PSYS 054 Study Guide for Exam 2
PSYS 054 Study Guide for Exam 2 PSYS 054
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This page Study Guide was uploaded by Delaney Row on Sunday February 28, 2016. The Study Guide belongs to PSYS 054 at University of Vermont taught by Keith Burt in Fall 2016. Since its upload, it has received 243 views. For similar materials see Statistics for Psychological Science in Psychlogy at University of Vermont.
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Date Created: 02/28/16
PSYS 054 Exam 2 Study Guide Delaney Row Sampling Distributions and Hypothesis Testing 83 87 811 Random sampling sampling a set of items so that each time an item is selected every item in the population has an equal opportunity of occurring Sampling error variability of a statistic from sample to sample due to chance Sampling distribution the variability of a statistic over repeated sampling from a specified population 0 In other words a sampling distribution tells us what values we will or will not expect to obtain for a certain statistic under a set of predefined conditions something will happen ifsomething else is true 0 You typically graph the sampling distribution on a histogram Discrete probability distributions If a give variable is a discrete variable then its probability distribution is referred to as a discrete probability distribution 0 Discrete variables are variables that can only take on a small set of possible values Continuous probability distributions If a given variable is a continuous variable then its probability distribution is referred to as a continuous probability distribution 0 Continuous variables are variables that take on any value Why are sampling distributions important for hypothesis testing 0 If we didn t have sampling distributions we would not have any statistical tests 0 In order to test a hypothesis we need sampling distributions I The sampling distribution generates a model of what things would look like given the null hypothesis was true and a statistic could be collected an infinite number of times Hypothesis testing steps State an alternative hypothesis the research hypothesis State the null hypothesis Set an alpha level for our purposes 00 5 Collect data Create a sampling distribution Find the critical value values to draw rejection region Reject or retain the null hypothesis based on if the target sample is in the rejection region or not Null hypothesis Ho the statistical hypothesis tested by the statistical procedure usually stating no difference or no relationship between variables Type I Error incorrectly rejecting a true null hypothesis aka false positivequot OOOOOOO 0 Think of a jury convicting an innocent person Type 1 Error failing to reject a false null hypotheses aka false negativequot 0 Think of a jury not convicting a guilty person Z test Directional onetailed tests test that rejects extreme outcomes in one specified tailed of the distribution negative or positive you need to determine this Nondirectional twotailed tests test that rejects extreme outcomes in either tail of the distribution positive and negative What are alpha and beta levels What is the standard alpha level in psychology 0 Alpha the probability of a Type I Error same as the rejection region so typically 5 0 Beta the probability of a Type 11 Error 0 The standard alpha level in psychology is 005 0 NOTE by reducing the probability of a Type I Error you will increase the probability of a Type 11 Error 0 What is the Central Limit Theorem What are the notation and terms used 0 This 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 o It creates sampling distribution of the mean without actually taking 10000 random samples 0 Given a population with mean u and variance oz the sampling distribution of the mean will I Have a mean equal to u I Have a variance equal to 02 N and SD equal to o N o It also states that the shape is essentially normal as long as Ngt30 it will get more normal the bigger the sample is o This theorem underlies formulas on most statistical tests 0 Notation I u sub xbar the mean of the sampling distribution of the mean I 0 sub xbar the standard deviation of the sampling distribution of the mean also called the standard error Ztest give conceptual understanding formula stepbystep application and the information needed for the test 0 Ztest is useful when I You want to compare a target sample mean to a population mean I You know the population mean I You know the population SD 0 Basic idea I Convert observed sample mean to a Zscore location on the sampling distribution using the Central Limit Theorem I Find probability of getting that Zscore using area under the curve Appendix E10 in the textbook I M Z 0 0 Formula m I Sample mean minus the population mean all divided by the standard error Standard error population SD divided by the square root of N 0 Steps I Establish alpha level rejection region and critical values here critical value is a Zscore I Find the standard error of the population I Convert observed sample mean to Zscore I Compare your calculated Zscore to the critical value I Reject or retain the null hypothesis What are the assumptions of the ztest o The data are independently sampled from a normally distributed population 0 Mean 1 and standard deviation 6 are known population What are the critical values of z for onetailed and twotailed tests at the standard alpha level 005 0 One tailed 5 cutoff on one side critical value z score of 165 or 165 0 Two tailed 25 cutoff on both sides critical value z score of 196 OneSample t test 1211 121 7 Onesample ttest give basic understanding formula stepbystep application and information required for the test 0 Ttest is used when the population variance SD is unknown 0 In the onesample ttest we are comparing a sample mean to a hypothesized population value 0 What you need to know I Target sample mean Xbar I Target sample SD 5 I Population mean assuming the null hypothesis 1 I Target sample size N t X 1 X u S 0 Formula V N o The steps to carry out a ttest are the same as the ones to carry out the z test except you use 5 sample SD to estimate the population SD I Establish what information you have I Decide if the ttest is appropriate and if yes I Calculate your tstatistic using the formula I Calculate your degrees of freedom Nl I Use Appendix E6 to find critical values I Compare your tstatistic to critical value If your t statistic is more extreme than the critical value then reject the null hypothesis if not retain the null hypothesis What is the calculation of degrees of freedom for the onesample ttest o N1 What is the basic idea of the shape of the t distributions How does it vary by degrees of freedom 0 This new distribution we call a t distributionit 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 at pushed down How do you use Appendix E6 to find critical values of the t distribution 0 The column you pick depends on I Your alpha level level of significance usually 005 I Whether your test is one tailed or two tailed 0 When picking a row this depends on your degrees of freedom DF I For a one sample t test it is N1 0 NOTE one issue is you might have to add a negative sign or a plus and minus sign to the critical value of t based on your distribution it if it negative or two tailed respectively Percentage Points of the fDistribution 0 052 052 I 0 t t 0 t OneTailed Test Two Tailed Test Level of Signi cance for One Tailed Test 25 20 15 10 05 025 01 005 0005 Level of Signi cance for TwoTailed Test df 50 40 30 20 10 05 02 01 001 1 1 000 1376 1963 3078 6314 12706 I 2920 4303 2 816 1061 1386 1886 12924 2353 3182 4541 5841 3 765 978 1250 1638 8 610 32 2776 3747 4604 4 741 941 1190 1533 21 I 5 727 920 1156 1476 2015 2571 3365 4032 6869 6 718 906 1134 1440 1943 2447 3143 373 7 711 896 1119 1415 1895 2365 2998 3455 5041 8 706 889 1108 1397 1860 2306 2896 2350 4781 9 703 883 1100 1383 1833 2262 2821 3169 4587 10 700 879 1093 1372 1812 2228 2764 11 697 876 1088 1363 1796 2201 i 1782 2 179 12 695 873 1083 1356 2 4 221 1771 2160 2650 301 13 694 870 1079 1350 77 4 140 1761 2145 2624 29 14 692 868 1076 1345 15 691 866 1074 1341 1753 2131 2602 2947 4073 16 690 865 1071 1337 1746 2120 63 1069 1333 1740 2110 17 gm 1 mm 1 farm 1 734 2101 2552 2878 3922 What are the assumptions of the onesample ttest 0 Data are independently sampled from a normally distributed population Paired t test 136 139 1311 1313 Paired ttest give basic understanding formula stepbystep application and information required for the test 0 We use this when we have paired or related dependent observations it is also called dependent samples ttestquot I We are no longer dealing with just one sample of data D SD 0 Formula V ND 0 With paired ttest we analyze difference scores I This is one score 1 minus the other 2 also can be thought of pretest minus posttest 0 Difference scores I Pretest Xsub 1 posttest Xsub2 I Then D Xsubl Xsub2 the difference score I Once we get the difference score we are basically following all the steps we do in a onesample ttest with the difference score being the main data we are looking at Degrees of freedom for paired ttests will always be Number of difference scores 1 Use the M and SD of difference scores in the formula 0 What you need to know I Mean of difference scores Dbar I SD of difference scores S sub D I Standard error of difference scores SDbar I Number of difference scores N sub D What are the assumptions of the paired ttest o It assumes that the distribution of difference scores is relatively normal 0 It loses the assumption of independence of observations What is the hypothesis notation for the paired ttest o In all cases in M1 M2 0 Twotailed setup I Ho up 0 null hypothesis for any paired ttest I H1 up 0 0 One tailed test in the positive direction 39 H0 MD0 39 H1 MD gt 0 0 One tailed test in the negative direction 39 H0 MD 0 39 H1 MD lt 0 What possible research situations would you use the paired ttest 0 Example researching any improvement in test scores of students at the beginning of a course and at the end of the course How do you report results of a ttest using APA style 0 Italicize notation letters but not numbers 0 Include degrees of freedom in parentheses o If calculated t falls in rejection region write p lt 05quot or report exact p if you are using SPSS I Or write ns meaning not significant eg quot t11 053 nsquot 0 So for our paired ttest example I t7 250 p lt 05 I NOTE lower case t should be italicized and then the 7 in parentheses Independent samples t test 148 14 9 Independent sample ttest give basic understanding formula stepby step application and information required for the test 0 This is the most common ttest used in actual research 0 This test compares independent sample means I For example the difference between mean of variables X1 and X2 when these represent scores from different unrelated individuals X1 X2 1 i i N1 N2 0 Formula I dfN1N22 0 Steps I Appropriate test independent samples ttest I t X1bar Xzbar sqr521N1 szzNz I dfN1N22 I Twotailed test 05 alpha level degrees of freedom use Appendix E6 I Compare calculated t statistic to critical values Note a twotailed test has two critical values positive and negative If calculated t is greater than positive or less than negative critical value reject the null hypothesis otherwise retain o What you need to know I M and SD from two independent samples What are the assumptions of independent sample ttest 0 Data are independently sampled o From a normally distributed population I Normally assumption 0 Variance of the two groups is the same in the population I quotHomogeneity of variance assumptionquot What is the hypothesis notation for independent sample ttest 0 Similarly to a paired ttest these hypotheses are written in the form of population mean differences I Ho M1 u2 0 null hypothesis This is saying that the mean scores from each group will be equal have no difference I H1 M1 uz 0 twotailed research hypothesis I H1 M1 uz lt 0 onetailed research hypothesis OR I H1 M1 W gt 0 onetailed research hypothesis Con dence intervals include only material reached by end of 225 lecture 1218 1219 1413 Conceptual understanding of a confidence interval 0 A confidence interval is an interval with limits at either end having a specified probability of including the parameter being estimated 0 Confidence intervals shift our attention to the population parameters I We calculate a range of potential population means based on our sample data Relationship between alpha level and confidence level 0 The confidence level is 1 a I Confidence level and alpha must always sum to one 0 We will often have a 95 confidence interval because we often use the critical value of 005 5 I We are saying that our obtained value would fall within the limits 95 of the time 0 Confidence level retaining null hypothesis 0 Alpha level rejecting null hypothesis Formulas for calculating a confidence interval onesample independentsamples versions S 0 Cl formula V N 0 CI of 111112 formula CI X1 X2 i Cos 5X1 X2 0 Confidence intervals are always twotailed critical values
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