PSYS 04, Statistics for Psychological Science: 2/18/16
PSYS 04, Statistics for Psychological Science: 2/18/16 PSYS 054
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This 10 page Class Notes was uploaded by Delaney Row on Saturday February 20, 2016. The Class Notes belongs to PSYS 054 at University of Vermont taught by Keith Burt in Fall 2016. Since its upload, it has received 15 views. For similar materials see Statistics for Psychological Science in Psychlogy at University of Vermont.
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Date Created: 02/20/16
PSYS 054 – Statistics For Psychological Science Notes for the week of 2/15/16 – 2/19/16 2/16/16 Directional vs. Non-‐directional: Decision Tree • With directional, you have to predict which direction it will be o Positive – target sample mean would be higher o Negative – target sample mean would be lower • Remember this is all based on the researchers hypothesis, not on the data! Quantifying the Probability of Mistakes: Alpha and Beta • NOTE: we never know the true state of nature • Alpha is under type I error, beta is under type II error Two-‐Tailed Test Setup • If our null hypothesis is true, there is a low probability of getting sample means in the two shaded gray regions (rejection regions), and a high probability of getting sample means that are in the middle white region • By setting this cutoff, we are saying that if we get sample means that fall in the gray regions, then they must be the result of an error Setting Rejection Regions • The area under the curve of rejection region is called “alpha level” (α), and it is expressed as a probability o Two tailed distribution = the sum of the area under both extremes o One tailed distribution = the area under one extreme o The score that falls at the alpha level is called the critical value • We will set the cutoff at 5% (this means there is a 5% chance of making a type I error) o Two tailed = the cutoff will be at the most extreme low 2.5% and high 2.5% o One tailed = the cutoff will be the most extreme 5% (either on the left or right) • A one tailed test gives you more power because the 5% is all located on one side One Tailed Test Setup (positive direction): Alpha of 0.05 • This is still a distribution of sample means • All the 5% (rejection region) will be on the right hand side because this is a one tailed test in the positive direction • We would reject our null hypothesis if our target sample mean is under the rejection area Directional vs. Non-‐directional Hypothesis • Let μ be the mean of whatever population our sample comes from • SAT prep example (500 was non-‐special mean): o Our expectation: Kaplan classes improves scores = directional and positive… § H 0 μ = 500 § H 1 μ > 500 • Non-‐directional example: o GPAs are significantly different § Shows it will be a two tailed test (either higher OR lower) § H 0 μ = 2.97 § H 1 μ ≠ 2.97 (this “does not equal” sign is characteristic of a two tailed test) The Story So Far… 1. Determine if it will be a directional or non-‐directional test 2. Set alpha level (for our purpose it is .05) 3. Create sampling distribution 4. Find critical values to draw rejection region (depends on one tail or two tail) 5. Reject or retain the null hypothesis based on whether the target sample is in rejection region Central Limit Theorem • The central limit theorem tells you what the sampling distribution looks like (includes the mean and SD of sampling distribution) with just knowing mean, SD of population, and N • It creates sampling distribution of the mean without actually taking 10,000+ random samples 2 • Given a population with mean μ and variance σ , the sampling distribution of the mean will: o Have a mean equal to μ o Have a variance equal to σ / N (and SD equal to σ / √N) o Approach a normal distribution as N increases • The central limit theorem also states that the shape is essentially normal as long as N>30 (it will get more normal the bigger the sample is) • This theorem underlies formulas on most statistical tests – very important! Central Limit Theorem: Notation • μ sub xbar = the mean of the sampling distribution of the mean • σ sub xbar = the standard deviation of the sampling distribution of the mean (also called the ‘standard error’) Z-‐test • The z-‐test makes use of everything we have used so far • NOTE: the z-‐test is different than the z-‐score • Z-‐test is useful when: o You want to compare a target sample mean to a population mean o You know the population mean o You know the population SD • Basic idea: o Convert observed sample mean to a Z-‐score (=location on the sampling distribution) using the Central Limit Theorem o Find probability of getting that Z-‐score using area under the curve (Appendix E.10 in the textbook) Z-‐test Formula • Sample mean minus the population mean, all divided by the standard error o Standard error = populations SD divided by the square root of N Z-‐test Example: SAT Data • N = 30 sample of high school students taking an SAT prep class; sample mean of 540 • What we know: population of students NOT taking a prep class has μ = 500, σ = 100 • We can plug these 4 numbers into the z statistic formula • This is a one tailed test (hypothesizing increase in scores) o Research question: does taking a prep class increase SAT scores? Z-‐test: Basic Steps • We have to figure out where rejection regions will be: 1. Establish alpha level, rejection region, and critical values (here, critical value is a Z-‐score) 2. Find the standard error of the population 3. Convert observed sample mean to Z-‐score 4. Compare your calculated Z-‐score to the critical value 5. Reject or retain the null hypothesis Critical Values of z • Appendix E.10 in the textbook • In a one-‐tailed test, we need to find the z-‐score that “cuts off” the upper 5% of the normal distribution • We have to find a z score that has exactly 5% of the area above it o Use positive z score o There is no z score exactly at .05 so we take the two surrounding it…z score of 1.65 o Anything above 1.65 = in rejection region • If our test was a one tailed negative, we would use z score of -‐1.65 and reject anything that fell below -‐1.65 Critical Values of z • In a two-‐tailed test, we need to find the z-‐score that “cuts off” the upper (and lower) 2.5% on each side of the normal distribution • We do the same thing we did for the one tailed test with the Appendix E.10 and we get a z-‐score of +/-‐ 1.96 Critical Values of z • These critical values never change for a z-‐test no matter what – memorize them! o +/-‐ 1.96 for two tailed tests o 1.65 (negative or positive, but never both) for one tailed tests Z-‐test of SAT Study: Step-‐by-‐Step • In a directional test, alpha = .05, therefore critical value is z-‐score that cuts off upper 5% of distribution o From previous slides: this is +1.65 o If we observe any z-‐score greater than 1.65, reject null hypothesis… otherwise we retain it • Population standard error = (σ/√N) = o 100 / √30 = 100 / 5.48 = 18.25 • Using formula: Z = (540 -‐ 500) / 18.25 = o 40 / 18.25 = 2.19 • Comparing observed z-‐score and critical value: o 2.19 > 1.65, therefore…. • Reject the null hypothesis! • NOTE: for our purposes, if you get a z-‐score that is exactly at the critical value…reject it One-‐Sample t-‐test • The z-‐test depends on a lot of information from the population, which isn’t always realistic • The t-‐test is for situations in which the population variance is unknown • T-‐test is one of the most common statistical tests used today • There are different versions of this test now 2/18/16 One-‐Sample t-‐test • T-‐test is used for circumstances in which the population SD is unknown o Most of the time we do not know the population SD anyway • In the one-‐sample t-‐test, we are comparing a sample mean to a hypothesized population value T-‐test Information • What you need for a one-‐sample t-‐test: o Target sample mean o Target sample SD o Population mean assuming the null hypothesis H 0 o Target sample size • We are computing a t-‐statistic in a similar way as the Z-‐statistic • BUT we are replacing the sample SD with what was the population SD (we don’t have to know the population SD) T-‐test formula Z-‐test vs. t-‐test • NOTE: the critical values 1.65 and +/-‐ 1.96 work ONLY for z-‐test (when you know the population SD) Z-‐test t-‐test Known population M, SD Hypothesized population M, unknown SD Denominator is population SD / √N Denominator is sample SD / √N Can compare calculated z-‐value with CAN’T use Appendix E.10 to critical value using normal compare t-‐value with critical value distribution table (Appendix E.10) More Z-‐test and t-‐test • Z-‐test requires known μ and σ (SD) • One-‐sample t-‐test allows unknown σ o Hypothesized μ (under H ) 0 o σ is estimated using s, the sample SD • Because of estimation, cannot use Z-‐distribution • “Student’s” solution: created a t-‐distribution o Actually many distributions, depending on sample size • T test causes for some error (because you are estimating) • T test makes it so the sampling distribution is no longer normally distributed…it takes on a new shape o This new distribution we call a t distribution…it still looks fairly like a normal distribution § As the sample size gets larger and larger, the t distribution looks more and more like a normal distribution (because its getting closer to a population) § As sample size gets small, t distribution starts getting platykurtic (flat, pushed down) Test Assumptions • The Z-‐test assumes: o The data are independently sampled from a normally distributed population o Mean (μ) and standard deviation (σ) are known • T-‐test assumes o The data are independently sampled from a normally distributed population Degrees of Freedom • With the t-‐test, we are estimating a population mean based on the sample mean o We have to account for error! • Degrees of freedom are a quality of any statistical test o It applies to every test we do except for the z-‐test o It is a correction factor • We know deviations from sample mean sum to zero • So, given a particular sample mean, only N-‐1 observations are really “free” to vary • We give our statistical test N-‐1 degrees of freedom T-‐test: Steps 1. Establish what information you have 2. Decide if the t-‐test is appropriate…if it is, 3. Calculate your t-‐statistic 4. Calculate your degrees of freedom (using N-‐1) 5. Use Appendix E.6 in the textbook to find critical values 6. Compare your t-‐statistic to critical value a. If your t statistic is more extreme than the critical value, then reject the null hypothesis (if not, retain the null hypothesis) Using Appendix E.6 in The Textbook • The column you pick depends on: o Your alpha level (level of significance) o Whether your test is one tailed or two tailed • In this class we will almost always be using the alpha level 0.05 • When picking a row, this depends on your degrees of freedom (DF) o For a one sample t test, it is N-‐1 • NOTE: one issue is you might have to add a negative sign or a plus or minus sign to the critical value of t, based on your distribution, it if it negative or two tailed respectively T-‐test Example: Developmental Milestones • Doctors use standard charts to measure the progress of children’s development during routine check-‐ups o Children average 15-‐word vocabularies by age 18 months o Target sample (N=24): walked early and ate solid foods early o Prediction to test: they are talking early as well § We have attained 15-‐word vocabulary at M = age 16.5 months, SD = 1 month • We want to know if this talking early sample of children sample mean is statistically significant T-‐test: Set Hypotheses • This example is DIRECTIONAL • Setting up the null (0 ) and alternative (H 1 hypotheses o H 0 sample drawn from population with 15-‐word vocabulary at mean of 18 months § H :0 μ = 18 o H 1 sample from different population, EARLY vocab § H :1 μ < 18 § If this were a non-‐directional test, the alternative hypothesis would be μ ≠ 18 Steps: Milestone Example • (1, 2) Info we have: o Xbar = 16.5 o S = 1 o Hypothesized population mean = μ = 18; o YES, t-‐test is appropriate • (3) t = (Xbar – Xbar • s Xbar = s / √N = 1 / √24 = 1/4.90 = 0.20 (this is the standard error) t = (16.5 – 18) / 0.20 = -‐1.5 / 0.20 = -‐7.50 • (4) Degrees of freedom (DF) = N-‐1 = 23 • (5) From Appendix E.6: need alpha & directionality o One-‐tailed test, alpha level = .05 o Appendix row number = critical value = -‐1.714 o NOTE: 1.714 is the t test score… but we have a one tailed test in the negative direction, so we have to change it to -‐1.714 • (6) Comparing our calculated t-‐value to critical value: o -‐7.50 < -‐1.714, therefore… reject null hypothesis because it falls in the rejection region Summary View of Two Tests • The commonality between Z-‐test and one sample t-‐test is that we are only dealing with one set of data Paired t-‐test: Overview • We use this when we have “paired” or related/dependent observations (it is also called “dependent samples t-‐test”) o We are no longer dealing with just one sample of data • The sets of data may be from the same person at two points in time, or from people that are related to each other Paired t-‐test: Assumptions • Our data depends on each other so we do not need to have the assumption of independent sampling • With paired t-‐test, we are analyzing different scores • Paired t-‐test assumes distribution of the difference scores is approximately normal Difference Scores • X is the first score of a pair and X is the second score then: 1 2 o D = (X1 -2‐ X ) = difference score • Once we transform our two sets of related scores into one set of difference scores, we can proceed with exactly the same steps as with the one-‐sample t-‐ test • D will be negative if the pre test is smaller than post test, and vice versa
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