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# Review for Exam 2 PSYC 3301

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This 12 page Study Guide was uploaded by Rachel Marte on Saturday March 7, 2015. The Study Guide belongs to PSYC 3301 at University of Houston taught by Dr. Perks in Fall. Since its upload, it has received 297 views. For similar materials see Introduction to Psychological Statistics in Psychlogy at University of Houston.

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Date Created: 03/07/15

Exam 2 Review Basic Layout Exam 2 will have a structure similar to that of the first exam 0 Six short answer questions only five of which you will have to answer 0 These will probably involve explaining the purpose or meaning of certain concepts expect a question about Type I and Type II errors although this might end up as a part of the hypothesis test 0 Six short calculations only five of which you will have to answer 0 These could include calculating zscores probabilities from the unit normal table zscores for sample means etc 0 One hypothesis test 0 You will likely have to complete a hypothesis test in its entirety all four steps and possibly have to compute an effect size although this might be under the short calculations What to Know Review Layout 0 Important terms definitions and significance 0 Zscores for raw scores 0 Calculate a zscore 0 Be familiar with the unit normal table I Calculate probabilities using the zscore I Calculate the zscore given the probability I Calculate the raw score given the probability 0 Zscores for sample means 0 Calculate a zscore population mean sample mean sample size or standard deviation using the zscore for sample means formula you will be given 4 of the 5 parts of the equation and be asked to find the missing piece 0 Be familiar with the unit normal table I Calculate probabilities using the zscore I Calculate the zscore given the probability I Calculate the sample mean given the probability 0 Hypothesis testing 0 State the null and alternative hypotheses Determine whether it is a onetailed positive or negative or twotailed test Calculate the statistics to test the hypothesis zscore and probability Decide which hypothesis to retain and which to reject using the alpha level Describe in words what your results mean Calculate the effect size and determine what size effect it is 00000 Important TermsDefinitions One of the two hypotheses stated when performing a hypothesis test states that there is no effect no difference or no relationship there is no difference between the population and sample mean they come from the same distribution 0 Symbolized Ho One of the two hypotheses stated when performing a hypothesis test states that there is an effect difference or relationship there is a difference between the population and sample mean they come from different distributions 0 Symbolized H1 or Ha The probability value that is used to define the very unlikely outcomes if the null hypothesis is true it is also the probability of committing a Type I error 0 Symbolized 0t 0 Note In a twotailed test the alpha level is split in half half for each of the two critical regions Consists of outcomes that are very unlikely to be obtained if the null hypothesis is true 0 Note There will only be one critical region in a onetailed test but there will be two critical regions in a twotailed test each half the size of what the critical region would be if it was a onetailed test The standard deviation of the distribution of sample means the standard distance between a sample mean and the population mean 0 Symbolized am or SE 0 Formula oxh When the data lead you to reject Ho even though it is in fact true you conclude that a treatment has an effect when it actually does not have an effect a false positive 0 Note The smaller the alpha level the less likely a Type I error is When the data lead you to retain Ho even though it is false you conclude that a treatment has no effect when it actually does have an effect a false negative 0 Note The bigger the alpha level the less likely a Type II error is Statistical procedure that uses data from a sample to test a hypothesis about a population 0 Note As of now we are using zscores for this procedure A result is statistically significant if it is very unlikely to occur when the null hypothesis is true 0 Note A result is statistically significant if it is su icient to reject H0 ie the decision from the hypothesis test is to reject H0 or it is greater or less than would be expected by chance A way of quantifying the difference between two groups that emphasizes the size of the difference rather than confounding this with sample size measures the strength of a relationship instead of just saying whether or not a result is significant it tells you how significant it is and removes the variability of sample size from the equation 0 Formula Cohen s d 0 Note E ect sizes are often reported along with the results of a hypothesis test 0 Note You need only calculate the e ect size if you reject the null hypothesis A standardized score which allows for the comparison of raw scores especially if they are on different scales consist of a number and a sign or where positive scores are above the mean and negative scores are below the mean and the number is the distance between the mean and the raw score in units of standard deviation 0 Symbolized z o Formulas 36 x u 39 For raw scores 2 or z SD 6 39 For sam le means 2 2 p 39 ONE Zdistributions use the normal curve and zscores are in units of standard deviation 0 Note For example a zscore of I would be 1 standard deviation from the mean 0 Note The mean of the data will always have a zscore of 0 A distribution of statistics as opposed to a distribution of scores 0 Note the distribution of sample means is an example of a sampling distribution A mathematical theorem that specifies the characteristics of the distribution of sample means says that the distribution of the sum or average of a large number of independent identically distributed variables will be approximately normal regardless of the underlying distribution in other words the sampling distribution of any statistic will be normal or nearly normal as long as the sample size is large enough A hypothesis test that includes a directional prediction e g more less higher lower in the statement of the hypotheses and places the critical region entirely in one tail of the distribution 0 Note Directional tests can be either positive or negative depending on which end of the distribution the critical region falls 0 Note Also called a onetailed test A hypothesis test that does not include a directional prediction in the statement of the hypotheses and splits the critical region in two so that half of it is located in each tail of the distribution 0 Note Also called a twotailed test See Directional Hypothesis Test above See NonDirectional Hypothesis Test above ZScores for Raw Scores How to calculate zscores x i o For a sample 2 2 5 1 0 For a population 2 x q c 0 Remember that x is the raw score Eu is the samplepopulation mean and SD 0 is the samplepopulation standard deviation Remember that zscores serve two main objectives 1 They make raw scores more meaningful o A zscore Will tell you the exact location of a raw score in a distribution 2 They can be used to transform and standardize Whole distributions 0 Helps us compare results across different tests 0 Makes different distributions equivalent allows comparisons between distributions Zdistributions use the normal curve and zscores are in units of standard deviation A z distribution l l I I I 3 2 1 HM I l l I u3a u ZO p io h d p 10 p 20 p 30 The Translation of X to Z by the Transtormation Z X 500 As shown by the graph above zdistributions are always normal curves regardless of what the original distribution looked like A zscore of 0 is assigned to the mean of the original distribution a or E A zscore of is assigned to the value of the mean 1 standard deviation 0 or say n1 a A zscore of I is assigned to the value of the mean 1 standard deviation uI o Other zscores 3 2 1 0 I 2 3 etc are determined in a similar fashion The Unit Normal Table For a more detailed explanation of the unit normal table and how to use it along with sample problems look back at the notes for 21615 We use the unit normal table to find the probability that a statistic is observed in a certain range ie below a score above a score or between two scores on the standard normal distribution Once you have determined a zscore and want to know the probability of obtaining a score above or below it or between z and the mean or z and another zscore it is easy to locate the zscore in question on the table A series of probabilities will be listed next to the zscore each corresponding to a different area of the normal curve in relation to the zscore It also possible to be given a probability and use it to find the corresponding zscore All you need to do is locate the correct probability in the table make sure you re looking for the probability in the right column and see which zscore it is listed under If you can t nd the exact probability find the closest one If two are equally close to the probability you want then take the smaller one Below is an excerpt of the unit normal table Til31E 31 THE LilNItlT NORMAL T ELEt Eolutttti it listt ElSEIZIIE valor5 A tortiooi lino oration through a normal distribution at o tooo ro iooot ion divides tho distribttiott into two tootioot Colotttt B itlo39oti ot tho proportion in tho lorgtt Sootion coll liht ootiit Column C idottti oo the proportion in tlio smoliot section called tho toil Column D identi es the proportion betwooo tlio moan and tho tsooro Notequot otation the normal distribution 15 tyilmuotriooi tho proportions for g il t otioortozt ore 11o S mo its tho o for positive gtooro5 ll EH Ci W itquots fFliI EC ED FmFIDEthti Proportion Propmoon Frogmrtion Fmioii 39P rmanoioo z ll39t Body in Tait outtom Mom and a 2 in Emily M lJl Tait Bollamt Mom and Dim S m I Willi 5513 4l1 13 393 ELM lH 4lil 1355 IEIl 3914 1925 1112 Eli i l Am Dllli 112 3935 105439 H113 5121 4331 Uli i 123 l i 339 1 1113 12134 5 l l 34451 31 I54 5 11129 5 1411 3359 l 1411 5199 4501 5119 0311 5111 3321 l 1119 Ell 35 5239 ATE DEEE 39 1 131 52 391quot NEWS 1211quot Elli 52 3721 EJE TE LEE 5255 3T45 1155 1113 5319 4531 li131 939 133 6193 BTU 1293 113393 5359 541 3359 LE4 i l EIEIEE 1331 Cl 1 I 5393 Il 39E 3398 H135 6355 3632 l As you can see the unit normal table is divided into four columns In order to understand the differences between the probabilities listed in the latter three columns we must first distinguish between the body and tail of a distribution If we draw a vertical line through the curve at any random point we divide the distribution into two parts the body and the tail The body of the distribution is the part of the distribution that contains the mean The tail of the distribution is the part of the distribution that does not contain the mean If you look at the graphs at the top of the excerpt from the unit normal table the bodies of the distributions are labeled B and the tails are labeled C Column A lists the zscores The zscores on the table range from 0 to 4 and count by 01 Note that only positive zscores are listed If you have a negative zscore you take the absolute value and use the probability listed for its positive counterpart The other three columns list different probabilities associated with zscores in column A Column B lists the proportion in the body of the distribution If you are looking for everything either above or below the zscore and that area happens to include the mean therefore making it the body of the distribution then you use the proportion listed in column B At the top of the unit normal table three graphs are displayed The first two show situations in which you would use column B because the shaded regions include the mean The two graphs below also illustrate situations in which you would use column B to find the probability of the shaded region Column C lists the proportion in the tail of the distribution If you are looking for everything either above or below the zscore and that area does not include the mean therefore making it the tail of the distribution then you use the proportion listed in column C If you look at the graphs at the top of the unit normal table again you would notice that none of them correspond to this situation However if you were looking for the area of the unshaded regions of graphs one and two you would use column C The two graphs below illustrate situations in which you would use column C to find the probability of the shaded region Column D lists the proportion between the relevant zscore and the mean The third graph at the top of the unit normal table shows a situation in which you would use column D because the shaded area is between 2 and the mean The two graphs below also illustrate situations in which you would use column D to find the probability of the shaded region ZScores for Sample Means You can compute a zscore for the probability of obtaining a certain sample mean These z scores are usually higher than the zscores we found previously because of the extra error the equation includes the standard error Equation for finding the zscore for the probability of obtaining sample means 2H z ONE 0 Remember that E is the sample mean u is the population mean a is the population standard deviation and n is the sample size the term oxh in the denominator is called the standard error Standard Error For a more detailed explanation of standard error and sample problems look back at the notes for 21815 The standard error of the mean is not the same thing as the standard deviation Standard deviation is the standard distance between a score and the mean while standard error is the standard distance between a sample mean and the population mean Standard error measures how much error we should expect between a sample mean and the population mean on average The larger the sample size the lower the standard error Standard error is represented am or SE Equation for finding the SEom o 039 0 Remember that o is the population standard deviation and n is the sample size this term is found in the denominator of the formula for nding zscores for the probability of obtaining sample means Sampling Distributions For a more detailed explanation of sampling distributions and distributions of sample means along with sample problems look back at the notes for 21815 We often use zscores to represent individual scores but they can also represent the position of entire samples in a population of scores Even though none of the samples are exactly alike we can still establish rules for the behavior of all the samples that make up a population The distribution of sample means is the collection of sample means for all the possible random samples of a particular size n that can be obtained from a population Distributions of sample means are not based on raw scores but on means this is why zscores will now represent entire samples instead of individual scores Sampling distributions are distributions of statistics that are obtained when you select all possible samples of a specific size from a population The sampling distribution centers on the population mean 11 while different sample means i are turned into zscores that make up the distribution Sample means will be normally distributed in accordance with the Central Limit Theorem Q or a more detailed explanation of the CLT than is o ered above under Important TermsDe nitions look back at the notes for 21815 Hypothesis Testing For a more detailed explanation of hypothesis testing and the concepts related to it along with sample problems look back at the notes for 223153615 The point of hypothesis testing is to use samples to make inferences about the population We use hypothesis testing when we want to know the likelihood of obtaining the results we got from the sample ie the sample mean from the population We can then compare this probability with a standard the most commonly used standards are 05 01 and 001the standard is called the alpha level or Ct to determine whether or not these findings are very unlikely to occur in the population If we determine that a sample mean is very unlikely to occur in a certain population we consider the possibility that that statistic does not in fact come from that population Such unlikely results suggest that the sample mean may actually come from a di erent distribution which would mean that there is a significant di erence between that distribution and the original distribution of the population In other words we are trying to determine whether or not a sample is part of the original distribution or is part of its own distribution If the sample is part of a different distribution then there must be a significant difference between the sample and the original population In this way we can tell if a treatment has a significant effect The Process of Hypothesis Testing As you may recall we follow four steps when performing a hypothesis test 1 State a hypothesis about a population both the null and alternative hypotheses 2 Use the hypothesis to predict characteristics the sample should have set criteria for a decision on whether or not a sample is significantly different from the population 3 Obtain a random sample from the population 4 Compare the obtained sample to the prediction about the population Step 1 The hypotheses 1 The null hypothesis Ho treatment has no effect there is no change or relationship present there is no difference between the population and sample mean they come from the same distribution 2 The alternative hypothesis H1 or Ha treatment has an effect on the dependent variable there is an effect come from a different distribution Recall the difference between a onetailed and twotailed hypothesis test Which type of test we are performing affects how you will state the alternative hypothesis We use a twotailed test when we just want to know whether or not there is a difference between the population mean and sample mean We can also perform directional hypothesis tests if we want to know what direction the difference is in We use a positive onetailed test if we want to know whether or not the sample mean is significantly higher than the population mean We use a negative one tailed test if we want to know whether or not the sample mean is significantly lower than the population mean In the graphs below the shaded areas are the critical regions Pmitiu um Lulu In 39c39nix c one tailed trxi T39NCAtazlcd tcs How to state the hypotheses 1 The null hypothesis 0 Sentence form e g There is no difference between the sample mean and the population mean 0 Symbolization o Hmiu or 0 Ho 2 p 0 2 The alternative hypothesis a Twotailed test 0 Sentence form e g There is a difference between the sample mean and the population mean 0 Symbolization o Huxiu or O H1ip 0 b Positive onetailed test 0 Sentence form e g The sample mean is significantly higher than the population mean 0 Symbolization o Huxgtu m o Huxugt0 c Negative onetailed test 0 Sentence form e g The sample mean is significantly lower than the population mean 0 Symbolization o Huxltu m o Huxult0 Step 2 Step 2 s purpose is to answer the question Which sample means would indicate that the H0 is true and which sample means would indicate that the H1 is true In order to decide which sample means indicate that H0 is true we choose an alpha 0L level The alpha level or level of significance informs us about high probability and low probability samples The most commonly used alpha levels are 05 01 and 001 when in doubt assume 0L05 An alpha level of 05 0L05 indicates that we will only reject Ho if the sample falls into the 5 most extreme sample means The alpha level we choose will determine the critical region The alpha level is a probability value that is used to define the very unlikely sample outcomes if H0 is true The critical region is composed of the sample values that are unlikely if Ho holds true If the sample data fall within the critical region then we reject Ho In other words an alpha level of 05 means that the critical region contains 5 of the most extreme scores in the distribution The alpha level is just the probability value that marks off the critical region and the critical region is the actual area of the graph made up of the extremely unlikely sample mean values we would use to reject Ho For your alpha level of choice you need to find the boundaries that separate the critical region from the rest of the distribution this will be a zscore One important point to keep in mind is that if you are performing a twotailed test you must divide your critical region in half For example if 105 and you are performing a twotailed test you need have a critical region at each extreme end of the graph that begins at a probability of 025 In order to determine the boundary of the critical regions you must use the unit normal table to find the zscore that matches your alpha level Look for the probability closest to your alpha level in column C the critical regions are always in the tails of the distribution The zscore that matches that probability marks the beginning of the critical regions and is denoted Zcrit Step 3 We will not actually go through the process of collecting data in class Instead you will be given any relevant information e g sample size population mean and statistics which you will then use to calculate zscores Once you have collectedbeen given the datastatistics you will use that information to calculate the test statistic As of now we are using zscores as our test statistic of choice Therefore we need the sample mean so that we can compare it to the population mean Since we are interested in comparing sample means to population means we will use the zscore for sample means formula 2 57 Remember that the zscore formula describes a ratio of the obtained 039 difference versus the difference due to chance The zscore you calculate is often referred to as Zobt the obtained zscore or zcalc the calculated zscore This just means that you calculated this zscore from the sample statistics instead of looking it up in the unit normal table like Zcrit Step 4 After getting zobt in step 3 you need to use it to decide whether to reject or retain H0 in accordance to the criteria you established in step 2 If zobt falls within the critical region you will reject H0 and accept H1 If zobt does not fall within the critical region you will retain Ho and reject H1 The easiest way to determine whether or not zobt is in the critical region is to sketch a graph where you label the population mean z0 the Zcrit values and shade in the critical regions and the zobt value If zobt falls in a shaded region reject the null Statistical Signi cance A result is statistically significant if it is very unlikely to occur when the null hypothesis is true In other words a result is statistically significant if 0 It is sufficient to reject Ho ie the decision from the hypothesis test is to reject Ho 0 It is greater or less than would be expected by chance Effect Size The effect size measures the strength of a relationship instead of just saying whether or not a result is significant it tells you how significant it is and removes the variability of sample size from the equation Hypothesis testing tells you if the result is significant or not but it doesn t tell you about the size of an effect Effect sizes are often reported along with the results of a hypothesis test You really only need to calculate the e ect size if you reject H0 We will calculate the effect size using Cohen s d o Cohen s d Size of the e ect 0 Small effect d i 2 0 Medium effect d i 5 0 Large effect d i 8 The scale above has a lot of gray areas in it Is an effect size of 3 small or medium Either pick one or the other if the context of the problem suggests one over the other or just say that it s a small to medium effect Error There are two types of errors we can make in hypothesis testing 0 Type I Error When the data lead you to reject Ho even though it is in fact true 0 This can also be called a false positive 0 Type I error is equivalent in size to the size of the critical region e g if 105 we will incorrectly reject Ho about 5 of the time o The smaller the alpha level the less likely a Type I error is 0 Type II Error When the data lead you to retain Ho even though it is false 0 This can also be called a false negative 0 The bigger the alpha level the less likely a Type 11 error is As you can see the alpha level a ects the chances of both a Type I and a Type II error If we choose a smaller alpha we will be less likely to get a Type I error but we will be more likely to get a Type 11 error If we choose a bigger alpha we will be less likely to get a Type 11 error but we will be more likely to get a Type I error In other words we need to nd a balance we can t make the alpha level either too big or too small

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