive.
(a) What percentage of the purity values would you
expect to be between 99.5 and 99.7?
6.21 The tensile strength of a certain metal compo-
(b) What purity value would you expect to exceed ex- nent is normally distributed with a mean of 10,000 kilo-
actly 5% of the population? grams per square centimeter and a standard deviation
of 100 kilograms per square centimeter. Measurements
6.17 The average life of a certain type of small motor are recorded to the nearest 50 kilograms per square
is 10 years with a standard deviation of 2 years. The centimeter.
manufacturer replaces free all motors that fail while (a) What proportion of these components exceed
under guarantee. If she is willing to replace only 3% of 10,150 kilograms per square centimeter in tensile
the motors that fail, how long a guarantee should be strength?
offered? Assume that the lifetime of a motor follows a
(b) If specifications require that all components have
normal distribution.
tensile strength between 9800 and 10,200 kilograms
6.18 The heights of 1000 students are normally dis- per square centimeter inclusive, what proportion of
tributed with a mean of 174.5 centimeters and a stan- pieces would we expect to scrap?
dard deviation of 6.9 centimeters. Assuming that the
heights are recorded to the nearest half-centimeter, 6.22 If a set of observations is normally distributed,
how many of these students would you expect to have what percent of these differ from the mean by
heights (a) more than 1.3σ?
(a) less than 160.0 centimeters? (b) less than 0.52σ?
(b) between 171.5 and 182.0 centimeters inclusive?
(c) equal to 175.0 centimeters? 6.23 The IQs of 600 applicants to a certain college
(d) greater than or equal to 188.0 centimeters? are approximately normally distributed with a mean
of 115 and a standard deviation of 12. If the college
requires an IQ of at least 95, how many of these stu-
6.19 A company pays its employees an average wage dents will be rejected on this basis of IQ, regardless of
of $15.90 an hour with a standard deviation of $1.50. If their other qualifications? Note that IQs are recorded
the wages are approximately normally distributed and to the nearest integers.
paid to the nearest cent,

6.5 Normal Approximation to the Binomial

Probabilities associated with binomial experiments are readily obtainable from the
formula b(x; n, p) of the binomial distribution or from Table A.1 when n is small.
In addition, binomial probabilities are readily available in many computer software
packages. However, it is instructive to learn the relationship between the binomial
and the normal distribution. In Section 5.5, we illustrated how the Poisson dis-
tribution can be used to approximate binomial probabilities when n is quite large
and p is very close to 0 or 1. Both the binomial and the Poisson distributions