Notes for 3/30/15-4/1/15
Notes for 3/30/15-4/1/15 PSYC 3301
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This 11 page Class Notes was uploaded by Rachel Marte on Monday April 6, 2015. The Class Notes belongs to PSYC 3301 at University of Houston taught by Dr. Perks in Fall. Since its upload, it has received 168 views. For similar materials see Introduction to Psychological Statistics in Psychlogy at University of Houston.
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Date Created: 04/06/15
33015 TTests for Two Independent Samples Using a TTest for Two Samples Most of the time in the real world researchers do studies involving two or more samples The t tests we have performed thus far have only used one sample to draw conclusions about a population Now that we are using two samples if there are more than two samples we cannot use a ttest at allthat requires a completely different kind of test we must remember to label which distribution our statistics come from e g SD1 and SD2 n1 and n2 SS1 and SS2 There are two different kinds of samples independent and dependent and we use a different kind of ttest for each one Right now we will be doing ttests for two independent samples We will learn how to do a ttest for two dependent samples later Independent vs Dependent Samples 0 Independent Samples 0 Samples come from completely separate samples I For example samples are drawn from women and men or children and adults 0 Also called a betweensubjects design 0 Dependent Samples 0 Sets of data come from the same sample 0 Also called a withinsubj ects design or repeated measures design Hypotheses The objective of a ttest for two independent samples is to evaluate mean differences between two populations or treatment conditions 0 The Null 0 H0 111 pg 0r H0 111 1120 I Ho mean of population 1 mean of population 2 0r Ho mean of population 1 mean of population 20 0 The Alternative 0 H1 uni H2 or H1 H1 u2 0 I H1 population means are not equal or H1 mean of population 1 mean of population 2 does not equal zero Note u21 X TStatistic The tstatistic for a single sample ttest gives the general structure for the two sample ttest s 21 iz U1 U2 Siliz x statistic t w single sample vs t 2 two samples Z S 0 2 vs i1 i2 0 In a single sample ttest we only have one sample mean 2 o In a two sample ttest we have two sample means i1 and i2 that we are trying to compare we subtract them because we are trying to compare them H VS 139 H2 o In a single sample ttest we are comparing our sample mean to one population m an u o In a two sample ttest we are comparing the difference between the two sample means i1 2 to the difference between two population means 111 m I In this class we will always assume that u 1 uz0 ie our hypothesis is that the population means are the samethere is no difference between the population means 0 Si vs 33142 0 In a single sample ttest the estimated standard error Si is determined by the S equatlon S V o In a two sample ttest we use the standard error of the difference between the S 2 S 2 sample means 331 zz wh1ch 1s determ1ned by the equatlon 33142 nil quotL2 I For an independent tstatistic standard error measures the amount of error that is expected for the sample mean difference I We need to account for the fact that error estimates based on larger samples represent the population error better than error estimates based on smaller samples that s why we split up the inside of the radical for the two samples and have n1 and n2 I sz is the pooled variance 2 sslss2 Sp df1df2 0 Remember that SS is the sums of squares SS 2 2x 502 in this class the SS values will be listed for you so that you do not have to calculate them e g SS65 0 Notice that you add the degrees of freedom from both samples in the denominator The Three Equations for Finding the TStatistic Find each of these components in the order listed so that you know their value before you need to plug it into the equation in the next step The main goal here is to calculate the tstatistic 1 Pooled Variance 2 sslss2 df1df2 2 Standard Error 2 2 S 2 11 Xl Xz n1 n2 3 TStatistic 21 iz P1 M2 Sil iz O Sp 0 t Example Calculate the tstatistic for a two independent sample ttest given the following information the first sample has a sample size of 6 a mean of 50 and a sums of squares value of 50 and the second sample has a sample size of 6 a mean of 30 and a sums of squares value of 30 List information i150 i230 SSi50 SSz30 n16 n26 Step I 39 Find the pooled variance 2 39391ss2 1 df1df2 2 5030 1 55 80 322 1 0 Simplify numerator and denominator Simplzm Step 2 Fin the standard error 0 Choose proper equation Plug in known quantities 0 Si1i2 51 1L2 Choose proper equation 8 8 0 331 zz g g Plug in known quantities 0 33142 267 Simplify inside of radical Step 3 Calculate the tstatistic 21 iz P1 l12 0 t S Choose proper equation Xl Xz 0 W Plug in known quantities 0 t 12 23 Simplify numerator 0 tobt 1227 Simplify Odds and Ends for Hypothesis Testing Note that the equation for the tstatistic is essentially equivalent to the following a t h th 39 t a a 8 1 eSlS 1e the X s are sample data the u s are our hypothes1s about the populat1on and the denominator is the standard error The tstatistic we learned to calculate above the one that takes three equations to find is tobt in a hypothesis test In other words this calculated tstatistic is what we will compare to a critical value to determine whether or not to reject the null hypothesis The critical value tont will be found using the tdistribution table the same table that we used to look up tscores for single sample ttests In order to look up a value in the tdistribution table we must know the degrees of freedom For two independent sample ttests dfde df2 or dfn11 n21 In other words the total degrees of freedom is equivalent to the degrees of freedom of the first sample plus the degrees of freedom of the second sample remember that finding the df of a sample is the sample size minus 1 which is where the second equation above comes from An Example Hypothesis Test Example An independentmeasures study is used to compare two experimental treatments The first sample has i50 SS100 and n10 The second sample has x45 SS50 and n15 Is there a significant difference between the two samples Alpha05 List information 150 i245 SSl100 SSz50 n1lO n215 105 twotailed test Step I 39 State a hypothesis about a population null and alternative hypotheses H0 There is no difference between the samples H1 There is a difference between the samples Step 2 Set the criteria for a decision dfdf1 df2 df9 14 df23 125 11 m G E15 Pmpartinn in ne Ta L LZE illi f L he mdi in Tm Tails 39Enmbim 1 tiff LEG HE H132 H m l H656 1321 25113 13 9 uEE i 3 i 1 EEG 416345 1 I 3 LM I 2492 239 Step I 39 Find the pooled variance o S 2 51 52 1 df1df2 2 10050 O S p 914 2 150 O S p 23 0 102 652 Step 2 Find the standard error 5 2 s 2 p p o s X1 X2 quot1 quot2 652 652 1 5 Si1i2 Step 3 Calculate the tstatistic 21 iz U1 M2 0 t 21 22 t 50 45 0 104 5 t 704 tobt 8 Step 4 Make a decision w i I 2069 tcrit 2069 4 8 tcrit tobt tcrit2069 Step 3 Collect data and compute sample statistics Choose the proper equation Plug in known quantities Simplify numerator and denominator SimplifY Choose the proper equation Plug in known quantities Simplify inside of radical SimplifY Choose the proper equation Plug in known quantities Simplify numerator SimplifY QL Because tobt falls Within the critical region we will say that there is a significant difference A A It39s 94 4 I4 I I Aul l qlll 4115 More about Independent Samples TTests Assumptions of Independent TTests 1 Observations within each sample must be independent 2 The two populations from which the samples are selected must be normal 0 There is no real way to know if the populations are normal or not but we assume they are 3 The two populations from which the samples are selected must have equal variances 0 This is called the homogeneity of variance assumption Testing the Homogeneity of Variance Assumption Fmax Test To test the homogeneity of variance assumption we use Hartley s Fmax test This is a hypothesis test in which we are testing whether or not the population variances are equal Since the homogeneity of variance assumption states that the population variances must be equal we want to obtain a nonsignificant result retain the null hypothesis from the F max test Instead of finding a zscore or a tscore we will find an Fmax value to test for this hypothesis test We still need a calculated FmaX value Fobt and a critical value Fem To find the critical value we will use the Critical values for the FmaX statistic table Table B 3 Below is an excerpt from the F max statistic table TABLE 33 ERITlllEAL VAlLlllJlES FUR T HlE FEMAX STA39TIS39TIE quot flirt cri39lica l iruluui or or US are in lighll39auu 1pr and lint u ILll hey are in boldface type k 2 Number of Samples 1 ll 2 3 4x 5 3quot E 9 11 39l M 1 lifi 201E 252 295 3313 3153 1 M distal 4150 5 H il 4 5quot Til it39ll 9 Mill i 39l L21 5 Tquot l 5 03 l 3 l 1523 I 3 2133 229 25L 2amp5 2312 29 1451 22 23 3393 42 4H3 51 5d 5 Eli E 533 333 1 ii I l ll ill lb ii 15 I Ed I 9STquot Ell 1 ll 155 ll ELl 22 25 it39ll i 52 341i 351 3 The column header is labeled k Number of Samples Since we are checking the assumptions for a two sample ttest we will always use the k 2 column the first column The row header is labeled nl ie the degrees of freedom Which row you use to nd the Fmax value will depend on the degrees of freedom This is not the total degrees of freedom Use the df value for the smaller of the two samples For example if the samples you were testing had n16 and n218 you would look for the row for df5 not df22 Notice the table has two entries listed for each df value a bolded one and a nonbolded one If 0i01 use the value that is bolded If a 05 use the value that is not bolded We will always use 105 in this class How to Calculate the F max Statistic F 52largest max 52 smallest o In other words divide the larger variance by the smaller variance The Hypotheses 0 Ho The variances are equal 0 H1 The variances are not equal Example A researcher wants to know if studying impacts exam scores To test this she selects two independent samples one that studied n10 and one that did not study n15 The sample that studied received exam scores with a mean of 90 SS50 and s220 The sample that didn t study received exam scores with a mean of 100 SS30 and s210 Are the assumptions for a ttest met List information 190 i2100 SS150 SSz30 s1220 s2210 n1lO n215 assume 105 Step I 39 State a hypothesis about a population null and alternative hypotheses H0 The variances are equal H1 The variances are not equal Step 2 Set the criteria for a decision II 3 Number of Samples 3 4 5 fr 393quot E 9 III I ll em T IS 3 I2 113339 93933 105 I I I I 1 I12 2quot ELI II 1 HIEJ 1455 153 ELISA HillJ 133 15135 21 13 II I II 30 34 I 395 99 I I I113 Ill 35 99 I 11 121 I ll Ill IldJ39 153 16le 166 l ELIE 1523511 1392 142 13 323 31556 91M 3 321 I4 Eli 91 quot14 LII II I H II 24 129 134 139 Note Use df9 because the sample size of the smallest sample is 10 Fcrit403 Step 3 Collect data and compute sample statistics 52largest 0 Fmax Sz smallest Choose the proper equation 0 Fmax Plug in known quantities Fmax 2 SimplifY Step 4 Make a decision lk 2 403 Fobt Fem Because Fobt does not fall Within the critical region we Will say that there is not a significant difference We Will retain Ho The variances are equal Because the variances are equal the homogeneity of variance assumption is met Variances that are Not Homogeneous If you perform an F maX test and reject the null hypothesis ie the variances are not equalhomogeneous you must use a different formula to determine What df total to use In other words you cannot just use add df1 and df2 to get the total df to use in looking up tent in the t distribution table Instead you must use the following formula V1 V22 df 2 2 V1 V2 quot1 1 n2 1 12 522 Where V1 n and V2 n 1 2 V is just a ratio of variance divided by sample size and doesn t stand for anything in particular V s are substituted into the original equation to make it look neater Use the sample variances and sample sizes to find V1 and V2 and then substitute your answers wherever you see V1 or V2 in the original equation Example Given the following information What is the dftotal if the homogeneity of variance assumption is not met s1220 s2216 11110 1128 Step I Find V1 0 V1 Choose proper equation 0 V1 Plug in known quantities Simply 0 V2 Choose proper equation 0 V2 Plug in known quantities Simply Step 3 Find dftotal V V 2 df V1 ii n1 1 n2 1 222 d f W Plug in known quantities 101 81 Choose proper equation 42 d f 4 4 Simplify main numerator and main denominator 9 7 16 d f Simplify main numerator and main denominator Effect Size for the TStatistic We will use two different measures of effect size for the tstatistic Cohen s d and percentage of variance explained r2 Cohen s d Recall that the equation for Cohen s d that we have used up to this point d The equation for Cohen s d for a two sample ttest is very similar 21 5 2 d 102 This new equation is still a mean difference divided by a measure of standard deviation Instead of using a sample mean and population mean we are finding the mean difference between the means of two samples Instead of using the population standard deviation we use the square root of the pooled variance Recall that we just learned how to calculate the pooled variance SP2 Recall that for Cohen s d o 2 small effect 0 5 medium effect 0 8 large effect r2 The measure of r2 tells us how much variability in the dependent variable scores is explained by the treatment effectindependent variable In other words it is the percentage of variance explained A r2 is almost always reported along With significant results t2 t2 df Please note that t in the equation is the tobt value that you calculate and df is the dftotal 12 Effect Size for r2 0 01 small effect 0 09 medium effect 0 25 large effect In many fields of psychology it is common to get low r2 values because psychologists deal with people and people are very variable In other fields it may be more common to get r2 values of 8 or 9 but that is unlikely in psychology Example Researchers found a significant effect of training on employee safety With t 30 and df 8 how much variance in employee safety can be attributed to the training What size effect is this 2 t2 0 r t2df Choose proper equation 2 32 0 328 Plug in known quantities 9 0 r2 Q Square 3 in both the numerator and denominator 9 0 r2 E Simplify denominator Simplify o This means that 53 of variance in employee safety can be attributed to the training 0 3953 is a large effect sizeI 0 Look at the above chart E ect Size for r2 to determine what size e ect this is Formally Reporting Results It is important that you know how to formally report the results of your research The format for reporting significant and nonsignificant results is different mainly because you must report much more information if your results are significant Below is an example of reporting each type of result significant and nonsignificant with the relevant information labeled Significant results rejected null 0 This study found a significant effect of drinking alcohol on memory impairment t28 325 p lt05 d 143 The group who drank alcohol recalled less words M 20 SD 55 than the group that did not drink alcohol M 25 SD 45 Drinking alcohol accounts for 43 of the variance in memory performance 1 2 43 o trefers to the fact that the researchers performed some kind of ttest I The 28 means that the ttest had a total df of 28 o p lt05 means that the results fell within the critical region at 0L05 the results were likely to occur less than 5 of the time o d refers to the effect size Cohen s d I Sometimes different measures of effect size may be reported instead of Cohen s 1 so there would be a different symbol instead of a d o M XX SDXX refers to the mean M and standard deviation SD of the sample in question report this for all relevant samples in the study 0 r2 is the percentage of variance explained Nonsignificant results retained null o The results indicate that wearing purple during an exam did not significantly increase exam scores t36 102 p gt 05 0 Notice that all the information reported is what kind of test was used t and the tobt value 102 the degrees of freedom 36 and that the results did not fall within the critical region at 0L05 p gt05 Note Notice that the statistical notations are italicized Make sure you include these italics when reporting results Math symbols are italicized so that they are easy to find in a paper and so that readers can easily find your numerical results Also notice that when reporting the mean of the samples we use M instead of 9 We use 9 when solving for the sample mean it is a math calculation notation and M when reporting results
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