Chapter 6: Probability

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.

Figure 6.1 The role of probability in inferential statistics. Probability is used to predict what kind of samples are likely to be obtained from a population. Thus, probability establishes a connection between samples and populations. Inferential statistics rely on this connection when they use sample data as the basis for making conclusions about populations.

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.

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.

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.

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.

Figure 6.6 A portion of the unit normal table. This table lists proportions of the normal distribution corresponding to each z-score value. Column A of the table lists z-scores. Column B lists the proportion in the body of the normal distribution up to the z-score value. Column C lists the proportion of the normal distribution that is located in the tail of the distribution beyond the z-score value. Column D lists the proportion between the mean and the z-score value.

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.

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.

Figure 6.7 The distributions for Example 6.3A–6.3C. Figure 6.8 The distributions for Examples 6.4A and 6.4B. Figure 6.9 The distribution of IQ scores. The problem is to find the probability or proportion of the distribution corresponding to scores less than 130.

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.

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.

Figure 6.16 Binomial distributions showing probabilities for the number of heads (a) in 4 tosses of a balanced coin and (b) in 6 tosses of a balanced coin. Figure 6.15 The binomial distribution showing the probability for the number of heads in 2 tosses of a balanced coin. Figure 6.17 The relationship between the binomial distribution and the normal distribution. The binomial distribution is always a discrete histogram, and the normal distribution is a continuous, smooth curve. Each X value is represented by a bar in the histogram or a section of the normal distribution.

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.

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.

Figure 6.19 A diagram of a research study. A sample is selected from the population and receives a treatment. The goal is to determine whether the treatment has an effect. Figure 6.20 Using probability to evaluate a treatment effect. Values that are extremely unlikely to be obtained from the original population are viewed as evidence of a treatment effect.