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- 1. CHAPTER 5

SPECIAL DISTRIBUTION

CHAPTER OUTLINE

5.1Discrete distribution

5.1.1 Binomial distribution

5.1.2 Poisson distribution

5.2Continuous distribution

5.2.1 Normal distribution

5.3Introduction to t-distribution

OBJECTIVES

Student should be able to:

Determine the distribution and also differentiate the distribution form.

Solve any problem related to Binomial, Poisson and Normal distribution.

5.1DISCRETE DISTRIBUTION

Consist of the values a random variable can assume and the corresponding probabilities of the values.

The probabilities are determined theoretically or by observation.

5.1.1BINOMIAL DISTRIBUTION

Binomial Distribution - the outcomes of a binomial experiment and the corresponding probabilities of these outcomes.

It is applied to find the probability that an outcome will occur x times in n performances of experiment.

For example:

The probability of a defective laptop manufactured at a firm is 0.05 in a random sample of ten.

The probability of 8 packages will not arrive at its destination.

To apply the binomial probability distribution, the random variable x must be a discrete dichotomous random variable.

Each repetition of the experiment must result in one of two possible outcomes.

Conditional of a Binomial Experiment

Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure.

There must be a fixed number of trials.

The outcomes of each trial must be independent of each other. In other words, the outcome of one trial does not affect the outcome of another trial.

The probability of success is denoted by p and that of failure by q, and p + q =1. The probabilities p and q remain constant for each trial.

Note:The success does not mean that the corresponding outcome is considered favorable or desirable and vice versaThe outcome to which the question refers is called a success; the outcome to which it does not refer is called a failure.

A.Calculating Binomial Probabilities by Using Binomial Formula

For a binomial experiment, the probability of exactly x successes in n trials is given by the binomial formula:

P(x) = nCx px qn-x

Where;

n=the total number of trials

p=probability of success

q= 1 - p = probability of failure

x=number of successes in n trials

n - x=number of failures in n trials

Example 1

Compute the probabilities of X successes, using the binomial formula.

a. n = 6,X = 3,p = 0.03

b. n = 4,X = 2,p = 0.18

Solution:

a. P(x = 3)=6C3(0.03)3(1-0.03)6-3

=6C3(0.03)3(0.97)3

=0.0005

b. P(x = 2)=4C2(0.18)2(1-0.18)4-2

=4C2(0.18)2(0.82)2

=0.1307

Example 2

A survey found that one out five Malaysian 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:

In this case, n = 10,x = 3,p = 1/5andq = 4/5

P(x = 3)=10C3(1/5)3(4/5)7

=0.2013

Exercise 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 will fail

b)Less than 2 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.

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)Atmost 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.

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.

B.Using Table of Binomial Probabilities

The probabilities for a binomial experiment can also be read from the table of binomial probabilities.

For any number of trials n:

The binomial probability distribution is symmetric ifp = 0.5.

The binomial probability distribution is skewed to the right if p is less than 0.5.

The binomial probability distribution is skewed to the left if p is greater than 0.5.

0172085

From the table also, it is easier to calculate various from of binomial distribution such as:

EquallyP(X = x)= P(X x) - P(X x - 1)At mostP(X x)= P(X x) (directly from the table)Less thanP(X < x)= P(X x - 1) At leastP(X x)= 1 P(X x - 1)Greater thanP(X > x)= 1 P(X x)From x1 to x2P(x1 X x2) = P(X x2) - P(X x1 - 1)Between x1 and x2P(x1 74)=P(z > 74 65)

15

=P(z > 0.6)

=1 P(z < 0.6)

=1 0.7257

=0.2743

c)P(71 < X < 56)=P(71 65 < z < 56 65)

15 15

=P(0.4 < z < -0.6)

=P(z < 0.4) P(z < -0.6)

=0.6554 0.2743

=0.3811

Exercise 6:

1.Let x denote the time takes to run a road race.Supposed x is approximately normally distributed with mean of 190 minutes and standard deviation of 21 minutes.If one runner is selected at random, what is the probability that this runner will complete this road race?

in less than 150 minutes

in 205 to 245 minutes

2.The mean number of hours a student spends on the computer is 3.1 hours per day. Assume the standard deviation is 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.The score of 6000 candidates in a certain examination are found to be approximately normal distributed with a mean of 55 and a standard deviation of 10:

If a score of 75 or more is required for passing the distinction, estimate the number of grades with distinction.

Calculate the probability that a candidate selected at random has a score between 45 and 65.

INTRODUCTION TO t-DISTRIBUTION

The t distribution is very similar to the standardized normal distribution.

Both distributions are bell-shaped and symmetrical.

However, the t distribution has more area in the tails and less in the center than does the standardized normal distribution.

This is becauseis unknown and S is used to estimate it.

Because the value ofis uncertain, the values of t that are observed will be more variable than for Z.

Standard normalt distribution for 5 degrees of freedom

As the number of degrees of freedom increases, the t distribution gradually approaches the standardized normal distribution until the two are virtually identical.

This happens because S becomes a better estimate ofas the sample size gets larger.

With a sample size of about 120 or more, S estimatesprecisely enough that there is little difference between the t and Z distributions.

For this reason, most statisticians use Z instead of t when the sample size is greater than 120.

EXERCISES

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

two bulbs are defective

at least one bulb is defective

In a university, 20 percent of the students fail the statistic test. If 20 students from the university are interviewed, what is the probability of getting:

less than 3 students who fail the test

more than 3 students who fail the test

exactly 4 students who fail the test

A financial institution in Kuala Lumpur has offer a job as a risk analyst. For the minimum qualification, the applicant must seats for writing test. Based on the management experience, 40% of the applicants will pass the test and qualified for the interview session. There are 20 applicants who have applied for the jobs. Find the probability that there are more than 50% applicants will pass the test.

An Elementary Statistic class has 75 members. If there is a 12% absentee rate per class meeting, find the mean, variance and standard deviation of the number of students who will be absent from each class.

Before an umbrella leaves the factory, it is given a quality control check. The probability that an umbrella contains zero, one or two defects is 0.88, 0.08 and 0.04 respectively. In a sample of 16 umbrellas, find the probability that:

9 will have no defect

4 will have one defect

3 will have two defect

En. Rostam is a credit officer at the Trust Bank. Based on his experience, he estimates that an average he will receive the loan application in a week is 3 applications. Find the probability that:

he receives none loan application in a week

he receives 2 until 5 loan applications in a week

at least 5 loan applications he receives in 14 days.

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. If indeed, there are five books with missing pages, find the probabilitythat the lot will be returned.

The numbers of customers who enter shop ABC independent of one another and at random intervals follow a Poisson distribution with an average rate 42 customers per hour. Find the probability that:

no customer enter the shop during a particular 1 minutes interval

at least 4 customers enter the shop during a particular 5 minutes interval

between 2 and 6 customers enter the shop during a particular 10-minute interval.

One research has been conduct