University:
California State University, Northridge
Course:
MATH 340 | Introduction to Probability and Statistics
Academic year:

2011
Views:

453

Pages:

20
Author:

Darren Villarreal

Chapter 6: Probability I t d ti Introduction to t Probability P b bilit • • • The role of inferential statistics is to use th sample the l data d t as the th basis b i for f answering questions about the population. To accomplish this goal, inferential procedures are typically built around the concept of probability. Specifically, the relationships between samples and populations are usually defined in terms of probability. P b bilit D Probability Definition fi iti • Definition: For a situation in which severall diff differentt outcomes t are possible, ibl the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. – In other words, if the possible outcomes are identified as A, B, C, D, and so on, then: • To simplify the discussion of probability, we use a notation system that eliminates a lot of the words. P b bilit Definition Probability D fi iti cont. t • • The probability of a specific outcome is expressed with a p (for probability) followed by the specific outcome in parentheses. – For example, example the probability of selecting a king from a deck of cards is written as p(king). – The p probability y of obtaining g heads for a coin toss is written as p(heads). Note that probability is defined as a proportion, or a part of the whole. – In other words, this definition makes it possible to restate any probability problem as a proportion problem. P b bilit D Probability Definition fi iti cont. t • • By convention, probability values most often are expressed as decimal values. But you should realize that any of these three forms is acceptable. – Fraction F ti – Proportion – Percentage R d Random S Sampling li • • For the preceding definition of probability to be accurate, accurate it is necessary that the outcomes be obtained by a process called random sampling. Definition: 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 formulas, states that the probabilities must stay constant from one selection to the next if more than one individual is selected. – To keep the probabilities from changing from one selection to the next, it is necessary to return each i di id l to individual t the th population l ti before b f you make the next selection. R d Random S Sampling li cont. t – This process is called sampling with replacement. • The second requirement for random samples (constant probability) demands that you sample with replacement. P b bilit and Probability d the th Normal N l Distribution Di t ib ti • • • • An example of a normal distribution is shown in Figure 6.3. Note that the normal distribution is symmetrical, with the highest frequency in the middle and frequencies tapering off as you move toward either extreme. Although the exact shape for the normal distribution is defined by y an equation q ((see Figure 6.3), the normal shape can also be described by the proportions of area contained in each section of the distribution. distribution Statisticians often identify sections of a normal distribution by using z-scores. Fig. 6-3, p. 170 Probability y and the Normal Distribution cont. • • • Figure 6.4 shows a normal distribution with several sections marked in z-score units. You should recall that z-scores measure positions in a distribution in terms of standard deviations from the mean. There are two additional points to be made about the distribution shown in Figure 6.4. – First, you should realize that the sections on the left side of the di ib i distribution have h exactly l the h same areas as the corresponding sections on the right side because the normal y distribution is symmetrical. Probability y and the Normal Distribution cont. • – Second, because the locations in the distribution are identified by z-scores, z scores the percentages shown in the figure apply to any normal distribution regardless of the values for the mean and the standard deviation. – Remember: When any distribution is transformed into z-scores, the mean b becomes zero and d the th standard t d d deviation becomes one. Because the normal distribution is a good model for many naturally occurring distributions and because this shape is guaranteed in some circumstances (as you will see in Chapter 7), we will devote considerable id bl attention tt ti to t this thi particular ti l distribution. Fig. 6-4, p. 171 Probability y and the Normal Distribution cont. • • The process of answering probability questions about a normal distribution is introduced in the following example. Example 6.2 – Assume A th thatt th the population l ti off adult d lt heights forms a normal-shaped distribution with a mean of μ = 68 inches and a standard deviation of σ = 6 inches. – Given this information about the population and the known proportions f a normall distribution for di ib i (see ( Figure Fi 6.4), we can determine the probabilities associated with specific samples. Probability y and the Normal Distribution cont. • • • For example, what is the probability of randomly selecting an individual from this population who is taller than 6 feet 8 inches (X = 80 inches)? Restating this question in probability notation, we get: We will follow a step-by-step process to find the answer to this question. – 1. First,, the probability p y question q is translated into a proportion question: Out of all possible adult heights, what proportion is greater than 80 inches? Probability y and the Normal Distribution cont. – 2. The set of "all possible adult heights" g is simply p y the p population p distribution. – This population is shown in Figure 6.5(a). – The mean is μ = 68, so the score X = 80 is to the right of the mean. – Because we are interested in all h i ht greater heights t th than 80 80, we shade h d iin the area to the right of 80. This area represents the proportion we are trying y g to determine. – 3. Identify the exact position of X = 80 by computing a z-score. For this example, Fig. 6-5, p. 172 Probability y and the Normal Distribution cont. • • • That is, a height of X = 80 inches is exactly 2 standard deviations above the mean and corresponds to a z-score of z = +2.00 [see Figure 6.5(b)]. 4 The proportion we are trying to 4. determine may now be expressed in terms of its z-score: According g to the p proportions p shown in Figure 6.4, all normal distributions, regardless of the values for μ and σ , will have 2.28% of the scores in the tail beyond z = +2 +2.00. 00 Probability y and the Normal Distribution cont. • Thus, for the population of adult heights, Th Unit The U it N Normall Table T bl • To make full use of the unit normal table, there are a few facts to keep in mind: – The body always corresponds to the larger part of the distribution whether it is on the right-hand right hand side or the leftleft hand side. • Similarly, the tail is always the smaller section whether it is on the right or the left. – Because the normal distribution is symmetrical, the proportions on the right-hand i h h d side id are exactly l the h same as the corresponding proportions on the left-hand side. – Th Unit The U it N Normall Table T bl cont. t – Although the z-score values change signs (+ and -) from one side to the other, the proportions are always positive. – Thus, Thus column C in the table always lists the proportion in the tail whether it is the right-hand tail or the lefthand tail.

Introduction to Probability

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