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# PSYS 04, Statistics for Psychological Science PSYS 054

UVM

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This 6 page Class Notes was uploaded by Delaney Row on Monday February 8, 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 29 views. For similar materials see Statistics for Psychological Science in Psychlogy at University of Vermont.

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Date Created: 02/08/16

PSYS 054 - Statistics for Psychological Science Notes for the week of: 2/8/16-2/12/16 Sampling Error • It is the idea that if we take multiple random samples from the same population, the statistics we acquire will vary slightly across all the samples • The term “error” here does not mean there was any systematic mistake made. Sampling error is universal o In the example on the PowerPoint there is a margin of error is 3 percentage points and we only have 1002 Americans as a sample, but if we had all Americans answering… § It would mean that between 82-88% of all Americans would answer yes (most likely) o If we had access to the population of all Americans and asked all of them this question…there would be NO margin of error! • It is impossible to have a sample without a margin of error (remember error does not refer to a systematic mistake) Sampling Error (Continued) • Population of all American high school students who took the SAT this past year (this is our population of interest) o Population parameters (mean, SD): µ = 500, σ = 100 o Remember that in this particular case, we know how large the population is because the SAT company keeps track of this, but this usually doesn’t happen • If you draw random samples from this population... o (of size N) o Means and SD’s will differ o Means will be around ~ 500 o SDs will be around ~ 100 More Sampling • We are trying to identify if a coin is far or not (we flip it 100 times) • Our population = the two possible outcomes (heads or tails) • Our statistic = the number of heads you get on each of the samples you take • Suppose we get… o 52 heads – we would conclude fair (not surprising) o 60 heads – still fair (still not surprising) o 75 heads - now starting to seem possibly not fair o 94 heads - most of us would think its not fair • There is a subjective line we have to draw, and what is beyond it would be saying, “Ok this coin is likely weighted” and below it is saying, “Ok this coin is probably fair” • Remember though, that no matter where we draw that line, there is always still a chance that a fair coin can show up beyond it o Say the line is at 75…a fair coin could be flipped and result in 80 heads (it’s possible) To the Rescue: Sampling Distributions • Sampling distribution allows us to be more precise in judging our individual results • Sampling distribution references a particular statistic (usually the sample mean) • It gives us the probability of getting different results when we randomly sample from a population • Why do we want a sampling distribution? o Now we can compare a sample mean with a distribution of sample means! (We used to just be comparing individual scores) Back to SAT Scores • Imagine we have a sample of high school students (N = 30) who have taken an SAT-Prep class (we will say a Kaplan course) • The statistics for our “prep class” sample: o M = 540, SD = 110 • We want to know whether or not this prep class is effective in raising scores (research Q) • Let’s pretend we can repeatedly randomly sample from the population o We can go back to what we know about the population “Flipping” the Research Question • We can’t really do anything else with the “prep class” data • But we can work with the general population data o Take repeated random samples from the (µ = 500, σ = 100) population, of the same size (N = 30) as our “prep class” sample • Compare the Kaplan sample and its mean to repeated random samples of the regular population o Then we can tell if there is a difference, and if the Kaplan class is helpful to raise scores o *Kaplan mean is 540* • New Question: how likely/unlikely is it that we would get (M = 540) from a random sample of the population? Assessing Sampling Error • Lets take multiple samples (each N=30) from the population • The population has a known mean (µ = 500), but sample our means will fluctuate • Now plot relative frequency of sample means as a histogram o This is a sampling distribution (distribution of sample means) Sampling Distribution Example • NOTE: this is a distribution of sample means (of random samples of N = 30) not of individual scores • Our arbitrary cutoff… 540 might not seem that unusual (but remember this is 10,000 random draws) o So there does seem to be something special going on about Kaplan scores (the class seems to raise scores) SAT Study Conclusions • American high school students (not partaking in a Kaplan course) generally average around 500 on their SAT • Our sample of 540 is out towards the upper tail of the sampling distribution o NOTE: we are comparing our sample mean to other sample means • Therefore, it is highly unlikely that our sample came from this “regular” population o Random sampling error is unlikely to be the cause of a 40 point increase in means between the two groups • SO… o Our conclusion: SAT prep class helped!!! Basic Logic of Hypothesis Testing 1. Let’s start with the assumption that your sample comes from a “non-special” population (meaning high school students who did not take a Kaplan class) a. This is the “null hypothesis” 2. Now find out what this population does 3. Compare our sample to that standard • So, the null hypothesis is the hypothesis of no difference/effect o 500 is going to be the mean (no difference in mean) o Then give a hypothesis that has a difference, 540 is going to be the mean The Null Hypothesis: H 0 • You usually are trying to reject the null hypothesis, so that you can accept the alternative hypothesis (H 1 o Rejecting it is usually desirable because that implies we have an effect • We also set up an alternative “research” hypothesis, often labeled… H 1 • NOTE: you should set up these hypotheses prior to doing the research Non-statistical Hypothesis Testing • Good analogy – Jury Trials • H 0 defendant is innocent • H 1 defendant is guilty • You have to start out with H 0 that they are innocent • You only have the evidence presented to you to make your decision o Like coin flip example…. a 90 is possible…. DNA evidence, a lot of evidence, BUT defendant could still be innocent • NOTE: terms for null hypothesis = reject or retain SAT Example: Hypothesis • H 0 our sample (M = 540) is no different than a population of high school students who did not take an SAT prep class (µ = 500) • H 1 our sample (M = 540) is different than a population of high school students who did not take an SAT prep class (µ > 500) o We can’t say µ = 540 because we don’t know if 540 is really accurate (remember 540 is the sample mean and we are looking at hypothesizing the population mean) o We just want to say it will be greater than 500; we are only interested if Kaplan increased the score, not by how much Sampling Distribution Notes • We can create sampling distributions of any statistic (mean, median, SD, variance) • The larger the sample size, the more tightly compressed the sample mean will be to the population mean o Give a more precise estimate of how likely the sample mean could come from the regular population Step-by-Step: SAT Hypothesis Testing 1. Based on what the research question is… a. Translate it into null and alternative hypotheses 2. Set a cutoff point (we will learn how next class) 3. Collect data and compute sample statistics 4. Make a decision! Decision-Making with Hypothesis Tests • Whenever we test hypotheses, we compare data to our prediction • Remember we can be correct or incorrect, in different ways Decision-Making: Type I or II Errors • We try to learn from errors • Type I error is seen as more problematic (think of testing medications) Directional vs. Non-directional Tests • Non-directional tests: reject H0if observed mean is either substantially higher or lower than population mean • Directional tests: reject 0 only if mean difference is in expected direction o Directional - alternative hypothesis has a specific direction to it (specifically predicting Kaplan classes would increase test scores) • Words like increase/decrease are clues for if a research setup is directional or non- directional! Examples of Directional vs. Non-directional Tests • Directional (greater than direction) • Directional (less than direction) • Non-directional (who is higher bullies or victims - not giving prediction over which group would be higher!)

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