SQQS1013 Chapter 5
SQQS1013 Chapter 5
SQQS1013 Chapter 5
5.1
SPECIAL
DISTRIBUTIONS
DISCRETE DISTRIBUTION
Consists of the values that a random variable can assume and the
corresponding probabilities of the values.
For example:
Where;
n =
p =
q =
x =
n-x =
Example 1
Compute the probabilities of X successes, using the binomial formula; n= 6, X= 3,
p=0.03
Solution:
a)
Example 2
A survey found that one out of five Malaysians says he or she has visited a doctor in
any given month. If 10 people are selected at random, find the probability that
exactly 3 will have visited a doctor last month.
Solution:
The probabilities for a binomial experiment can also be read from the table of
binomial probabilities.
How to read the probability value from the table of binomial probabilities
Determining P (X 3) for n = 6 and p = 0.30
The table of cumulative binomial probabilities (less than)
x
P( X x) nCr p r (1 p)n r
p = 0.30
r 0
p
n
6
x
0
1
2
3
4
5
6
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.5314
0.3771
0.2621
0.1780
0.1176
0.0754
0.0467
0.8857
0.7765
0.6554
0.5339
0.3191
0.2333
0.9842
0.9527
0.9011
0.8306
0.4202
0.7443
0.6471
0.5443
0.9987
0.9941
0.9830
0.9624
0.8826
0.8208
0.9999
0.9996
0.9984
0.9954
0.9295
0.9891
0.9777
0.9590
1.0000
1.0000
0.9999
0.9998
0.9993
0.9982
0.9959
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
n=6
X3
P (X 3) = 0.9295
If you are using cumulative binomial probabilities table; P(X x); it is easier to calculate
values from a binomial distribution.
Less than
At least
Greater than
From x1 to x2
Between x1 and x2
Between x1 to x2
Example 3
Compute the probability of X successes using the Binomial Table where n=2, p=0.30
and X 1
Solution:
P(X 1) = 0.9100
Example 4
25% of all VCR manufactured by a large electronics company are defective. A quality
control inspector randomly selects three VCRs from the production line. What is the
probability that,
a) exactly one of these three VCRs is defective.
b) at least two of these VCRs are defective.
Solution:
EXERCISE 5.1
1. A burglar alarm system has 6 fail-safe components. The probability of each failing is
0.05. Find these probabilities:
a. Exactly 3 components will fail
b. Less than 2 components will fail
c. None will fail
2.
A survey from Teenage Research Unlimited found that 30% of teenage consumers
receive their spending money from part-time jobs. If 5 teenagers are selected at
random, find the probability that at least 3 of them will have part-time jobs.
3.
R. H Bruskin Associates Market Research found that 40% of Americans do not think
that having a college education is important to succeed in the business world. If a
random sample of five American is selected, find these probabilities.
a. Exactly two people will agree with that statement.
b. At most three people will agree with that statement
c. At least two people will agree with that statement
d. Fewer than three people will agree with that statement.
4. It was found that 60% of American victims of health care fraud are senior citizens. If
10 victims are randomly selected, find the probability that exactly 3 are senior citizens.
5. If 65% of the people in a community use the gym facilities in one year, find these
probabilities for a sample of 10 people.
a. Exactly four people used the gym facilities.
b. At least six people not used the gym facilities.
6. In a poll of 12 to 18 year old females conducted by Harris Interactive for the Gillette
Company, 40% of the young female said that they expected the US to have a female
president within 10 years. Suppose a random sample of 15 females from this age
group selected. Find the probabilities that of young female in this sample who expect
a female president within 10 years is;
a. at least 9
b. at most 5
c. 6 to 9
d. in between 4 and 8
e. less than 4
Mean = = np
Variance = 2 = npq
Standard deviation = =
npq
where;
n = the total number of trials
p = probability of success
q = 1-p = probability of failure
Example 5
Find the mean, variance and standard deviation for each of the values of n and p when
the conditions for the binomial distribution are met.
a. n=100, p=0.75
Solution:
a. = np = 100 (0.75) = 75
EXERCISE 5.2
1. It has been reported that 83% of federal government employees use e-mail. If
a sample of 200 federal government employees is selected, find the mean,
variance and standard deviation of the number who use e-mail.
2. A survey found that 25% of Malaysian watch movies at the cinema. Find the
mean, variance and standard deviation of the number of individuals who watch
movies at the cinema, if a random sample of 1000 Malaysian is selected.
The number of television sets sold at a department store during a given week.
A certain type of fabric made contains an average 0.5 defects per 500 yards.
x
P(X)= e
x!
where;
: mean number of occurrences in that interval
e : approximately 2.7183
Find ex on your calculator
Example 7
Find probability P(X; ), using the Poisson formula for P(5;4)
Solution:
e 4 (45 )
P(X=5) =
5!
= 0.1563
Example 8
If there are 200 typographical errors randomly distributed in a 500-page manuscript, find
the probability that a given page contains exactly three errors.
Solution:
The probabilities for a Poisson distribution can also be read from the table
of Poisson probabilities.
How to read the probability value from the table of Poisson probabilities
Determining P (X 3) for = 1.5
The table of cumulative Poisson probabilities (less than)
e k
k!
k 0
x
P( X x)
= 1.5
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0.3329
0.3012
0.2725
0.2466
0.2231
0.2019
0.1827
1
2
0.6990
0.6626
0.6268
0.5918
0.5578
0.5249
0.4932
0.9004
0.8795
0.8571
0.8335
0.7834
0.7572
3
4
0.9743
0.9662
0.9569
0.9463
0.8088
0.9344
0.9212
0.9068
0.9946
0.9923
0.9893
0.9857
0.9814
0.9763
0.9704
0.9990
0.9985
0.9978
0.9968
0.9955
0.9940
0.9920
0.9999
0.9997
0.9996
0.9994
0.9991
0.9987
0.9981
1.0000
1.0000
0.9999
0.9999
0.9998
0.9997
0.9996
X3
P (X 3) = 0.9344
If you are using a cumulative Poisson probabilities table; P(X x); it is easier to
calculate.
Example 9
Find the probability P(X; ); using Poisson table; P(10;7)
Solution:
P(X = 10) = P(X 10) P(X 9)
= 0.9015 0.8305
= 0.0710
Example 10
A sales firm receives an average of three calls per hour on its toll-free number. For any
given hour, find the probability that it will receive the following:
a)
b)
c)
d)
at most 3 calls
at least 3 calls
five or more calls
between 1 to 4 calls in 2 hours
Solution:
= 3 calls per hour
a) P(X 3) = 0.6472
b) P(X 3) = 1 P(X 2)
= 1 0.4232
= 0.5768
c) P(X 5) = 1 P(X 4)
= 1 0.8153
= 0.1847
= 6 calls in 2 hours
d) P(1< X 4) = P(X 4) P(X 1)
= 0.2851 0.0174
= 0.2677
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3.
a)
b)
c)
no breakdown in 2 months
In an airport between 19:00 and 20:00 hours, the number of flights that land follows
a Poisson distribution with a mean 0.9 per five minutes interval. Find the probability
that the number of flights that land is:
a) one or less between 19:00 and 19:05 hours
b) more than three between 19:15 and 19:30 hours
4.
6.
An average of 4.8 customers visit Malaysia Savings and Loan every half hour. Find
the probability that during a given hour, the number of customer is
a)
exactly two
b)
at most 2
c)
none
d)
Sports Score Jay receives, on average, eight calls per hour requesting the latest
sports score. For any randomly selected hour, find the probability that the company
will receive
a)
b)
c)
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Mean, =
Variance, 2 =
Standard Deviation, =
Example 11
An auto salesperson sells an average of 0.9 cars per day. Find the mean, variance
and standard deviation of cars sold per day by this salesperson.
Solution:
= = 0.9
= = 0.9 = 0.9487
2 = = 0.9
Example 12
An insurance salesperson sells an average of 1.4 policies per day.
a)
Find the probability that this salesperson will sell no insurance policy on a
certain day.
b)
Find the mean, variance and standard deviation of the probability this
salesperson will sell the policies per day.
Solution:
12
5.2
CONTINUOUS DISTRIBUTION
2. The total probability of all the (mutually exclusive) intervals within which X can
assume a value is 1.0.
13
The normal distribution is the most important and most commonly used
among all of probability distributions.
c)
14
The two tails of the curve extend indefinitely which means that the curve never
touch x- axis.
The value under the curve indicates the proportion of area in each section.
(example figure 2; pg 13)
The units for the standard normal distribution curve are denoted by z and
are called the z values or z scores.
15
The standard normal distribution table lists the areas under the
standard normal curve to the left of z-values from 3.49 to 3.49.
Although the z-values on the left side of the mean are negative, the area
under the curve is always positive.
How to read the probability value from the standard normal distribution table
Determine the area under the standard normal curve to the left of z = 1.95
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Example 13
Find the area under the standard normal curve:
a) To the left of z = 1.56
b) To the right of z = -1.32
c) From z = 0.85 to z = 1.95
Solution:
a) To the left of z = 1.56
P(z < 1.56) = 0.9406
0
1.56
0.85
1.95
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EXERCISE 5.4
Find the area under the standard normal curve:
c) Between z = -2.15 and z=1.67
B.
z=
Where and are the mean and standard deviation of the normal distribution of
X, respectively.
Remember!
The z value for the mean of a normal
distribution is always zero.
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55
b)
35
Solution:
For the given normal distribution, = 50 and =10
a) The z value for X = 55 is computed as follows:
b)
z = 35 50 =
10
Example 15
Let X be a continuous random variable that is normally distributed with a mean of 65 and
a standard deviation of 15. Find the probability that X assumes a value:
a)
less than 43
b)
greater than 74
c)
between 56 and 71
Solution:
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EXERCISE 5.5
1. Let X denote the time taken to run a road race. Suppose X is approximately normally
distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If
one runner is selected at random, what is the probability that this runner will
complete this road race:
a)
b)
2.
The mean number of hours a student spends on the computer is 3.1 hours per day.
Assuming a standard deviation of 0.5 hour, find the percentage of students who
spend less than 3.5 hours on the computer. Assume the variable is normally
distributed.
3.
b)
5.3
INTRODUCTION TO t- DISTRIBUTION
However, the t distribution has larger area in the tails and smaller area in
the centre than does the standardized normal distribution.
Because the value of is uncertain, the values of t that are observed will be
more variable than for z.
Standard normal
t distribution for 5
degrees of freedom
20
For this reason, most statisticians use z instead of t when the sample size is
greater than 120.
EXERCISE 5.6
1. 10 % of the bulbs produced by a factory are defective. A sample of 5 bulbs is
selected randomly and tested for defect. Find the probability that
a) two bulbs are defective
b) at least one bulb is defective
2. In a university, 20 percent of the students fail statistics test. If 20 students from the
university are interviewed, what is the probability of getting:
a) less than 3 students who fail the test
b) more than 3 students who fail the test
c) exactly 4 students who fail the test
3. A financial institution in Kuala Lumpur has a job vacancy for a risk analyst and each
applicant must seat for a written test. Based on the management experience, only
40% of the applicants pass the test and qualify for the interview session. There are
20 applicants who have applied for the jobs. Find the probability that there are more
than 50% of the applicants will pass the test.
4. An Elementary Statistics class has 75 students. If there is a 12% absentee rate per
lesson, find the mean, variance and standard deviation of the number of students
who will be absent from each class.
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b)
c)
6. En. Rostam is a credit officer at the Trust Bank. Based on his experience, he
estimates that on average he will receive 3 loan applications in a week. Find the
probability that he receives:
a)
b)
c)
7. A bookstore owner examines 5 books from each lot of 25 to check for missing pages.
If he finds at least two books with missing pages, the entire lot is returned back to the
supplier. If indeed, there are five books with missing pages, find the probability that
the lot will be returned.
8. The numbers of customers who enter shop ABC are independent of one another and
at random intervals follow a Poisson distribution with an average rate of 42
customers per hour. Find the probability that:
a) no customer enters the shop during a particular 1 minute interval
b) at least 4 customers enter the shop during a particular 5-minute interval
c) between 2 and 6 customers enter the shop during a particular 10-minute
interval.
9. A research has been conducted by Students Affair Department of Menara University
about the PNGK obtained by final semester students for 2002/2203 session. The
outcome of the research showed that the PNGK of the students are normally
distributed with a mean 2.80 and a standard deviation 0.40. If one final semester
student has been selected at random;
a) Calculate the probability that the student gets PNGK from 2.00 until 3.00
b) Find the percentage that the student gets PNGK of less than 2.00
c) Calculate the probability that the student gets PNGK of at least 3.00
d) Calculate the probability that the student gets PNGK of more than 3.70
(first class honors). If the university has 1000 final semester students, find
the number of first class honors students.
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Encik Ahmad works as a lawyer at his own law firm which is situated at Bandar
Kenangan. He drives to his work place every day. The estimated time taken to his
work place is normally distributed with a mean of 24 minutes and a standard
deviation of 3.8 minutes.
a) Calculate the probability of estimated time taken of at least half and hour.
b) Find the probability of estimated time taken from 20 minutes until 25
minutes
c) Find the percentage that the estimated time taken of more than 25
minutes.
d) Find the probability of estimated time taken of less than 10 minutes.
11. Assume X is the time for a runner to finish his 2 km run. Given X is normally
distributed with a mean of 15 minutes and a standard deviation of 3 minutes. If one
runner is selected at random, find the probability that the runner can finish his 2km
run in;
a) less than 13 minutes
b) not more than 16 minutes
c) within 14 minutes and 17 minutes.
12.
Given the systolic blood pressure for the obesity group has a mean 132 mmHg
and a standard deviation 8 mmHg. Assuming the variable normally distributed,
find the probability an obese person that has been selected at random have a
systolic blood pressure:
a) More than 130 mmHg.
b) Less than 140 mmHg.
c) Between 131 mmHg and 136 mmHg.
13.
The number of passenger for domestic flight from Alor Setar to Kuala Lumpur is
normally distributed with mean 80 and standard deviation 12. If one domestic
flight is selected at random, find the probability the flight carries:
a) less than 90 passengers
b) at least 75 passengers
c) between 79 to 95 passengers
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