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PROBABILITY

KULIAH 8

OLEHISMAIL KAILANI

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INTRODUCTION TO PROBABILITY

The idea of research studies begin with a generalquestion about the entire population, but the

actual research is conducted using a sample. In thissituation the role of inferential statistics is to usethe sample data as the basis for answeringquestions about population. To accomplish this

goal, inferential procedures are typically builtaround the concept of probability. Specifically, therelationship between samples and populations areusually defined in term of probability.

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INTRODUCTION TO PROBABILITY

Basic idea of probability

1. Suppose you are selecting a single marble from a jar that contains 50 black and 50white marbles. ( In this example, the jar of marbles is the population, and thesingle marble is the sample). Although we cannot guarantee the exact outcome of

our sample, it is possible to talk about the potential outcomes in term ofprobabilities. In this case we have 50-50 chance of getting either color.

2. Consider another jar that has 90 black and 10 white marbles. Again we cannotspecify the exact outcome of the sample, but now we know that the sampleprobability will be a black marble.

By knowing the make up of a population, we can determine the probability ofobtaining specific samples. In this way,probability gives us a connection betweenpopulations and samples, and this connection will be the foundation for theinferential statistics.

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THE ROLE OF PROBABILITY IN INFERENTIAL

STATISTICS

In the preceding examples we begin with a population andthen use probability to describe the samples that could beobtained. This is exactly backward from what we want to do

with inferential statistics where we begin with a sample andthen answer general question about the population. To dothat, first we develop probability as a bridge from populationsto samples. This stage involves identifying the types ofsamples that probably would be obtained from a specificpopulation. Once this bridge is established, we will reversethe probability rules to allow us to move from samples topopulation Figure 6.1).

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THE ROLE OF PROBABILITY IN INFERENTIAL

STATISTICS

Insert fig 6.1 p163 G

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WHAT IS PROBABILITY

For a situation in which several different outcome arepossible, the probability for any specific outcome isdefined as a fraction or a proportion of all the possibleoutcomes. If the possible outcome are identified as

A,B,C,D and so on, then:

Probability of A =

The probability of a specific outcome will be express with ap (for probability) followed by the specific outcome inparentheses. For example, the probability of obtainingheads for a coin toss will be written asp (heads).

outcomespossibleofnumbertotal

Aasclassifiedoutcomesofnumber

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RANDOM SAMPLING

A random sample requires that each individual

in the population has an equal chance of

being selected. A second requirement,

necessary for many statistical formulas, states

that the probabilities must stay constant from

one selection to the next if more than one

individual is selected.

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PROBABILITY AND FREQUENCY DISTRIBUTIONS

The situation in which we are concerned with probability

usually will involve a population of scores that can be displayed

in a frequency distribution graph. If you think of the graph as

representing the entire population, then different portions of

the graph will represent different portions of the population.

Because probabilities and proportions are equivalent, a

particular portion of the graph corresponds to a particular

probability in the population. Thus, whenever a population is

presented in a frequency distribution graph, it will be possible

to represent probabilities as portion of the graph.

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PROBABILITY AND FREQUENCY

DISTRIBUTIONS

Example.

A simple population that contains only N=10 scores withvalues 1,1,3,3,4,4,4,5,6. This population is shown in thefrequency distribution graph in Figure 6.2. What is theprobability of obtaining an individual with a score greaterthan 4?

p(X>4) = ?

The answer is the shaded part of the distribution that is 2squares out of the total 10 squares in the distribution.

Notice that we are now defining probability as a proportionof area in the frequency distribution graph.

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PROBABILITY AND FREQUENCY

DISTRIBUTIONS

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PROBABILITY AND THE NORMAL

DISTRIBUTION

Now that you have better understanding of the concept ofprobability, and we are learning how to use z-score todetermine probabilities. Learning how to determine probabilityusing the normal distribution will help us master other skillsrequired in the future.

Example

Assume that the population of adult heights forms a normal-shaped distribution with a mean = 68 inches and a standard

deviation of = 6 inches. Given this information (see Figure6.4), we can determine the probabilities associated with specificsamples. For example, what is the probability of randomlyselecting an individual from this population who is taller than 80inches?

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DISTRIBUTION

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DISTRIBUTION

Restating this question in probability notation, we getp(X>80) = ?

Following are the steps to find the answer:

1. Translate the probability question into a proportion question. Out of allpossible adult heights, what proportion is greater than 80 inches?

2.The set of all possible adult heights is simply the populationdistribution. Because we are interested in all heights greater than 80, weshade in the area to the right of 80. This area represents the proportionwe are trying to determine.

3. Identify the exact position of X=80 by computing z-score.

z = = = = 2.0

4. The proportion we are trying to determine may now be express in termof its z-score, i.e. p(z>2.0) = ? According to the proportion shown in thenormal distribution graph, we have 2.15% of the scores in the tail beyondz = +2.00. So the answer is 2.15%.

X

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6880

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NORMAL DISTRIBUTION AND AREA

Insert fig 6.2 p 120 R

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THE UNIT NORMAL TABLE

The graph of standard normal distribution shows proportions foronly a few selected z-score values. A more complete listing of z-scores and proportions is provided in the unit normal table.

This table lists proportion of the normal distribution for a fullrange of possible z-score values. The vertical lines separate thedistribution into two section, the bodyand the tail. Part of thetable is reproduced in Figure 6.6. Notice that the table isstructured in a four-column format. The z-score values listed incolumn A. Column B presents the body, column C presents theportion in the tail, and column D identifies the proportion of thedistribution between the mean and the z-score.

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THE UNIT NORMAL TABLE

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PROBABILITY,PROPORTIONS, AND z-

SCORES

The unit normal table lists relationships between z-score

locations and proportions in a normal distribution. For

any z-score location, we can use the table to look up the

corresponding proportion. Because we have defined

probability as equivalent to proportion, we can also use

the unit normal table to look up probabilities for normal

distribution.

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SCORES

The unit normal table lists relationships

between z-score locations and proportions in a

normal distribution. For any z-score location,we can use the table to look up the

corresponding proportions, and since we have

defined probability as equivalent to proportion,we can also use the unit normal table to look

up probabilities for normal distribution.

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SCORES

Insert fig 6.7 p172

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FINDING PROPORTIONS/PROBABILITIES FOR SPECIFIC z-

SCOREVALUES

Example A

What proportion of the normal distribution corresponds to z-score values greater than Z = 1.00?

Solution:

1. Sketch the distribution and shade the area you are trying todetermine (refer fo Figure 6.7A).

2. Look up z=1.00 in column A of the unit normal table, than readcolumn C (tail) for the proportion. We find the answer is 0.1587.

3. This problem could be rephrased as What is the probability ofselecting a z-score value greater than z = 1.00+ ?

The answer is p(z>1.00) = 0.1587 (or 15.87%)

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SCOREVALUES

EXAMPLE B

For normal distribution, what is the probability of selecting a z-score less than z = 1.50 (In symbol p(z

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SCOREVALUES

Example C

Many problems will require that we find proportions for negative z-scores. For example, what proportion of the normal distributioncorresponds to the tail beyond z =-1.50? That is p(z

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PROBABILITIES AND PROPORTIONS

FOR SCORES FROM A NORMAL

DISTRIBUTIONIn the preceding examples, we used the unit normal table to find probabilitiesand proportions corresponding to specific z-score values. In most situations,however, it will be necessary to find probabilities for specific X values.

Example

It is known that IQ scores from a normal distribution with = 100 and =15.Given this information, what is the probability of randomly selecting anindividual with an IQ score less than 130? (p(X

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PROBABILITIES AND PROPORTIONS FOR

SCORES FROM A NORMAL DISTRIBUTION

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PROBABILITIES AND PROPORTIONS

FOR SCORES FROM A NORMAL

DISTRIBUTIONEXAMPLEThe highway department conducted a study measuring driving speeds on a local

highway. They found an average speed of =58 km per hour with a standarddeviation of =10. The distribution is approximately normal. Given this information,what proportion of the cars are traveling between 55 and 65 km per hour? Step 1.Determine the z=score corresponding to the X value at each end of the interval.

For X=55; z = = = = -0.30

For X=65; z = = = = 0.70

The proportion we are seeking can be divided into two section; the area on the left ofthe mean and the area on the right of the mean. Using column D of the unit normaltable

p(55 < X < 65) = p(-0.30 < z < +0.70) = 0.1179 + 0.2580 = 0.3759 (or 37.59%)

X

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5855

10

3

X

10

5865

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PROBABILITIES AND PROPORTIONS FOR

SCORES FROM A NORMAL DISTRIBUTION

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PROBABILITIES, PROPORTIONS, AND

PERCENTILE RANKS

Thus far we have discussed parts of distribution in term of

proportions and probabilities. However, there is another set of

terminology that deals with many of the same concepts, i.e.

percentile ranks. As we have defined, the percentile rank for aspecific score as the percentage of the individuals in the

distribution who have scores that are less than or equal to the

specific score. Using this terminology, it is possible to rephrase

some of the probability problems, for example, the problemWhat is the probability of randomly selecting an individual with

IQ of less than 130? could be rephrased as What is the

percentile rank for IQ score of 130?27

PROBABILITY AND INFERENTIAL

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STATISTICSProbability forms a direct link between samples and populations from whichthey come. This link will be the foundation for the inferential statistics. The

following example provides a brief preview of how probability will be used inthe context of inferential statistics.

Example

A research begins with a population that forms a normal distribution with a

mean of = 400 and a standard deviation = 20. A sample is selected from thepopulation and a treatment is administered to the sample (Figure 6.19). Thegoal for the study is to evaluate the effect of the treatment. To determinewhether or not the treatment has an effect, the researcher simply comparesthe treated sample with the original population. If the individual in the sample

have scores around 400, then we must conclude that the treatment appears tohave no effect. On the other hand, if the treated individuals have scores thatare noticeably different from 400, then the researcher has evidence that thetreatment does have an effect. The problem for the researcher is determiningexactly what is mean by noticeably different from 400.

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STATISTICS

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STATISTICS

Notice that the study is using a sample to help answer a question about apopulation; this is the essence of inferential statistics. Figure 6.20 shows theoriginal population from our hypothetical research study. We use z-scores toresolve this problem and select z-score value beyond z=+1.96 (or z=-1.96) as an

extreme value and therefore noticeable different. The boundaries provideobjective criteria for deciding whether or not our sample is noticeable differentfrom the original population. Specifically, a sample that falls in this extremearea is extremely unlikely and is defined as a probability that is 5% or less.Note that probabilities allow us to separate a distribution into those scores thatare likely to be obtained (high probability) and those scores that are extremely

unlikely (low probability). So if our treated individual has a score that is locatedin the extreme 5% area, we can reach a logical conclusion that the treatmenthad an effect.

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STATISTICS

Insert fig 6.20 p188G

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