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Psychology (2,628)
PSYC 2002 (80)
Lecture 7

8 Pages
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School
Carleton University
Department
Psychology
Course
PSYC 2002
Professor
Steven Carroll
Semester
Winter

Description
Lecture 7: distributions? Distribution of the diced - Let’s do some bad stats! - When you roll a die, 6 things can happen: • 1, 2, 3, 4, 5, 6 - This is our population of scores. - What is the mean? 3.5 - What is the standard deviation? 1.71 Bad stats - What z-score would you associate with a score of 3? • Z = (3 – 3.5) / 1.71 = -.29 - What z-score would you associate with a score of 6? • Z = (6 – 3.5) / 1.71 = 1.46 Bad stats - What, according to the z-table in the back of the book, is the probability of rolling a score less than or equal to “3” if the z-score for a “3” is -.29 • .3859 - What, according to the z-table in the back of the book, is the probability of rolling a score less than or equal to “6” if the z-score for a “3” is 1.46 • .9279 Bad stats - So, according to the z-table, you are more likely to roll a “4, 5, or 6” than a “1, 2, or 3” - And, according to the z-table, there is 7.21% chance that you’ll roll something other than a “1, 2, 3, 4, 5, or 6” Huh?! - Ok, so what’s wrong with what we did? - We’re our calculations wrong? - So wtf happened there? I’ve been saying it all along - What assumption is being made with regards to the z-table? • That the z-scores are normally distributed!!! Distribution of the diced - This is NOT a normal distribution - This is a platykurtic distribution - Changing these to z-scores doesn’t suddenly make the distribution normal Let’s try something... - Instead of just rolling the die once, let’s get a sample of size n = 2 • We’ll roll the die two times, and take the average for the roll • We’ll get all possible averages Sample size n = 2 - How many things can happen? X = 36 Sample size n = 2 - What is the probability of obtaining a sample where the average = “3”? } (1 + 5) / 2 = 6 / 2 = 3 } (2 + 4) / 2 = 6 / 2 = 3 } (3 + 3) / 2 = 6 / 2 = 3 } (4 + 2) / 2 = 6 / 2 = 3 } (5 + 1) / 2 = 6 / 2 = 3 Sample size n = 2 - What is the probability of obtaining a sample where the average = “3”? • p ( M ≤ 3 ) = 5 / 36 = .1389 Now look at this: M f 1 1 1.5 2 2 3 2.5 4 3 5 3.5 6 4 5 4.5 4 5 3 5.5 2 6 1 - What is the mean of this distribution of sample means? • Hint: What is it between? - What do you notice about this value? And check this out... And check this out... Wow! - So what is the difference between the distribution of sample means when n = 1, n = 2, and the distribution of sample means when n = 3? n = 1 The central limit theorem - For any population with a mean µ and a standard deviation , the distribution of sample means for samples of size n will have a mean of µ and a standard deviation of  / n and will approach a normal distribution as n approaches infinity - Actually, by the time n = 30 the distribution is pretty normal What does that mean? - It means that we tend to do our stats on the means of samples - The location of a score in a sample or in a population can be represented with a z-score • Z = ( X - µ ) /  - BUT: researchers don’t want to study single scores • We aren’t interested in one person or one particular rat. We’re interested in groups of people or groups of rats Definitions - The distribution of sample means: a collection of sample means for all possible random samples of a particular size (n) that can be obtained from a population - Asampling distribution: a distribution of statistics obtained by selecting all of the possible samples of a specific size from a population • The distribution of sample means is an example of a sampling distribution How we roll in the brain sciences - Samples provide estimates of what is going on in the population! That’s good! - Samples are variable. No two samples are identical. That’s bad!
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