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Psychology (2,710)
PSYC 2002 (84)
Lecture 5

7 Pages
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School
Department
Psychology
Course
PSYC 2002
Professor
Steven Carroll
Semester
Winter

Description
Lecture 5: Z test/score Where we are going... - Z-scores requires a knowledge of standard deviation - Probability theory requires a knowledge of frequency distributions, proportions - The distribution of sample means requires an understanding of the mean and of the standard deviation - Hypothesis testing requires an understanding of z-scores, probability theory, and the distribution of sample means Consider the following... - Imagine you wrote two different tests TEST 1 TEST 2 X = 60 X = 60 µ = 50 µ = 50 - Are you happier with your mark on the first test, or with your mark on the second test? - What if I included this information? TEST 1 TEST 2 X = 60 X = 60 µ = 50 µ = 50  = 20  = 5 - Are you happier with your mark on the first test, or with your mark on the second test? Why? Calculate X−µ Z= ❑ - Calculate z when X= 60, µ= 50, and = 20 - Calculate z when X= 60, µ= 50, and = 5 What is a z-score? - It describes the exact location of a score in a distribution - Az-score of +.5 indicates X was exactly one-half standard deviations above the mean of the distribution of scores - Az-score of +2 indicates X was exactly two standard deviations above the mean of the distribution of scores Calculate X−µ Z= ❑ - Calculate z when X= 50, µ= 60, and = 20 - Calculate z when X= 50, µ= 60, and = 5 What is a z-score? - Az-score of -.5 indicates X was exactly one-half standard deviations below the mean of the distribution of scores - Az-score of -2 indicates X was exactly two standard deviations below the mean of the distribution of scores Imagine the test scores were normally distributed - Az-score of “1” indicates that the raw score X is one standard deviation above the mean of the distribution of scores - Since the height of this distribution indicates the likelihood of a score occurring, it should be noted that more extreme z-scores are comparatively rare - So you should be quite happy with a z-score of +2.0! Why? 50 If =. 5 60 If =.2 60 Try it! X 1 1 1 2 5 - Transform this distribution of raw scores into a distribution of z-scores Consider the following distribution of scores X X – µ (X – µ)2 z 1 1 1 2 5 - What is µ? - What is ? X X – µ (X – µ)2 z 1 -1 1 -.65 1 -1 1 -.65 1 -1 1 -.65 2 0 0 0 5 3 9 1.94 - What is µ? (2) - What is ? (1.55) Let’s do some more math! z -.65 -.65 -.65 0 1.94 - What is the mean of the distribution of z-scores? 0 - What is the standard deviation of the distribution of z-scores? 1 z z – µ (z – z µz )² -.65 -.65 .42 -.65 -.65 .42 -.65 -.65 .42 0 0 0 1.94 1.94 3.76 Always true!! - The mean of a distribution of z-scores is ALWAYS 0 • All positive z-scores are above the mean, and all the negative z-scores are below the mean - The standard deviation of a distribution of z-scores is ALWAYS 1 Let’s draw graphs! - First draw a frequency distribution of the raw scores 1, 1, 1, 2, 5 - Then, draw a
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