Chapter 9 Notes
Chapter 9 Notes PSYCH 2220 - 0020
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This 7 page Class Notes was uploaded by Emma Dahlin on Thursday October 22, 2015. The Class Notes belongs to PSYCH 2220 - 0020 at Ohio State University taught by Anna Yocom in Summer 2015. Since its upload, it has received 30 views. For similar materials see Data Analysis in Psychology in Psychlogy at Ohio State University.
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Date Created: 10/22/15
Chapter 9 The t Distributions Used when the parameters are NOT known Use tdistributions to 0 Estimate a population standard deviation from a sample Sample standard deviation SD M N Estimated population standard deviation S 2X M2 Nl Example S We want to determine if multitasking has any in uence on productivity Population average for time spent on one task is 11 minutes We design an intervention to see if it has any effect on productivity and time spent on one task We sample 5 employees and see scores of 8 12 16 12 and 14 minutes spent on one task Calculating the Estimated Population SD 1 Calculate the sample mean 2 Use the sample mean in the correct standard deviation formula 2X M2 Nl Steps to calculating 5 8 44 1936 12 O4 016 16 36 1296 12 04 016 14 16 256 M 812161214 124 5 352 5 1 88 297 Calculating Standard Error for the t Statistic Using the standard error S SM W o The t statistic t distance of sample mean from pop mean in terms of estimated std error M MM SM t Steps to calculating t statistic using standard error From prvious example as 29 SIE l E Y 2133 IAssun le population mean is HM jquot 3 1 m I a 1 1735 l The t Statistic The tdistributions 0 Distribution of differences bt means 0 As sample size increases 0 s approaches a o tand zbecome more equal Wider and Flattert Distributions Standard iidrmal z distributun t distribution ED individuals t distributidn E individuald distributidng 2 individuals z underestimates variance and would have too many Type errors Hypothesis Tests The Single Sample tTest The single sample ttest 0 When we know the population mean but NOT the standard deviation 0 Degrees of freedom 0 dfN1 where N is sample size 0 gt of scores free to vary in estimate of population TABLE 91h Eaerpt frem the t Table 1il39ilheri eanauetiha hyaatheeie teethg we tree the rtahle te illE lElilTlil l39 eritieal tialaee ter a giaeri ja leael haeerl ah the degreee at treeaem and whether the Eat ie are er taretaller it j39l El 3W3 1355 1533 1533 LariWE the Tailed Teete BLUE il ii 292 2353 21 33 Elfl l E U i 31 EE l 595 45 3M 3335 llil i 6312 all 3920 2353 EquotlEiE Etii Twirl Tailed Teete quotll 215 133 312 ENE 251 331 3355 9923 5341 l l 413 Which is more conservative Onetailed or twotailed Two tailed bC you are putting less in each tail smaller tail makes it harder to reject The ttest The six steps of hypothesis testing 0 1 0 U1 2 o 3 4 Identify critical values 0 dfN1 Calculate test statistic tobs Decision 0 reject or fail to reject null Identify population distributions assumptions State the hypotheses Characteristics of the comparison distribution Example Clients at a counseling center attend an average of 46 sessions It is hypothesized that having patients sign a contract to attend at least 10 sessions will make a difference in the average number of sessions that are attended Five students sign the contract and the number of sessions attended for each student is recorded Data6 6 12 7 8 N5 p 46 0 STEP 1 Identify population distribution assumptions 0 Population 1 All clients at this counseling center who sign a contract to attend at least 10 session 0 Population 2 All clients at this counseling center who do not sign a contract to attend at least 10 sessions 0 Comparison distribution distribution of means 0 Use a singlesample t test one sample know population mean but not population standard deviation 0 Assumptions Dependent variable is scale random selection population normally distributed STEP 2 State the hypotheses 0 H0 11 12 O H1I11 12 We use a tdistribution instead of the zdistribution when sampling requires us to estimate the population standard deviation from the sample standard deviation 0 When we do not know the population standard deviation and we are comparing only two groups tdistributions help us specify how con dent we can be about research ndings Want to know if we can generalize from sample to larger population The ttest based on the tdistributions tells us how con dent we can be that the sample differs from the larger population Uncertainty of small sample size means that tdistributions become atterand more spread out As sample size gets larger the tdistribution begins to merge with the zdistribution bc we gain more con dence as more participants are added to the study Estimating Population Standard Deviation from a Sample Before conducting singlesample ttest we must estimate pop standard dev by using the sample standard dev Instead of diving by N we divide by N1 to get the mean of the standard deviationsthis corrects for the probability that the sample SD slightly underestimates the actual pop SD We call this 5 instead of SD bc it is a statistic from a sample rather than a parameter from a population STEPS 1 Calculate sample mean 2 Use sample mean in corrected formula for the standard deviation S 2X M2 Nl Standard Error for the tStatistic The formula for standard error when we estimate from a sample is we use 5 instead of a bc we re working from a sample LStatistic tstatistic indicates the distance of a sample mean from the population mean in terms of the estimated standard error 0 Formula is identical to that for the zstatistic except that it uses estimated standard error M T MM SM Corrected denominator makes tstatistic smaller therefore reducing the probability that we would have an extreme t t statistic o tstatistic is more conservative than zstatistic less extreme THE SINGLESAMPLE tTEST singIesample t test hypothesis test in which we compare a sample form from which we collect data to the population for which we know the mean but not the standard deviation degrees of freedom df the number of scores that are free to vary when we estimate a population parameter from a sample The tdistributions become closer to the zdistribution as sample size increases 0 Think of zstatistic as a singleblade Swiss Army knife and the tstatistic as a multiblade Swiss Army knife that still includes the single blade HYPOTHESIS TEST STEPS 1 Identify populations distributions assumptions 2 State the nullresearch hypotheses 3 Determine characteristics of the comparison distribution 4 Determine critical values or cutoffs df 5 Calculate test statistic 6 Make a decision CONFIDENCE INTERVAL STEPS 1 Draw a picture of a tdistribution that includes the con dence interval centersampe mean 2 Indicate the bounds of the con dence interval on the drawing in tail 3 Look up the tstatistics that fall at each line marking the middle 95 tcritical 4 Convert the tstatistics back into raw means 0 Mlower 39tSM Msample O Mupper tSM Msample 5 Verify that the con dence interval makes sense 0 Sample mean should fall exactly in middle of interval CALCULATING EFFECT SIZE Cohen s d is based on the spread of distribution of individual scores rather than the distribution of means This tells us how many standard deviations apart the sample mean is from the population mean dM II S