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MTH3003 PJJ
SEM I 2015/2016

ASSIGNMENT :25%
Assignment 1 (10%)
Assignment 2 (15%)

Mid exam :30%
Part A (Objective)
Part B (Subjective)

Final Exam: 40%
Part A (Objective)
Part B (Subjective - Short)
Part C (Subjective – Long)
oDefinition
oGraphing
MEASURES OF CENTER
- Arithmetic Mean or Average
- Median
- Mode
Group and ungrouped data
•
•
•
•
Range
Interquartile Range
Variance
Standard Deviation
Group an ungrouped data
interpret
Calculate
 Q1, Q2 and Q3, IQR, Upper fence,
lower fence, outlier
• The lower and upper quartiles (Q1 and Q3), can
be calculated as follows:
0.25(n + 1)
• The position of Q1 is
•The position of Q3 is
0.75(n + 1)
once the measurements have been ordered.
If the positions are not integers, find the
quartiles by interpolation.
Example
The prices ($) of 18 brands of walking shoes:
40 60 65 65 65 68 68 70 70
70 70 70 70 74 75 75 90 95
Position of Q1 = 0.25(18 + 1) = 4.75
Position of Q3 = 0.75(18 + 1) = 14.25
• Basic
concept
• The probability of an event - how
to find prob
• Counting rules
• Calculate probabilities
 Event
Relations: Union, Intersection, Complement
 Calculating Probabilities for
Unions
The Additive Rule for Unions
A Special Case – Mutually Exclusive
Complements
Intersections
Independent and Dependent Events
Conditional Probabilities
The Multiplicative Rule for Intersections
Probability Distributions for
Discrete Random Variables
 Properties
for Discrete Random
Variables
 Expected Value and Variance
 The
properties for a discrete probability
function (PMF) are:
p( x)  P( X  x)
0  p( x)  1 x
 p( x)  1
all x
 Cumulative
Distribution Function (CDF)
F ( x)  P( X  x)
F (b)  P ( X  b) 
b
 p( x)
y  
F ()  0
F ( )  1
 Toss
a fair coin three times and
define X = number of heads.
x
HHH
1/8
3
HHT
1/8
2
HTH
1/8
2
THH
1/8
2
HTT
1/8
1
THT
1/8
1
TTH
1/8
1
TTT
1/8
0
P(X = 0) =
P(X = 1) =
P(X = 2) =
P(X = 3) =
1/8
3/8
3/8
1/8
X
0
1
p(x)
2
3
3/8
1/8
1/8
3/8
Discrete distributions:

The binomial distribution

The Poisson distribution

The hypergeometric distribution
 To
find probabilities
formula
cumulative table
I. The Binomial Random Variable
1. Five characteristics: n identical independent trials,
each resulting in either success S or failure F;
probability of success is p and remains constant from
trial to trial; and x is the number of successes in n
trials.
2. Calculating binomial probabilities
nk
a. Formula: P( x  k )  Ck p q
b. Cumulative binomial tables
3. Mean of the binomial random variable: m  np
4. Variance and standard deviation: s 2  npq and
n
s  npq
k
A marksman hits a target 80% of the
time. He fires five shots at the target. What is the
probability that exactly 3 shots hit the target?
P(x = 3) = P(x  3) – P(x  2)
= .263 - .058
= .205
Check from
formula:
P(x = 3) = .205
II. The Poisson Random Variable
1. The number of events that occur in a period of time
or space, during which an average of m such events are
expected to occur. Examples:
•
•
The number of calls received by a switchboard during a given
period of time.
The number of machine breakdowns in a day
2. Calculating Poisson probabilities
m k e m
P( x  k ) 
a. Formula:
k!
b. Cumulative Poisson tables
3. Mean of the Poisson random variable: E(x)  m
4. Variance and standard deviation: s 2  m and
s m
III. The Hypergeometric Random Variable
1. The number of successes in a sample of size n from a
finite
population containing M successes and N  M failures
2. Formula for the probability of k successes in n trials:
CkM CnMk N
P( x  k ) 
CnN
3. Mean of the hypergeometric random variable:
M 
m  n 
N
4. Variance and standard deviation:
 M  N  M  N  n 



N
N
N

1
 


s 2  n
A package of 8 AA batteries contains 2
batteries that are defective. A student
randomly selects four batteries and replaces
the batteries in his calculator. What is the
probability that all four batteries work?
Success = working
battery
N=8
M=6
n=4
6
4
2
0
CC
P( x  4) 
8
C4
6(5) / 2(1)
15


8(7)(6)(5) / 4(3)( 2)(1) 70
The Standard Normal Distribution
1. The normal random variable z has mean 0 and standard
deviation 1.
2. Any normal random variable x can be transformed to a
standard normal random variable using
z
xm
s
3. Convert necessary values of x to z.
4. Use Normal Table to compute standard normal
probabilities.
The weights of packages of ground beef are
normally distributed with mean 1 pound and
standard deviation 0.1. What is the probability
that a randomly selected package weighs
between 0.80 and 0.85 pounds?
P(.80  x  .85) 
P(2  z  1.5) 
.0668  .0228  .0440
We can calculate binomial probabilities using
 The binomial formula
 The cumulative binomial tables
 When n is large, and p is not too close to zero or one,
areas under the normal curve with mean np and
variance npq can be used to approximate binomial
probabilities.

Make sure to include the entire rectangle
for the values of x in the interval of interest.
That is, correct the value of x by 0.5 This
is called the continuity correction.
Standardize the values of x using
( x  0.5)  np
z
npq
Make sure that np and nq are both
greater than 5 to avoid inaccurate
approximations!
Suppose x is a binomial random variable
with n = 30 and p = .4. Using the normal
approximation to find P(x  10).
n = 30
np = 12
p = .4
q = .6
nq = 18
The normal
approximation
is ok!
Calculate
m  np  30(.4)  12
s  npq  30(.4)(.6)  2.683
10.5  12
P( x  10)  P( z 
)
2.683
 P( z  .56)  .2877
 Sampling


Sampling distribution of the sample mean
Sampling distribution of a sample proportion
 Finding


Distributions
Probabilities for the
Sample Mean
Sample Proportion
A random sample of size n is selected from a
population with mean m and standard deviation s.
The sampling distribution of the sample mean
have mean m and standard deviation s / n .
x
will
If the original population is normal, the sampling
distribution will be normal for any sample size.
If the original population is non normal, the sampling
distribution will be normal when n is large.
The standard deviation of x-bar is sometimes called the
STANDARD ERROR (SE).
If the sampling distribution of x is normal or
approximately normal, standardize or rescale the
interval of interest in terms of
z
x m
s/ n
Find the appropriate area using Z Table.
Example: A random
sample of size n = 16 from
a normal distribution with
m = 10 and s = 8.
12  10
P ( x  12 )  P ( z 
)
8 / 16
 P ( z  1)  1  .8413  .1587
A random sample of size n is selected from a binomial
population with parameter p.
x
The sampling distribution of the sample proportion, pˆ 
n
pq
will have mean p and standard deviation
n
If n is large, and p is not too close to zero or one, the
sampling distribution of p̂ will be approximately
normal.
The standard deviation of p-hat is sometimes
called the STANDARD ERROR (SE) of p-hat.
If the sampling distribution of p̂ is normal or
approximately normal, standardize or rescale the
p̂  p
interval of interest in terms of
If both np > 5 and
z
pq
n
np(1-p) > 5
Find the appropriate area using Z Table.
Example: A random
sample of size n = 100
from a binomial
population with p = 0.4.
.5  .4
P ( pˆ  .5)  P ( z 
)
.4(.6)
100
 P ( z  2.04)  1  .9793  .0207