Chapter 7: Continuous Probability Distributions
TRUE/FALSE
1.True or FalseIn any normal distribution, the mean, median,
mode, and standard deviation are all at the same position on the
horizontal axis.
ANS:FPTS:1OBJ:Section 7.2
2.True or FalseIn the normal distribution, the curve is
asymptotic but never intercepts the horizontal axis either to the
left or right.
ANS:TPTS:1OBJ:Section 7.2
3.True or FalseIn the normal distribution, the total area
beneath the curve represents the probability for all possible
outcomes for a given event.
ANS:TPTS:1OBJ:Section 7.2
4.True or FalseUnder certain conditions, the normal distribution
can be used to approximate the binomial distribution.
ANS:TPTS:1OBJ:Section 7.4
5.True or FalseThere exists a different normal curve for every
possible pair of and .
ANS:TPTS:1OBJ:Section 7.3
6.True or FalseThe standard deviation of the binomial
distribution with n = 25 and = .50 is 6.25.
ANS:FPTS:1OBJ:Section 7.4
7.True or FalseThe mean of the exponential distribution is the
inverse of that of the normal distribution.
ANS:FPTS:1OBJ:Section 7.5
8.True or FalseIf Excel and Minitab packages are both used to
generate its own sample of 10,000 observations from a theoretical
continuous distribution, the two sets of observations are expected
to be identical.
ANS:FPTS:1OBJ:Section 7.6
9.True or FalseAccording to the law of large numbers, frequency
distributions generated by statistical packages will more closely
approach the theoretical distribution from which they were drawn
whenever the number of observations becomes larger.
ANS:TPTS:1OBJ:Section 7.6
10.True or FalseContinuous probability distributions describe
probabilities associated with random variables that can take on any
value along a given range or continuum and for which there are no
gaps between these possible values.
ANS:TPTS:1OBJ:Section 7.7
11.True or FalseFor any specified interval of values, the
probability that a continuous random variable x will assume a value
within the interval is the area beneath the curve between the two
points describing the interval.
ANS:TPTS:1OBJ:Section 7.7
12.True or FalseAny normal distribution is a symmetrical,
bell-shaped curve, with mean = 0.0 and standard deviation =
1.0.
ANS:FPTS:1OBJ:Section 7.7
13.True or FalseRegardless of the shape of a particular normal
curve, about 75% of the area is within the interval , where and are
the mean and the standard deviation, respectively.
ANS:FPTS:1OBJ:Section 7.7
14.True or FalseThe standard normal distribution has a mean of
0.0 and standard deviation of 1.0.
ANS:TPTS:1OBJ:Section 7.7
15.True or FalseThe exponential distribution is a family of
discrete distributions, all of which are negatively skewed.
ANS:FPTS:1OBJ:Section 7.7
16.True or FalseComputer statistical packages, such as Excel and
Minitab, can determine the exact area beneath a specified portion
of the continuous distribution curve, or probability density
function, and eliminate the need for calculations or table
references.
ANS:TPTS:1OBJ:Section 7.7
MULTIPLE CHOICE
1.A continuous probability distribution represents a random
variable:a.having outcomes, which occur in counting numbers.
b.having an infinite number of outcomes which may assume any
number of values within an interval.
c.which is best described in a histogram.
d.which has a definite probability for the occurrence of a given
integer.
e.None of these is correct.
ANS:BPTS:1OBJ:Section 7.1
2.Which of the following are not correct concerning the
probability distribution for any continuous random variable?a.The
vertical coordinate is the probability density function.
b.The range of the random variable is found on the y-axis.
c.The total area represented under the curve will be equal to
1.00.
d.The probability that x will take on a value between a and b
will be the area under the curve between points a and b.
e.The area under the curve represents the sum of probabilities
for all possible outcomes.
ANS:BPTS:1OBJ:Section 7.1
3.In a continuous probability distribution, the probability that
x will take on an exact value:a.is equal to the height of the curve
at that value.
b.is calculated using the probability density.
c.is always equal to 0.
d.is always greater than 0.
e.None of these is correct.
ANS:CPTS:1OBJ:Section 7.1
4.Which of the following statements are true regarding the areas
beneath any normal curve?a.About 68.3% of the area is in the
interval to .
b.About 95.5% of the area is in the interval to .
c.Nearly all of the area (about 99.7%) is in the interval to
d.All of these are true.
e.None of these is true.
ANS:DPTS:1OBJ:Section 7.2
5.Which of the following is not a characteristic for a normal
distribution?a.It is symmetrical distribution
b.The mean is always zero
c.The mean, median, and mode are all equal
d.It is a bell-shaped distribution
e.The area under the curve always equals 1.0
ANS:BPTS:1OBJ:Section 7.2
6.A videotape store has an average weekly gross of $1,158 with a
standard deviation of $120. Let x be the store's gross during a
randomly selected week. If this is a normally distributed random
variable, then the number of standard deviations from $1,158 to
$1,360 is:a.0.4535.
b.0.0465.
c.20.98.
d.1.683.
e.none of these.
ANS:DPTS:1OBJ:Section 7.3
7.Using the standard normal table, the total area between z =
-0.75 and z = 1.21 is:a.0.2734
b.0.3869
c.0.3397
d.0.2266
e.0.6603
ANS:EPTS:1OBJ:Section 7.3
8.Let z1 be a z score that is unknown but identifiable by
position and area. If the area to the right of z1 is 0.8413, then
the value of z1 must be:a.1.00.
b.-1.00.
c.0.00.
d.0.41.
e.-0.41.
ANS:BPTS:1OBJ:Section 7.3
9.Using the standard normal table, the total area between z =
-1.82 and z = -1.38 is:a.0.0494
b.0.4656
c.0.1554
d.0.4162
e.0.1005
ANS:APTS:1OBJ:Section 7.3
10.If the area to the right of a positive z1 is 0.0869, then the
value of z1 must be:a.-1.36.
b.1.36.
c.1.71.
d.1.80.
e.0.22.
ANS:BPTS:1OBJ:Section 7.3
11.The z-score representing the 10th percentile of the standard
normal distribution is:a.0.255.
b.-0.255.
c.-1.645.
d.1.28.
e.-1.28.
ANS:EPTS:1OBJ:Section 7.3
12.The z-score representing the 75th percentile of the standard
normal distribution is:a.0.67.
b.-0.67.
c.1.28.
d.-1.28.
e.1.645.
ANS:APTS:1OBJ:Section 7.3
13.The z-score representing the 90th percentile of the standard
normal distribution is:a.1.28.
b.-1.28.
c.0.67.
d.-0.67.
e.1.645.
ANS:APTS:1OBJ:Section 7.3
14.For a normal curve, if the mean is 20 minutes and the
standard deviation is 5 minutes, then the area between 22 and 25
minutes is:a.0.3413.
b.0.1554.
c.0.4967.
d.0.1859.
e.0.2248.
ANS:DPTS:1OBJ:Section 7.3
15.For a normal curve, if the mean is 20 minutes and the
standard deviation is 5 minutes, then the area between 11 and 19
minutes is:a.0.4641.
b.0.0359.
c.0.3848.
d.0.8848.
e.0.9641.
ANS:CPTS:1OBJ:Section 7.3
16.A bakery firm finds that its average weight of the most
popular package of cookies is 32.06 ounces with a standard
deviation of 0.1 ounces. What portion of cookie packages will weigh
less than 31.9 ounces or more than 32.3 ounces?a.0.8000
b.0.4452
c.0.4918
d.0.9370
e.0.0630
ANS:EPTS:1OBJ:Section 7.3
17.A retailer finds that the demand for a very popular board
game averages 100 per week with a standard deviation of 20. If the
seller wishes to have adequate stock 95% of the time, how many of
the games must she keep on hand?a.139.2
b.100.0
c.132.9
d.195.0
e.125.6
ANS:CPTS:1OBJ:Section 7.3
18.A manufacturer of tow chains finds that the average breaking
point is at 3,500 pounds and the standard deviation is 250 pounds.
If you pull a weight of at least 4200 pounds with this tow chain,
what percentage of the time would you expect the chain to
break?a.2.8%
b.0.26%
c.49.74%
d.99.74%
e.50%
ANS:BPTS:1OBJ:Section 7.3
19.The average labor time to sew a pair of denim jeans is 4.2
hours with a standard deviation of 30 minutes. If the distribution
is normal, then the probability of a worker finishing a pair of
jeans in less than 3.5 hours is:a.0.0808.
b.0.4192.
c.0.5808.
d.0.9192.
e.0.9808.
ANS:APTS:1OBJ:Section 7.3
20.A salesman who uses his car extensively finds that his
gasoline bills average $125.32 per month with a standard deviation
of $49.51. The probability that his bill will be less than $50 a
month or more than $150 for a single month is:a.0.4357.
b.0.1915.
c.0.6272.
d.0.3728.
e.0.6915.
ANS:DPTS:1OBJ:Section 7.3
21.Given that z is a standard normal random variable and that
the area to the right of z is 0.1949, then the value of z must
be:a.0.51.
b.-0.51.
c.0.86.
d.-0.86.
ANS:CPTS:1OBJ:Section 7.3
22.Using the standard normal table, the total area between z =
0.45 and z = 1.05 is:a.0.1736
b.0.3531
c.0.3264
d.0.6469
e.0.1795
ANS:EPTS:1OBJ:Section 7.3
23.Given that z is a standard normal random variable, a negative
value of z indicates that:a.The value z is to the left of the
mean.
b.The value z is to the right of the median.
c.The standard deviation of z is negative.
d.The area between zero and z is negative.
e.The probability associated with z is negative.
ANS:APTS:1OBJ:Section 7.3
24.Given that z is a standard normal random variable, and that
the area to the right of z is 0.9066, then the value of z must
be:a.1.32.
b.-1.32.
c.0.66.
d.-0.66.
ANS:BPTS:1OBJ:Section 7.3
25.The proportion of the data from a standard normal
distribution that falls within two standard deviations from the
mean is:a.0.3413.
b.0.4772.
c.0.6826.
d.0.9544
ANS:DPTS:1OBJ:Section 7.3
26.If the z-value for a given value x of the random variable x
is z = 1.96, and the distribution of x is normally distributed with
a mean of 60 and a standard deviation of 6, then the x-value that
this z-value corresponds to is:a.71.76.
b.67.96.
c.61.96.
d.48.24.
ANS:APTS:1OBJ:Section 7.3
27.Your high school graduating class had 564 members.
Thirty-three percent of these are expected to attend college. If
the number of students who attend college follows the normal
distribution, the probability that less than 160 will attend
college is:a.0.3610.
b.0.4900.
c.0.0090.
d.0.0832.
e.0.2500
ANS:CPTS:1OBJ:Section 7.4
28.A large mail house which mails such items as catalogues,
magazines, and other bulk mailings guarantees that there will be no
more than a 3% error rate on its mailing labels. A customer who
contracted a mailing to 190,000 individuals experienced a return of
5,900 items, which had incorrect addresses. Which of the following
statements is true?a.There is a 3% chance of an incorrect
return.
b.There is a 0.0054 probability that a return of 5900 or more
incorrect addresses could occur if the true error rate is 3%.
c.There is a 0.4964 probability that a return of 5900 or more
incorrect addresses could occur if the true error rate is 3%.
d.There is a 2.69 percent possibility that a return of 5900
incorrect addresses could occur if the true error rate is 3%.
e.None of these statements are correct.
ANS:BPTS:1OBJ:Section 7.4
29.In your college campus, 30% of the students use tobacco in
some form. In your statistics class of 40 students, the probability
of finding at least 15 students who use tobacco is:a.0.3051.
b.0.1949.
c.0.8051.
d.0.1515.
e.0.3485.
ANS:BPTS:1OBJ:Section 7.4
30.A zipper manufacturer has found that 1.5% of its zippers are
defective. The company averages 4982 zippers per day. The
probability of finding between 90 and 100 defects in a given day
is:a.0.4981.
b.0.0310.
c.0.0360.
d.0.5310.
e.0.4671.
ANS:BPTS:1OBJ:Section 7.4
31.Which of the following is not a correct statement?a.The
exponential distribution describes the Poisson process as a
continuous random variable.
b.The exponential distribution is a family of curves, which are
completely described by the mean.
c.The mean of the exponential distribution is the complement of
the mean of the Poisson.
d.The Poisson is a probability distribution or a discrete random
variable while the exponential distribution is continuous.
e.All of these are correct.
ANS:CPTS:1OBJ:Section 7.5
32.A local radio station maintains a very popular phone service,
which provides callers with current weather reports. The incoming
calls have a Poisson distribution with an average of 5 calls for
each 15 minute-period. If x = time between calls, then the
probability of receiving four or fewer calls in the next 15 minutes
is:a.0.2341.
b.0.7659.
c.0.2636.
d.0.7364.
e.0.9502.
ANS:DPTS:1OBJ:Section 7.5
33.A dispatcher for an airport shuttle averages sending a van to
the airport 2 times every 60 minutes. The distribution is Poisson,
and the driver must take a 15-minute lunch break. The probability
that he can finish lunch before getting a call is:a.0.1350.
b.0.6065.
c.0.3935.
d.1.6490.
e.0.8650.
ANS:CPTS:1OBJ:Section 7.5
34.A very large logging operation has serious problems keeping
their skidders operating properly. The equipment fails at the rate
of 3 breakdowns every 48 hours. Assume that x is time between
breakdowns and is exponentially distributed. The probability of two
or less breakdowns in the next 48-hour period is:a.0.9672.
b.0.0307.
c.0.2231.
d.0.7769.
ANS:BPTS:1OBJ:Section 7.5
35.If the mean of an exponential distribution is 2, then the
value of the parameter is:a.4.0.
b.2.2.
c.1.0.
d.0.5.
ANS:DPTS:1OBJ:Section 7.5
36.If the random variable x is exponentially distributed with
parameter = 4, then P(x 0.25), up to 4 decimal places,
is:a.0.6321.
b.0.3679.
c.0.2500.
d.0.5000.
ANS:APTS:1OBJ:Section 7.5
37.If the random variable x is exponentially distributed with
parameter = 1.5, then P(2 x 4), up to 4 decimal places,
is:a.0.6667.
b.0.0473.
c.0.5000.
d.0.2500.
ANS:BPTS:1OBJ:Section 7.5
38.Which of the following is not true for an exponential
distribution with parameter ?a.Mean = .
b.Standard deviation = .
c.The distribution is completely determined once the value of is
known.
d.The distribution is a two-parameter distribution since the
mean and standard deviation are equal.
ANS:DPTS:1OBJ:Section 7.5
39.Like the normal distribution, the exponential density
function f(x):a.is bell-shaped.
b.is symmetrical.
c.approaches infinity as x approaches zero.
d.approaches zero as x approaches infinity.
ANS:DPTS:1OBJ:Section 7.5
40.Which of the following distributions is suitable to model the
length of time that elapses before the first telephone call is
received by a switchboard?a.Exponential
b.Normal
c.Poisson
d.Binomial
e.Hypergeometric
ANS:APTS:1OBJ:Section 7.5
41.The mean of the exponential distribution equals the mean of
the Poisson distribution only when the former distribution has a
mean equals:a.1.00.
b.0.50.
c.0.25.
d.2.00.
ANS:APTS:1OBJ:Section 7.5
42.Which of the following distributions is appropriate to
measure the length of time between arrivals at a grocery checkout
counter?a.Binomial
b.Normal
c.Exponential
d.Poisson
e.Hypergeometric
ANS:CPTS:1OBJ:Section 7.5
43.Which of the following statements are true regarding the
normal distribution curve?a.It is symmetrical.
b.It is bell-shaped.
c.It is asymptotic in that each end approaches the horizontal
axis, but never reaches it.
d.Its mean, median, and mode are located at the same
position.
e.All of these statements are true.
ANS:EPTS:1OBJ:Section 7.7
NUMERIC RESPONSE
1.Assume x is normally distributed with mean = 15 and standard
deviation = 3. Use the approximate areas beneath the normal curve,
as discussed in this section, to find P(x 15).
ANS:0.50
PTS:1OBJ:Section 7.2
2.Assume x is normally distributed with mean = 15 and standard
deviation = 3. Use the approximate areas beneath the normal curve,
as discussed in this section, to find P(12 x 18).
ANS:0.683
PTS:1OBJ:Section 7.2
3.Assume x is normally distributed with mean = 15 and standard
deviation = 3. Use the approximate areas beneath the normal curve,
as discussed in this section, to find P(x 9).
ANS:0.0225
PTS:1OBJ:Section 7.2
4.Assume x is normally distributed with mean = 15 and standard
deviation = 3. Use the approximate areas beneath the normal curve,
as discussed in this section, to find P(x = 20).
ANS:0
PTS:1OBJ:Section 7.2
5.Assume x is normally distributed with mean = 15 and standard
deviation = 3. Use the approximate areas beneath the normal curve,
as discussed in this section, to find P(9 x 21).
ANS:0.955
PTS:1OBJ:Section 7.2
6.Assume x is normally distributed with mean = 15 and standard
deviation = 3. Use the approximate areas beneath the normal curve,
as discussed in this section, to find P(x 12).
ANS:0.8415
PTS:1OBJ:Section 7.2
7.In 2000, the average charge of tax preparation was $95.
Assuming a normal distribution and a standard deviation of $10, use
the approximate areas beneath the normal curve, as discussed in
this section, to answer: What proportion of tax preparation fees
were more than $95?
ANS:0.50
PTS:1OBJ:Section 7.2
8.In 2000, the average charge of tax preparation was $95.
Assuming a normal distribution and a standard deviation of $10, use
the approximate areas beneath the normal curve, as discussed in
this section, to answer: What proportion of tax preparation fees
were between $75 and $115?
ANS:0.955
PTS:1OBJ:Section 7.2
9.In 2000, the average charge of tax preparation was $95.
Assuming a normal distribution and a standard deviation of $10, use
the approximate areas beneath the normal curve, as discussed in
this section, to answer: What proportion of tax preparation fees
were between $85 and $105?
ANS:0.683
PTS:1OBJ:Section 7.2
10.In 2000, the average charge of tax preparation was $95.
Assuming a normal distribution and a standard deviation of $10, use
the approximate areas beneath the normal curve, as discussed in
this section, to answer: What proportion of tax preparation fees
were more than $115?
ANS:0.0225
PTS:1OBJ:Section 7.2
11.In 2000, the average charge of tax preparation was $95.
Assuming a normal distribution and a standard deviation of $10, use
the approximate areas beneath the normal curve, as discussed in
this section, to answer: What proportion of tax preparation fees
were between $80 and $90?
ANS:0.2417
PTS:1OBJ:Section 7.2
12.Using the standard normal curve, find the area to the left of
z = -2.26.
ANS:0.0119
PTS:1OBJ:Section 7.3
13.Using the standard normal curve, find the area between z = 0
and z = 3.3.
____________________.
ANS:0.50
PTS:1OBJ:Section 7.3
14.Determine P(z 1.58) for the standard normal curve.
ANS:0.0571
PTS:1OBJ:Section 7.3
15.For the standard normal curve, determine the area to the
right of z = 0.34.
ANS:0.3669
PTS:1OBJ:Section 7.3
16.Assuming a standard normal distribution, determine P(z
-0.75).
ANS:0.2266
PTS:1OBJ:Section 7.3
17.Determine P(-2.45 z 1.69), assuming a standard normal
distribution.
ANS:0.9474
PTS:1OBJ:Section 7.3
18.Determine P(-2.45 z 0), assuming a standard normal
distribution.
ANS:0.4929
PTS:1OBJ:Section 7.3
19.Assume the standard normal curve and determine the
probability or area between z = -1.28 and z = 1.28.
ANS:0.7994
PTS:1OBJ:Section 7.3
20.Find z1 if the area to the left of a positive z1 is
0.9131.
ANS:1.36
PTS:1OBJ:Section 7.3
21.Find the z-score that determines that the area to the left of
z is 0.9292.
ANS:1.47
PTS:1OBJ:Section 7.3
22.Find the z-score that determines that the area to the right
of z is 0.8264.
ANS:-0.94
PTS:1OBJ:Section 7.3
23.A continuous random variable x is normally distributed with a
mean of 120 grams and a standard deviation of 30 grams. Convert x =
106 into its corresponding z-score.
ANS:-0.47
PTS:1OBJ:Section 7.3
24.A continuous random variable x is normally distributed with a
mean of $250 and a standard deviation of $50. Find the z-score for
x = 175.
ANS:-1.50
PTS:1OBJ:Section 7.3
25.A continuous random variable x is normally distributed with a
mean of $250 and a standard deviation of $50. Find the z-score for
x = 285.
ANS:0.70
PTS:1OBJ:Section 7.3
26.For a normal curve, if the mean is 20 minutes and the
standard deviation is 5 minutes, find the area to the left of 10
minutes.
ANS:0.0228
PTS:1OBJ:Section 7.3
27.For a normal curve, if the mean is 20 minutes and the
standard deviation is 5 minutes, find the area to the right of 12
minutes.
ANS:0.9452
PTS:1OBJ:Section 7.3
28.The age of customers for a particular retail store follows a
normal distribution with a mean of 37.5 years and a standard
deviation of 7.6 years. What is the probability that the next
customer who enters the store will be more than 31 years old?
ANS:0.8051
PTS:1OBJ:Section 7.3
29.The age of customers for a particular retail store follows a
normal distribution with a mean of 37.5 years and a standard
deviation of 7.6 years. What is the probability that the next
customer who enters the store will be less than 42 years old?
ANS:0.7224
PTS:1OBJ:Section 7.3
30.The age of customers for a particular retail store follows a
normal distribution with a mean of 37.5 years and a standard
deviation of 7.6 years. What is the probability that the next
customer who enters the store will be between 40 and 45 years
old?
ANS:0.2096
PTS:1OBJ:Section 7.3
31.The average waiting time at the checkout counter for a large
grocery chain is 2.45 minutes with a standard deviation of 24
seconds (0.40 minutes). Assume that the distribution of waiting
time is normal. What is the probability that a customer must wait
more than 3 minutes for check out?
ANS:0.0838
PTS:1OBJ:Section 7.3
32.The average waiting time at the checkout counter for a large
grocery chain is 2.45 minutes with a standard deviation of 24
seconds (0.40 minutes). Assume that the distribution of waiting
time is normal. What proportion of the customers are served in
between 1 minute and 2.5 minutes?
ANS:0.5517
PTS:1OBJ:Section 7.3
33.Suppose the monthly demand for automobile tires at a tire
dealer is normally distributed with a mean of 250 tires and a
standard deviation of 50 tires. How many tires must the store have
in inventory at the beginning of each month in order to meet demand
95 percent of the time?
ANS:332
PTS:1OBJ:Section 7.3
34.A company sells toothpaste in a tube advertised to contain 8
ounces. The tube filling process is set with a mean of 8.21 ounces.
In this continuous production process, the amount of toothpaste put
in a tube is normally distributed with a mean of 8.21 ounces and a
standard deviation of 0.09 ounces. If the actual capacity of the
tubes used is 8.45 ounces, what proportion of the tubes will be
filled beyond capacity?
ANS:0.0038
PTS:1OBJ:Section 7.3
35.A manufacturer of nails packages 16-penny nails in 25-pound
boxes. Filling of the boxes is automated with the dispenser set to
an average of 25.1 pounds with a standard deviation of 0.25 pounds.
One 25-pound box of 16-penny nails is included in the estimate for
a remodeling contract by a local builder. What is the probability
that the box purchased will have less than the required 25
pounds?
ANS:0.3446
PTS:1OBJ:Section 7.3
36.The Independent Bank surveyed the status of student accounts
and found that the average overdraft was $21.22 with a standard
deviation of $5.49. If the distribution is normal, find the
probability of a student being overdrawn by more than $18.75.
ANS:0.6736
PTS:1OBJ:Section 7.3
37.A circus performer who gets shot from a cannon is supposed to
land in a safety net positioned at the other end of the arena. The
distance he travels is normally distributed with a mean of 175 feet
and a standard deviation of 15 feet. His landing net is 50 feet
long and the mid-point of the net is positioned 175 feet from the
cannon. What is the probability that the performer will hit the net
on a given night?
ANS:0.9050
PTS:1OBJ:Section 7.3
38.A circus performer who gets shot from a cannon is supposed to
land in a safety net positioned at the other end of the arena. The
distance he travels is normally distributed with a mean of 175 feet
and a standard deviation of 15 feet. His landing net is 50 feet
long and the mid-point of the net is positioned 175 feet from the
cannon. What is the probability that the performer will miss the
net on a given night?
ANS:0.0950
PTS:1OBJ:Section 7.3
39.The recent average starting salary for new college graduates
in computer information systems is $47,500. Assume salaries are
normally distributed with a standard deviation of $4,500. What is
the probability of a new graduate receiving a salary between
$45,000 and $50,000?
ANS:0.4246
PTS:1OBJ:Section 7.3
40.The speed of cars passing through a checkpoint follows a
normal distribution with a mean of 62.6 miles per hour and a
standard deviation of 3.7 miles per hour. What is the probability
that the next car passing by will be exceeding 65.5 miles per
hour?
ANS:0.2177
PTS:1OBJ:Section 7.3
41.The speed of cars passing through a checkpoint follows a
normal distribution with a mean of 62.6 miles per hour and a
standard deviation of 3.7 miles per hour. What is the probability
that the next car passing by will be exceeding 58.1 miles per
hour?
ANS:0.8880
PTS:1OBJ:Section 7.3
42.The speed of cars passing through a checkpoint follows a
normal distribution with a mean of 62.6 miles per hour and a
standard deviation of 3.7 miles per hour. What is the probability
that the next car passing by will be traveling between 61 and 70
miles per hour?
ANS:0.6436
PTS:1OBJ:Section 7.3
43.The recent average starting salary for new college graduates
in computer information systems is $47,500. Assume salaries are
normally distributed with a standard deviation of $4,500. What is
the probability of a new graduate receiving a salary between
$45,000 and $50,000?
ANS:0.4246
PTS:1OBJ:Section 7.3
44.A manufacturer of washing machines has experienced a 2%
repair rate. In a city where 120,000 of its machines are located,
the company plans to open its own repair service and needs to
determine the number of repair workers to hire. Each worker can
handle calls for 1000 machine repairs. If the company wants to
cover calls 90% of the time, how many workers must it hire for the
new repair facility?
ANS:2.46
PTS:1OBJ:Section 7.4
45.An elementary school teacher learned that 40 percent of
school age children have at least 3 cavities. The teacher has 30
students in his class. How many students would he expect in his
class to have at least three cavities?
ANS:12
PTS:1OBJ:Section 7.4
46.An elementary school teacher learned that 40 percent of
school age children have at least 3 cavities. The teacher has 30
students in his class. What is the standard deviation for the
number of school age children that have at least 3 cavities?
ANS:2.683
PTS:1OBJ:Section 7.4
47.The selling price of various homes in a community follows the
normal distribution with a mean of $176,000 and a standard
deviation of $22,300. What is the probability that the next house
will sell for less than $190,000?
ANS:0.7357
PTS:1OBJ:Section 7.3
48.The selling price of various homes in a community follows the
normal distribution with a mean of $176,000 and a standard
deviation of $22,300. What is the probability that the next house
will sell for less than $158,000?
ANS:0.2090
PTS:1OBJ:Section 7.3
49.The selling price of various homes in a community follows the
normal distribution with a mean of $176,000 and a standard
deviation of $22,300. What is the probability that the next house
will sell for between $150,000 and $168,000?
ANS:0.2384
PTS:1OBJ:Section 7.3
50.An elementary school teacher learned that 40 percent of
school age children have at least 3 cavities. The teacher has 30
students in his class. What is the probability that 10 students
from this class have at least 3 cavities?
ANS:0.1115
PTS:1OBJ:Section 7.4
51.Suppose that x is a binomial random variable with n = 100 and
= 0.18. Employ the normal approximation to find P(15 x 21).
ANS:0.6976
PTS:1OBJ:Section 7.4
52.Let x be a binomial random variable. Find P(x 40) if n = 100
and = 0.32.
ANS:0.0537
PTS:1OBJ:Section 7.4
53.Women make up 58% of the U. S. civilian workforce of 124
million. The U. S. Department of Commerce randomly selects 100
workers for a conference on national health care. What is the
probability that more than 20 of these workers are female?
ANS:0.9999 or approximately 1
PTS:1OBJ:Section 7.4
54.Women make up 58% of the U. S. civilian workforce of 124
million. The U. S. Department of Commerce randomly selects 100
workers for a conference on national health care. If there are
fewer than 45 or more than 65 females invited, severe political
ramifications are possible. What is the probability that the
conference planners will be criticized for the representation by
women at the conference?
ANS:0.0674
PTS:1OBJ:Section 7.4
55.A coin is flipped 14 times. Using the normal distribution,
calculate the probability of observing either 4, 5, or 6 heads.
ANS:0.3629
PTS:1OBJ:Section 7.4
56.An efficiency expert makes periodic checks for weighting
errors for a long distance shipping firm. The expert inspects for
errors in weighing, in recording weights, and errors in processing
the bills of lading. Based on past records, the number of weekly
errors for all shipments averages 5.30 with a standard deviation of
1.23, and the frequency histogram approximates a normal
distribution. Suppose x is the number of weighing errors that will
occur next week. Compute the approximate probability for (x =
6).
ANS:0.2729
PTS:1OBJ:Section 7.4
57.A statistics class is composed of 60 percent females. If 15
students are selected randomly, what is the probability that this
group will include either 8, 9, 10, or 11 females? Use the normal
distribution.
ANS:0.6916
PTS:1OBJ:Section 7.4
58.An efficiency expert makes periodic checks for weighting
errors for a long distance shipping firm. The expert inspects for
errors in weighing, in recording weights, and errors in processing
the bills of lading. Based on past records, the number of weekly
errors for all shipments averages 5.30 with a standard deviation of
1.23, and the frequency histogram approximates a normal
distribution. Suppose x is the number of weighing errors that will
occur next week.
Compute the approximate probability for (4 x 7).
ANS:0.8912
PTS:1OBJ:Section 7.4
59.An efficiency expert makes periodic checks for weighting
errors for a long distance shipping firm. The expert inspects for
errors in weighing, in recording weights, and errors in processing
the bills of lading. Based on past records, the number of weekly
errors for all shipments averages 5.30 with a standard deviation of
1.23, and the frequency histogram approximates a normal
distribution. Suppose x is the number of weighing errors that will
occur next week.
Compute the approximate probability for (x 2).
ANS:0.0113
PTS:1OBJ:Section 7.4
60.In a paper mill, which produces newsprint from Southern Pine,
the production manager finds a flaw for each 1000 feet of
newsprint. He knows that the outcome is a Poisson process with a
continuous distribution. If x = feet between flaws, what is the
probability of finding 600 or more feet between the next two
flaws?
ANS:0.5488
PTS:1OBJ:Section 7.5
61.In a paper mill, which produces newsprint from Southern Pine,
the production manager finds a flaw for each 1000 feet of
newsprint. He knows that the outcome is a Poisson process with a
continuous distribution. A group of visitors is touring the plant.
After arriving at the final stage of production, if they remain
long enough to observe 330 feet of newsprint produced, what is the
probability that they will miss seeing the next defect?
ANS:0.2811
PTS:1OBJ:Section 7.5
62.In 1997, private planes had 1.5 fatal crashes per 100
thousand flying hours. For a continuous random variable with an
exponential distribution: What is the probability that the time
between the next two crashes will fall between 50 and 70 thousands
hours, i.e. P(50 x 70); x = thousands of flying hours between fatal
crashes.
ANS:0.1224
PTS:1OBJ:Section 7.5
63.In 1997, private planes had 1.5 fatal crashes per 100
thousand flying hours. Assume a continuous random variable with a
exponential distribution. A national association of private plane
owners has 560 members who fly 1.5 million miles each year. The
mileage crash rate has been estimated to be 20 per 100 million
miles of air travel. What is the probability that the next crash by
a member of the association will not occur until more than one year
from now?
ANS:0.2592
PTS:1OBJ:Section 7.5
64.Grades for a statistics exam for a certain class follow the
normal probability distribution with a mean of 82 and a standard
deviation of 12. What percentage of the students in the class had a
grade 81 or more?
ANS:0.5319
PTS:1OBJ:Section 7.3
65.Grades for a statistics exam for a certain class follow the
normal probability distribution with a mean of 82 and a standard
deviation of 12. What percentage of the students in the class
earned a C grade (70-79)?
ANS:0.2426
PTS:1OBJ:Section 7.3
66.Grades for a statistics exam for a certain class follow the
normal probability distribution with a mean of 82 and a standard
deviation of 12. What percentage of the students in the class
earned a B grade (80-89)?
ANS:0.2865
PTS:1OBJ:Section 7.3
67.The time it takes a technician to fix a computer problem is
exponentially distributed with a mean of 15 minutes. What is the
probability that it will take a technician less than 10 minutes to
fix a computer problem?
ANS:0.4866
PTS:1OBJ:Section 7.5
68.The time it takes a technician to fix a computer problem is
exponentially distributed with a mean of 15 minutes. What is the
probability that it will take a technician between 10 to 15 minutes
to fix a computer problem?
ANS:0.1455
PTS:1OBJ:Section 7.5
69.A manufacturer produces Walkman tap players/radios. The
electronic components are positioned in the outer case after the
case has been given an epoxy finish. The production manager must
time the drying process with the total assembly line and therefore
must determine the appropriate drying time. An intensive cost
analysis indicated that a risk of just 0.3% of the cases being wet
and therefore smudged was acceptable. If drying times are normally
distributed with a mean of 3.6 minutes and a standard deviation of
0.6 minutes, what time setting should the production manager set
for the automatic timer which controls the production line?
____________________ minutes
ANS:5.25
PTS:1OBJ:Section 7.3
70.The time it takes a technician to fix a computer problem is
exponentially distributed with a mean of 15 minutes. What is the
variance of the time it takes a technician to fix a computer
problem?
____________________ minutes
ANS:225
PTS:1OBJ:Section 7.5
71.The total area beneath the curve of any continuous
distribution is ____________________.
ANS:1.0
PTS:1OBJ:Section 7.7
72.The probability that any continuous random variable x will
take on any specific value along an interval is
____________________.
ANS:0.0
PTS:1OBJ:Section 7.7
COMPLETION
1.In the normal distribution, the total area under the curve is
equal to ____________________.
ANS:one1
PTS:1OBJ:Section 7.2
2.In the normal distribution, the right half of the curve is a
mirror image of the _________________________, since the
distribution is ____________________.
ANS:left half; symmetric
PTS:1OBJ:Section 7.2
3.If we ____________________ the normal curve, we express the
original x values in terms of their number of standard deviations
away from the mean.
ANS:standardize
PTS:1OBJ:Section 7.3
4.The binomial distribution is symmetrical whenever the
population proportion is ____________________, and approaches
symmetry for values that are close to ____________________.
ANS:0.5; 0.5
PTS:1OBJ:Section 7.4
5.For a Poisson process, the ____________________ distribution
describes the continuous random variable x, where x is the amount
of time, space, or distance between occurrences of these rare
events.
ANS:exponential
PTS:1OBJ:Section 7.5
SHORT ANSWER
1.In 2000, the average charge of tax preparation was $95.
Assuming a normal distribution and a standard deviation of $10, use
the approximate areas beneath the normal curve, as discussed in
this section, to answer: What proportion of tax preparation fees
were exactly $100?
ANS:P(x = 100) = 0
PTS:1OBJ:Section 7.2
2.If z is a standard normal random variable, find the value z1
for which:
A) P(0 zz1) = 0 .276
B) P(zz1) = 0.341
C) P(zz1) = 0.819
D) P(-z1 Zz1) = 0.785
ANS:0.76; 0.41; -0.91; 1.24
PTS:1OBJ:Section 7.3
3.If x is a normal random variable with a mean of 100 and a
standard deviation of 10, find the following probabilities:
A) P(x 128)
B) P(x 113)
C) P(87 x 98)
ANS:0.0026; 0.9032; 0.3239
PTS:1OBJ:Section 7.3
4.A computer statistical package has simulated 1000 random
observations from a normal distribution with mean = 50 and standard
deviation = 10. Sketch the approximate box-and-whisker plot for the
resulting data.
ANS:
PTS:1OBJ:Section 7.6
5.If a computer statistical package were to simulate 500 random
observations from a normal distribution with mean = 100 and
standard deviation = 50, what percentage of these observations
would you expect to have a value of 200 or more? Do you think the
actual number in the " 200" range would equal the expected number
in this range? If so, why? If not, why not?
ANS:2.28%; No. We would expect 11.4 of the 500 observations
(2.28%) to have a value of 200 or more. The actual number would not
be equal to the expected number. However, the more observations we
select, the closer we will tend to come to what we expect.
PTS:1OBJ:Section 7.6
6.Scores of high school students on a national mathematics exam
in Egypt were normally distributed with a mean of 86 and a standard
deviation of 4.
A) What is the probability that a randomly selected student will
have a score of 80 or higher?B) If there were 97,680 students with
scores higher than 91, how many students took the test?
____________________ (Remove all commas from your answer before
submitting).
ANS:0.9332; 925000
PTS:1OBJ:Section 7.3
7.The time it takes a technician to fix a computer problem is
exponentially distributed with a mean of 15 minutes. What is the
probability density function for the time it takes a technician to
fix a computer problem?
ANS:f(x) = (1/15) e-x/15, x 0
PTS:1OBJ:Section 7.5
8.What are the mean and standard deviation of a normally
distributed random variable, which has been "standardized"?
Mean = ____________________
SD = ____________________
ANS:0.0; 1
PTS:1OBJ:Section 7.3
9.A certain brand of flood lamps has a lifetime that is normally
distributed with a mean of 3,750 hours and a standard deviation of
300 hours.
A). What proportion of these lamps will last for more than 4,000
hours?
B).What lifetime should the manufacturer advertise for these
lamps in order that only 2% of the lamps will burn out before the
advertised lifetime?
____________________ hours (Remove all commas from your answer
before submitting).
ANS:0.2033; 3132
PTS:1OBJ:Section 7.3
10.The normal distribution is a very good approximation to the
binomial distribution whenever ____________________ and
____________________ are 5.
ANS:n; n(1-)
PTS:1OBJ:Section 7.4
11.Complete the following table indicating which procedure to
use in calculating binomial probabilities:
nProcedure
Large0.2
Large0.0
Large0.9
Small0.0
ANS:
nProcedure
Large0.2 Use the normal approximation
Large0.0 Use the Poisson approximation
Large0.9 Use the Poisson approximation
Small0.0 Use the binomial probability formula
PTS:1OBJ:Section 7.4
12.Although the binomial distribution is discrete and the normal
distribution is continuous, the normal distribution is a good
approximation to the binomial whenever both ____________________
and ____________________ are 5, where n = number of trials, and =
the probability of success in any given trial, are the parameters
of the binomial distribution.
ANS:n; n(1-
PTS:1OBJ:Section 7.7
13.Regardless of the shape of a particular normal curve, about
what percentage of the area is within , respectively, where is the
mean and is the standard deviation.
ANS:95.5% and 99.7% respectively
PTS:1OBJ:Section 7.7
ESSAY
1.What are continuous probability distributions?
ANS:Continuous probability distributions describe probabilities
associated with random variables that are able to assume any of an
infinite number of values along an interval.
PTS:1OBJ:Section 7.1
2.Why is the probability that a continuous random variable takes
on any specific value equal to zero?
ANS:The probability that a continuous random variable takes on
any specific value is equal to zero because there is an infinite
number of possible values.
PTS:1OBJ:Section 7.1
3.Explain why the total area beneath a probability density
function is equal to 1.0.
ANS:The area beneath the probability density function represents
the probability of the random variable, x, being between . Since x
must be between (this is a certain event), the area must be equal
to 1.0.
PTS:1OBJ:Section 7.1
4.Identify the distribution shown in the following graph.
Indicate the approximate areas that will lie beneath the curve for
the intervals shown.
ANS:
PTS:1OBJ:Section 7.2
5.What does it mean when we say a random variable is
standardized?
ANS:A random variable x is standardized when each value of x has
(the mean of x) subtracted from it, and the difference is divided
by (the standard deviation of x.)
PTS:1OBJ:Section 7.3
6.Explain what we mean by saying a random variable is
approximately normally distributed.
ANS:If the probabilities for the outcomes of the random variable
are approximately equal to the areas under the normal curve, its
distribution is approximately normal.
PTS:1OBJ:Section 7.3
7.Express in your own words the procedure for finding the area
under the standard normal curve between z = 0 and z = 1.53.
ANS:Use the standard normal table in text. Trace down the left
hand column to "1.5" and go across the row until reaching the
column headed at the top with ".03". The value is 0.4370, which is
the area between z = 0 and z = 1.53.
PTS:1OBJ:Section 7.3
8.Describe the method of finding the area under the standard
normal curve between z = 0 and z =- 1.33.
ANS:Recall the property of symmetry. The area to the left of 0
to - 1.33 is the same as for the area to the right of 0 to 1.33. Go
down the left most column of the standard normal table to the row
with "1.3" and right across the rows until in the column headed at
the top with ".03". The value is 0.4082, which is the area between
z = 0 and z = -1.33.
PTS:1OBJ:Section 7.3
9.Use the properties of the standard normal curve to describe
the method of finding the area to the right of z = 1.42.
ANS:Since the standard normal table in text gives areas between
0 and positive values of z, to find the area to the right of a z
value requires the use of the symmetry property which places .5 of
the whole area to the right of z = 0. Trace down the left most
column to the value "1.4" and across the row to the column headed
by the value ".02". The value is 0.4222, which is the area between
z = 0 and z = 1.42. Subtract this area from 0.5 to obtain the
desired area of 0.0778.
PTS:1OBJ:Section 7.3
10.For a value of z = -1.59, explain the steps needed to find
the area to the left of this value.
ANS:By symmetry the area to the right of 0 is the same as the
area to the left of 0 for values of z. In finding the area to the
left of a negative value of z use the symmetry property to reason
that .5 of the area lies to the left of 0. Go down the left most
column to the value "1.5" and across the row until reaching the
column headed by "9". The value is 0.4441, which is the area
between z = 0 and z = -1.59. Subtract this area from 0.5 to obtain
the desired area of 0.0559.
PTS:1OBJ:Section 7.3
11.Why is it important to use the correction for continuity when
approximating binomial probabilities with the normal
distribution?
ANS:Because the binomial distribution has gaps between possible
values of x (since it is discrete), while the normal distribution
is continuous, the normal approximation to the binomial involves a
correction for continuity. The correction consists of expanding
each possible value of the discrete random variable x by 0.5 in
each direction.
PTS:1OBJ:Section 7.4
12.A continuous random variable x has the following probability
density function:
f(x) = 2e-2x , x 0
What is the distribution of the random variable x?
ANS:Exponential distribution with = 2
PTS:1OBJ:Section 7.5