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Statistical Hypothesis Testing and Variance Estimation in Psychological Research: A Comprehensive Analysis Using t Tests

A comprehensive analysis of statistical hypothesis testing and variance estimation in psychological research using t-tests.

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    Statistical Hypothesis Testing and Variance Estimation in Psychological Research: A Comprehensive Analysis Using t Tests WEEK FOUR INDIVIDUAL ASSIGNMENT Last week, we worked on comparisons involving either one individual’s score being compared to a population or a sample of people’s scores being compared to a population. We called these comparisons z tests and in both situations we had information about the population’s mean and variance. Remember also, that I said that this was rarely the kind of research that was done in the field of psychology but a good foundation that we needed to understand. This week, we’ll move closer to the type of research that you’ll read in the professional literature. We’re still making comparisons but in real life, we seldom have information on the population’s mean and variance. If the population’s variance is not known, a solution is to estimate it based on the sample’s variance. While it might seem easy then to just use the sample’s variance directly; mathematically, it can be shown that the sample’s variance will, on average, be a bit smaller than its population’s variance so we tweak it just a little. Last week, we calculated variance as the sum of square deviations from the mean divided by the number of participants in the sample: SD 2 = SS/N. But the estimated population variance is figured as the sum of squared deviations from the mean divided by the number of participants minus one : SD 2 = SS/N - 1. Not a big tweak, just a little one. This “tweak” is called degrees of freedom. With these new comparisons (where we don’t know the population’s variance), some things to keep in mind: • The comparisons are called t tests (not z tests) and there are two kinds of t tests. One, a t Test for Dependent Means and Two, a t Test for Independent Means. I’ll leave it to your assigned readings to explain the difference between the two. • Estimating the population variance loses some accuracy so we make up for that by setting the cutoff score a little more extreme • You will use the t table at the end of our textbook to determine your cutoff score. So for this week’s Individual Assignment, Part One will require you to demonstrate that you can calculate the estimated population variance. This is Formula (7 - 1) on your Major Formulas Handout. Part Two, will require you to go through the Five - Step Hypothesis Testing Process for a t Test for Dependent Means and Part Three will require you to go through the Five - Step Hypothesis Testing Process for Independent Means. Again, I’ll walk you through an example of each one and then give you a scenario to do on your own. PART ONE: CALCULATING AN ESTIMATED POPULATION VARIANCE FORMULA (7 - 1) First, my EXAMPLE:

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    5 . 6 4 26 1 ) ( 2 2 = = = − − =  df SS N M X S df SS N SS N M X S = − = − − =  1 1 ) ( 2 PARTICIPANT SCORE DEVIATION DEV. SQU. X X – M ( X – M ) 2 1 8 2 4 2 6 0 0 3 4 – 2 4 4 9 3 9 5 3 – 3 9 Σ X = 30 SS = 26 M = 6 SS = 26 df = N – 1 = 5 – 1 = 4 Now , your turn. Student Participation Points A 4 B 10 C 6 D 8 E 0 F 4 G 2 PARTICIPANT SCORE DEVIATION DEV. SQU. X X – M ( X – M ) 2 A 4 - 1 0.73 B 10 5 26.45 C 6 1 1.31 D 8 3 9.88 E 0 - 5 23.59 F 4 - 1 0.73 G 2 - 3 8.16 Σ X = 34 SS = 70.86 M = 6 SS = 70.86 df = N – 1 = 7 – 1 = 6

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