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Chapter 6: Probability

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Probability

• Probability is a method for measuring and quantifying the likelihood of obtaining a specific sample from a specific population.

• We define probability as a fraction or a proportion.

• The probability of any specific outcome is determined by a ratio comparing the frequency of occurrence for that outcome relative to the total number of possible outcomes.

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Probability (cont.)

• Whenever the scores in a population are variable it is impossible to predict with perfect accuracy exactly which score or scores will be obtained when you take a sample from the population.

• In this situation, researchers rely on probability to determine the relative likelihood for specific samples.

• Thus, although you may not be able to predict exactly which value(s) will be obtained for a sample, it is possible to determine which outcomes have high probability and which have low probability.

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Probability (cont.)

• Probability is determined by a fraction or proportion.

• When a population of scores is represented by a frequency distribution, probabilities can be defined by proportions of the distribution.

• In graphs, probability can be defined as a proportion of area under the curve.

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Probability and the Normal Distribution

• If a vertical line is drawn through a normal distribution, several things occur.

1. The exact location of the line can be specified by a z-score.

2. The line divides the distribution into two sections. The larger section is called the body and the smaller section is called the tail.

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Probability and the Normal Distribution (cont.)

• The unit normal table lists several different proportions corresponding to each z-score location. – Column A of the table lists z-score values. – For each z-score location, columns B and C list the

proportions in the body and tail, respectively. – Finally, column D lists the proportion between the

mean and the z-score location.

• Because probability is equivalent to proportion, the table values can also be used to determine probabilities.

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Probability and the Normal Distribution (cont.)

• To find the probability corresponding to a particular score (X value), you first transform the score into a z-score, then look up the z-score in the table and read across the row to find the appropriate proportion/probability.

• To find the score (X value) corresponding to a particular proportion, you first look up the proportion in the table, read across the row to find the corresponding z-score, and then transform the z-score into an X value.

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Percentiles and Percentile Ranks

• The percentile rank for a specific X value is the percentage of individuals with scores at or below that value.

• When a score is referred to by its rank, the score is called a percentile. The percentile rank for a score in a normal distribution is simply the proportion to the left of the score.

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Probability and the Binomial Distribution

• Binomial distributions are formed by a series of observations (for example, 100 coin tosses) for which there are exactly two possible outcomes (heads and tails).

• The two outcomes are identified as A and B, with probabilities of p(A) = p and p(B) = q.

• The distribution shows the probability for each value of X, where X is the number of occurrences of A in a series of n observations.

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Probability and the Binomial Distribution (cont.)

• When pn and qn are both greater than 10, the binomial distribution is closely approximated by a normal distribution with a mean of μ = pn and a standard deviation of σ = npq.

• In this situation, a z-score can be computed for each value of X and the unit normal table can be used to determine probabilities for specific outcomes.

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Probability and Inferential Statistics

• Probability is important because it establishes a link between samples and populations.

• For any known population it is possible to determine the probability of obtaining any specific sample.

• In later chapters we will use this link as the foundation for inferential statistics.

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Probability and Inferential Statistics (cont.)• The general goal of inferential statistics is to use

the information from a sample to reach a general conclusion (inference) about an unknown population.

• Typically a researcher begins with a sample. • If the sample has a high probability of being

obtained from a specific population, then the researcher can conclude that the sample is likely to have come from that population.

• If the sample has a very low probability of being obtained from a specific population, then it is reasonable for the researcher to conclude that the specific population is probably not the source for the sample.