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Introduction to Statistics and Probability: Understanding Hypothesis Testing and Z-Scores

This introductory guide to statistics and probability outlines the fundamental concepts of probability theory and hypothesis testing. It explains how to form hypotheses about populations, test them using samples, and determine if observed effects are likely due to chance. Key topics covered include the properties of the normal distribution, the significance of z-scores, and the process of comparing data sets. Emphasis is placed on using standard scores for meaningful comparisons and understanding the risk of error in hypothesis testing, particularly in psychology research.

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Introduction to Statistics and Probability: Understanding Hypothesis Testing and Z-Scores

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Presentation Transcript

  1. Intro to Stats Probability

  2. Probability theory • Provides a basis for thinking about the probability of possible outcomes • & can be used to determine how confident we can be in an effect • (not due to chance)

  3. Remember the problem • The hypothesis is formed with respect to a population --- but tested with a sample • We rely on what is known about populations to estimate whether effect is due to chance or not

  4. Assumptions • Large sets of data should approximate a normal curve • (it is assumed that the population is normally distributed)

  5. Normal Distribution

  6. Properties of Normal Distribution • 1. Mean = median = mode • 2. Symmetrical • 3. Asymptotic tails

  7. Standard Scores • Can therefore compare between samples if standardized with respect to the mean and SD • Standard scores: comparable because in standard deviation units • The z score is a common standard score • The number of SDs above the mean • Are comparable across different distributions

  8. Z Scores z = (X - X) s z = z score X = individual score X = mean of distribution s = SD of distribution

  9. Parts of the z-score • The z-score can be broken into the sign and the magnitude of the value • The sign (+/-) tell us the position of the score relative to the mean • A positive z score means that the value is above the mean • A negative z score means that the value is below the mean • The number (magnitude) itself tells us the distance the score is from the mean in terms of the number of standard deviations

  10. Properties of z The shape of the distribution will be the same as the original The mean will always be zero The standard deviation will always be one (always reliable axes in z-score units) Z-scores can be used to compare across different data sets

  11. Jump to Probability • A certain % of scores fall below or above each z score in a predictable way (as seen earlier) • This can give us the probability of attaining a particular score • Does the sample differ from the population (is there a difference between groups?) • We have to specify the risk we’re willing to take of being wrong *Use a table for exact percents

  12. Testing Hypotheses • If the probability of an event is extremely low, it is unlikely that it is due to chance • Psychology uses 5% (p < .05) as the extreme probability • z-score > 1.96 • If significant (p < .05), the research hypothesis is a more likely explanation than the null

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