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Chapter Six

Chapter Six. z -Scores and the Normal Curve Model. The absolute value of a number is the size of that number, regardless of its sign. For example, the absolute value of +2 is 2 and the absolute value of -2 is 2. The symbol means “plus or minus.” Therefore, 1 means +1 and/or -1.

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Chapter Six

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  1. Chapter Six z-Scores and the Normal Curve Model

  2. The absolute value of a number is the size of that number, regardless of its sign. For example, the absolute value of +2 is 2 and the absolute value of -2 is 2. The symbol means “plus or minus.” Therefore, 1 means +1 and/or -1. New Statistical Notation Chapter 6 - 2

  3. Understanding z-Scores Chapter 6 - 3

  4. Frequency Distribution of Attractiveness Scores Chapter 6 - 4

  5. z-Scores A z-score is the distance a raw score is from the mean when measured in standard deviations. A z-score is a location on the distribution. A z-score also communicates the raw score’s distance from the mean. Chapter 6 - 5

  6. z-Score Formula The formula for computing a z-score for a raw score in a sample is Chapter 6 - 6

  7. When a z-score and the associated and are known, this information can be used to calculate the original raw score. The formula for this is Computing a Raw Score Chapter 6 - 7

  8. Interpreting z-ScoresUsing The z-Distribution Chapter 6 - 8

  9. A z-Distribution A z-distribution is the distribution produced by transforming all raw scores in the data into z-scores. Chapter 6 - 9

  10. z-Distribution of Attractiveness Scores Chapter 6 - 10

  11. Characteristics of the z-Distribution A z-distribution always has the same shape as the raw score distribution The mean of any z-distribution is 0 The standard deviation of any z-distribution is1 Chapter 6 - 11

  12. Comparison of Two z-Distributions, Plotted on the Same Set of Axes Chapter 6 - 12

  13. Relative frequency can be computed using the proportion of the total area under the curve The relative frequency at particular z-scores will be the same on all normal z-distributions Relative Frequency Chapter 6 - 13

  14. The Standard Normal Curve The standard normal curve is a perfect normal z-distribution that serves as our model of any approximately normal raw score distribution Chapter 6 - 14

  15. Proportions of Total Area Under the Standard Normal Curve Chapter 6 - 15

  16. Relative Frequency For any approximately normal distribution, transform the raw scores to z-scores and use the standard normal curve to find the relative frequency of the scores Chapter 6 - 16

  17. Percentile The standard normal curve also can be used to determine a score’s percentile. Chapter 6 - 17

  18. Proportions of the Standard Normal Curve at Approximately the 2nd Percentile Chapter 6 - 18

  19. Using z-Scores to Describe Sample Means Chapter 6 - 19

  20. Sampling Distribution of Means The frequency distribution of all possible sample means that occur when an infinite number of samples of the same size N are randomly selected from one raw score population is called the sampling distribution of means. Chapter 6 - 20

  21. Central Limit Theorem The central limit theorem tells us the sampling distribution of means forms an approximately normal distribution, has a m equal to the m of the underlying raw score population, and has a standard deviation that is mathematically related to the standard deviation of the raw score population. Chapter 6 - 21

  22. Standard Error of the Mean The standard deviation of the sampling distribution of means is called the standard error of the mean. The formula for the true standard error of the mean is Chapter 6 - 22

  23. z-Score Formula for a Sample Mean The formula for computing a z-score for a sample mean is Chapter 6 - 23

  24. Example Using the following data set, what is the z-score for a raw score of 13? What is the raw score for a z-score of -2? Chapter 6 - 24

  25. Example z-Score Assume we know and Chapter 6 - 25

  26. ExampleRaw Score from a z-Score Again, assume we know and Chapter 6 - 26

  27. Examplez-Score for a Sample Mean If = 13 , N = 18, m = 12, and = 2.5, what is the z-score for this sample mean? Chapter 6 - 27

  28. Key Terms central limit theorem relative standing sampling distribution of means standard error of the mean standard normal curve standard score z-distribution z-score Chapter 6 - 28

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