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Probability
- 1. Chapter 6 Probability PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
- 2. Chapter 6 Learning Outcomes • Understand definition of probability1 • Explain assumptions of random sampling2 • Use unit normal table to find probabilities3 • Use unit normal table to find scores for given proportion4 • Find percentiles and percentile rank in normal distribution5
- 3. Tools You Will Need • Proportions (Math Review, Appendix A) – Fractions – Decimals – Percentages • Basic algebra (Math Review, Appendix A) • z-scores (Chapter 5)
- 4. 6.1 Introduction to Probability • Research begins with a question about an entire population. • Actual research is conducted using a sample. • Inferential statistics use sample data to answer questions about the population • Relationships between samples and populations are defined in terms of probability
- 5. Figure 6.1 Role of probability in inferential statistics
- 6. Definition of Probability • Several different outcomes are possible • The probability of any specific outcome is a fraction or proportion of all possible outcomes outcomespossibleofnumbertotal Aasclassifiedoutcomesofnumber Aofyprobabilit
- 7. Probability Notation • p is the symbol for “probability” • Probability of some specific outcome is specified by p(event) • So the probability of drawing a red ace from a standard deck of playing cards could be symbolized as p(red ace) • Probabilities are always proportions • p(red ace) = 2/52 ≈ 0.03846 (proportion is 2 red aces out of 52 cards)
- 8. (Independent) Random Sampling • A process or procedure used to draw samples • Required for our definition of probability to be accurate • The “Independent” modifier is generally left off, so it becomes “random sampling”
- 9. Definition of Random Sample • A sample produced by a process that assures: – Each individual in the population has an equal chance of being selected – Probability of being selected stays constant from one selection to the next when more than one individual is selected • Requires sampling with replacement
- 10. Probability and Frequency Distributions • Probability usually involves population of scores that can be displayed in a frequency distribution graph • Different portions of the graph represent portions of the population • Proportions and probabilities are equivalent • A particular portion of the graph corresponds to a particular probability in the population
- 11. Figure 6.2 Population Frequency Distribution Histogram
- 12. Learning Check • A deck of 52 cards contains 12 royalty cards. If you randomly select a card from the deck, what is the probability of obtaining a royalty card? • p = 1/52A • p = 12/52B • p = 3/52C • p = 4/52D
- 13. Learning Check - Answer • A deck of 52 cards contains 12 royalty cards. If you randomly select a card from the deck, what is the probability of obtaining a royalty card? • p = 1/52A • p = 12/52B • p = 3/52C • p = 4/52D
- 14. Learning Check TF • Decide if each of the following statements is True or False. • Choosing random individuals who walk by yields a random sampleT/F • Probability predicts what kind of population is likely to be obtainedT/F
- 15. Learning Check - Answers • Not all individuals walk by, so not all have an equal chance of being selected for the sample False • The population is given. Probability predicts what a sample is likely to be like False
- 16. 6.2 Probability and the Normal Distribution • Normal distribution is a common shape – Symmetrical – Highest frequency in the middle – Frequencies taper off towards the extremes • Defined by an equation • Can be described by the proportions of area contained in each section. • z-scores are used to identify sections
- 17. Figure 6.3 The Normal Distribution 22 2/)( 2 2 1 X eY
- 18. Figure 6.4 Normal Distribution with z-scores
- 19. Characteristics of the Normal Distribution • Sections on the left side of the distribution have the same area as corresponding sections on the right • Because z-scores define the sections, the proportions of area apply to any normal distribution – Regardless of the mean – Regardless of the standard deviation
- 20. Figure 6.5 Distribution for Example 6.2
- 21. The Unit Normal Table • The proportion for only a few z-scores can be shown graphically • The complete listing of z-scores and proportions is provided in the unit normal table • Unit Normal Table is provided in Appendix B, Table B.1
- 22. Figure 6.6 Portion of the Unit Normal Table
- 23. Figure 6.7 Proportions Corresponding to z = ±0.25
- 24. Probability/Proportion & z-scores • Unit normal table lists relationships between z-score locations and proportions in a normal distribution • If you know the z-score, you can look up the corresponding proportion • If you know the proportion, you can use the table to find a specific z-score location • Probability is equivalent to proportion
- 25. Figure 6.8 Distributions: Examples 6.3a—6.3c
- 26. Figure 6.9 Distributions: Examples 6.4a—6.4b
- 27. Learning Check • Find the proportion of the normal curve that corresponds to z > 1.50 • p = 0.9332A • p = 0.5000B • p = 0.4332C • p = 0.0668D
- 28. Learning Check - Answer • Find the proportion of the normal curve that corresponds to z > 1.50 • p = 0.9332A • p = 0.5000B • p = 0.4332C • p = 0.0668D
- 29. Learning Check • Decide if each of the following statements is True or False. • For any negative z-score, the tail will be on the right hand sideT/F • If you know the probability, you can find the corresponding z-scoreT/F
- 30. Learning Check - Answer • For negative z-scores the tail will always be on the left sideFalse • First find the proportion in the appropriate column then read the z-score from the left column True
- 31. 6.3 Probabilities/Proportions for Normally Distributed Scores • The probabilities given in the Unit Normal Table will be accurate only for normally distributed scores so the shape of the distribution should be verified before using it. • For normally distributed scores – Transform the X scores (values) into z-scores – Look up the proportions corresponding to the z- score values.
- 32. Figure 6.10 Distribution of IQ scores
- 33. Figure 6.11 Example 6.6 Distribution
- 34. Box 6.1 Percentile ranks • Percentile rank is the percentage of individuals in the distribution who have scores that are less than or equal to the specific score. • Probability questions can be rephrased as percentile rank questions.
- 35. Figure 6.12 Example 6.7 Distribution
- 36. Figure 6.13 Determining Normal Distribution Probabilities/Proportions
- 37. Figure 6.14 Commuting Time Distribution
- 38. Figure 6.15 Commuting Time Distribution
- 39. Learning Check • Membership in MENSA requires a score of 130 on the Stanford-Binet 5 IQ test, which has μ = 100 and σ = 15. What proportion of the population qualifies for MENSA? • p = 0.0228A • p = 0.9772B • p = 0.4772C • p = 0.0456D
- 40. Learning Check - Answer • Membership in MENSA requires a score of 130 on the Stanford-Binet 5 IQ test, which has μ = 100 and σ = 15. What proportion of the population qualifies for MENSA? • p = 0.0228A • p = 0.9772B • p = 0.4772C • p = 0.0456D
- 41. Learning Check • Decide if each of the following statements is True or False. • It is possible to find the X score corresponding to a percentile rank in a normal distribution T/F • If you know a z-score you can find the probability of obtaining that z- score in a distribution of any shape T/F
- 42. Learning Check - Answer • Find the z-score for the percentile rank, then transform it to XTrue • If a distribution is skewed the probability shown in the unit normal table will not be accurate False
- 43. 6.4 Looking Ahead to Inferential Statistics • Many research situations begin with a population that forms a normal distribution • A random sample is selected and receives a treatment, to evaluate the treatment • Probability is used to decide whether the treated sample is “noticeably different” from the population
- 44. Figure 6.16 Research Study Conceptualization
- 45. Figure 6.17 Research Study Conceptualization
Editor's Notes
- 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.
- FIGURE 6.2 A frequency distribution histogram for a population that consists of N = 10 scores. The shaded part of the figure indicates the portion of the whole population that corresponds to scores greater than X = 4. The shaded portion is two-tenths (p = 2/10) of the whole distribution.
- FIGURE 6.3 The normal distribution. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency). The equation is provided on the slide. (Pi and e are mathematical constants.) In simpler terms the normal distribution is symmetrical with a single mode in the middle. The frequency tapers off as you move farther from the middle in either direction.
- FIGURE 6.4 The normal distribution following a z-score transformation.
- FIGURE 6.5 The distribution of SAT scores described in Example 6.2.
- 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.
- FIGURE 6.7 Proportions of a normal distribution corresponding to z = +0.25 (a) and -0.25 (b).
- FIGURE 6.8 The distribution for Example 6.3a—6.3c
- FIGURE 6.9 The distributions for Examples 6.4a and 6.4b.
- FIGURE 6.10 The distribution of IQ scores. The problem is to find the probability or proportion of the distribution corresponding to scores less than 120.
- FIGURE 6.11 The distribution for Example 6.6.
- FIGURE 6.12 The distribution for Example 6.7.
- FIGURE 6.13 Determining probabilities of proportions for a normal distribution is shown as a two-step process with z-scores as an intermediate stop along the way. Note that you cannot move directly along the dashed line between X values and probabilities or proportions. Instead, you must follow the solid lines around the corner.
- FIGURE 6.14 The distribution of commuting time for American workers. The problem is to find the score that separates the highest 10% of commuting times from the rest.
- FIGURE 6.15 The distribution of commuting times for American workers. The problem is to find the middle 90% of the distribution.
- FIGURE 6.16 A diagram of a research study. A sample is selected from the populations and receives a treatment. The goal is to determine whether the treatment has an effect.
- FIGURE 6.17 Using probability to evaluate a treatment effect. Values that are extremely unlikely to be obtained form the original population are viewed as evidence of a treatment effect.
- FIGURE 6.18 A sketch of the distribution for Demonstration 6.1.