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PSYC 2002 (84)
Lecture 9

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

Description
Lecture 9 t-test Z-test assumptions - Random sampling - Anormal sampling distribution - Independent observations - Constant  - Two of these are actually pretty hard to justify in the real world. Which? • Normal sampling distribution and Z-test assumptions - Constant  • You often don’t know ² • In fact, you often run experiments specifically so that you can estimate the population parameters - Anormal sampling distribution • If you can get a large enough sample, that’s fine • But what if you can’t? • This problem prompted a pilgrimage which I undertook a few years ago Gosset’s revolutionary discovery - The mean of a small n sample, that produces an estimate of , can be compared to a theoretical distribution of sample means that is NOT normally distributed - The distribution would be relatively platykurtic for small n’s, but would still be symmetrical and unimodal - The distribution would approach the shape of a normal distribution as n increased The student’s t-test - Used instead of a z-test whenever you don’t know  - Used instead of a z-test whenever you have small samples and, as such, can’t be certain that the distribution of sample means is normally distributed “New” formulae S = S M √n SS Where: S = √ n−1 M−µ t= S M Let’s have a flashback… - Remember in Quiz #1 when I asked you to find the missing X value given a particular mean? - Many of you found unique formulae that let you do this • That’s great! You got full marks for that - The problem is: those unique formulae missed the point of the question… Find the missing X - If X = {1, 2, 3, ???} and M = 5 what is the missing value? - One of the values is NOT FREE TO VARY - This means that, although most of the scores are free to vary, the sample has lost one degree of freedom - For any t-test: df = n – 1 df = n – 1 - A t distribution will approach a normal distribution as df increases - So how do you increase df for a t-test? The t-table - The table is a list of critical values to be used in hypothesis testing - This one is a bit more complicated than the z-table. Fortunately, you’ll only have to deal with an abbreviated version: • Only a few  values are listed - What critical t score would be associated with the following: • I tested 10 people who I believe were made better by my treatment.  = .05 • Do men perform differently than women? I tested 15 men.  = .05 Let’s do a t-test!! - I suspect that students will do better on Test 2 than on Test 1 because they will have had more exposure to the subject matter - I get a sample of 5 test takers - The mean grade for test 1 was µ = 5 - My sample had the following scores: • 6, 8, 4, 9, 8 - Test my hypothesis, maintaining a Type I error rate of 2.5% Getting the necessary values X (X-M)2 6 1 s2 = 16/(5-1) = 4 8 1 s = 2 4 9 9 4 8 1 M = 7 SS = 16 Step 1 - State the hypothesis H µ - 0 = treatmen≤ 5 - H 1 = µtreatmen> 5 Step 2 - State your critical value - At  = .025 one-tailed, and with df = 4, tcritic= 2.776 Step 3 - Run the test - T = (7 – 5) / (2 / 5) - = 2 / .89 - 2.2361 - 2.24 Step 4 - Compare obtained and critical values - T- observed (2.24) is less extreme than
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