MTH3003 PJJSEM 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)

oDefinitionoGraphing

MEASURES OF CENTER- Arithmetic Mean or Average- Median- Mode

Group and ungrouped data

• Range• Interquartile Range• Variance• Standard Deviation

Group an ungrouped data

interpretCalculateQ1, Q2 and Q3, IQR, Upper fence, lower fence, outlier

• The lower and upper quartiles (Qlower and upper quartiles (Q1 1 and Qand Q33), ), can be calculated as follows:

• The position of Qposition of Q11 is

0.75(n + 1)

0.25(n + 1)

•The position of Qposition of Q33 is

once the measurements have been ordered. If the positions are not integers, find the quartiles by interpolation.

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

Example

• 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 UnionsThe Additive Rule for Unions

A Special Case – Mutually Exclusive

Complements

Intersections

Independent and Dependent EventsIndependent and Dependent Events

Conditional Probabilities

The Multiplicative Rule for Intersections

Probability Distributions forDiscrete Random VariablesProperties for Discrete Random VariablesExpected Value and Variance

The properties for a discrete probability function (PMF) are:

Cumulative Distribution Function (CDF)

1)(

1)(0

)()(

all

x

xp

xxp

xXPxp

1)(0)(

)()()(

)()(

FF

xpbXPbF

xXPxFb

y

Toss a fair coin three times and define X = number of heads.

1/8

1/8

1/8

1/8

1/8

1/8

1/8

1/8

P(X = 0) = 1/8P(X = 1) = 3/8P(X = 2) = 3/8P(X = 3) = 1/8

P(X = 0) = 1/8P(X = 1) = 3/8P(X = 2) = 3/8P(X = 3) = 1/8

HHHHHH

HHTHHT

HTHHTH

THHTHH

HTTHTT

THTTHT

TTHTTH

TTTTTT

x

3

2

2

2

1

1

1

0

X p(x)

0 1/8

1 3/8

2 3/8

3 1/8

Discrete distributions: The binomialbinomial distribution

The PoissonPoisson distribution

The hypergeometrichypergeometric distributionTo find probabilities

formulacumulative table

I. The Binomial Random VariableI. The Binomial Random Variable1. 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

a. Formula:b. Cumulative binomial tables

3. Mean of the binomial random variable: np 4. Variance and standard deviation: 2 npq and

knknk qpCkxP )(

npq

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(P(xx = 3) = 3) = P(x 3) – P(x 2)= .263 - .058= .205

P(P(xx = 3) = 3) = P(x 3) – P(x 2)= .263 - .058= .205

Check from formula:

P(x = 3) = .205

II. The Poisson Random VariableII. The Poisson Random Variable 1. The number of events that occur in a period of time

or space, during which an average of such events are expected to occur. Examples: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

a. Formula:b. Cumulative Poisson tables

3. Mean of the Poisson random variable: E(x) 4. Variance and standard deviation: 2 and

( )!

keP x k

k

III. The Hypergeometric Random VariableIII. The Hypergeometric Random Variable1. The number of successes in a sample of size n from a finite population containing M successes and N M failures2. Formula for the probability of k successes in n trials:

3. Mean of the hypergeometric random variable:

4. Variance and standard deviation:

N

Mn

N

Mn

12

N

nN

N

MN

N

Mn

12

N

nN

N

MN

N

Mn

Nn

NMkn

Mk

C

CCkxP

)( N

n

NMkn

Mk

C

CCkxP

)(

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?

84

20

64)4(C

CCxP

Success = working battery

N = 8

M = 6

n = 470

15

)1)(2)(3(4/)5)(6)(7(8

)1(2/)5(6

The Standard Normal DistributionThe Standard Normal Distribution1. 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

3. Convert necessary values of x to z.4. Use Normal Table to compute standard normal probabilities.

x

z

x

z

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?

)85.80(. xP

)5.12( zP

0440.0228.0668.

We can calculate binomial probabilities usingThe binomial formulaThe 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 This is called the continuity correction.continuity correction. Standardize the values of x using

( 0.5)x npz

npq

( 0.5)x npz

npq

Make sure that np and nq are both greater than 5 to avoid inaccurate approximations!

0.50.5

Suppose x is a binomial random variable with n = 30 and p = .4. Using the normal approximation to find P(x 10).

n = 30 p = .4 q = .6

np = 12 nq = 18

683.2)6)(.4(.30

12)4(.30

Calculate

npq

np

683.2)6)(.4(.30

12)4(.30

Calculate

npq

np

The normal approximation is ok!

)683.2

125.10()10(

zPxP

2877.)56.( zP

Sampling DistributionsSampling distribution of the sample

meanSampling distribution of a sample

proportion Finding Probabilities for the

Sample MeanSample Proportion

A random sample of size n is selected from a population with mean and standard deviation

he sampling distribution of the sample mean will have mean and standard deviation .

If the original population is normalnormal, , the sampling distribution will be normal for any sample size.

If the original population is non normal, 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).

xn/

1587.8413.1)1(

)16/8

1012()12(

zP

zPxP

1587.8413.1)1(

)16/8

1012()12(

zP

zPxP

If the sampling distribution of is normal or approximately normal standardize or rescale the interval of interest in terms of

Find the appropriate area using Z Table.

If the sampling distribution of is normal or approximately normal standardize or rescale the interval of interest in terms of

Find the appropriate area using Z Table.

Example: Example: A random sample of size n = 16 from a normal distribution with = 10 and = 8.

x

/

xz

n

The standard deviation of p-hat is sometimes called the STANDARD ERROR (SE) of p-hat.

A random sample of size n is selected from a binomial population with parameter p.

The sampling distribution of the sample proportion,

will have mean p and standard deviation

If n is large, and p is not too close to zero or one, the sampling distribution of will be approximately approximately normal.normal.

n

xp ˆ

n

pq

p̂

0207.9793.1)04.2(

)

100)6(.4.

4.5.()5.ˆ(

zP

zPpP

0207.9793.1)04.2(

)

100)6(.4.

4.5.()5.ˆ(

zP

zPpPExample: Example: A random sample of size n = 100 from a binomial population with p = 0.4.

If the sampling distribution of is normal or approximately normal, standardize or rescale the interval of interest in terms of

Find the appropriate area using Z Table.

If the sampling distribution of is normal or approximately normal, standardize or rescale the interval of interest in terms of

Find the appropriate area using Z Table.

p̂

p̂ pz

pq

n

If both np > 5 and

np(1-p) > 5