Sample Exam Paper Answer Guide

Sample Exam Paper Answer Guide Notes This handout provides perfect answers to the sample exam paper. I would not expect you to be able to produce such perfect answers in an exam. So, use this document as a guide and not as a cause for panic. Question 1 Cohen s d is the difference between two means expressed in standard deviation units. Where there is an obvious control group, it is expressed relative to the control group standard deviation, but it can also be expressed as a function of the pooled standard deviation which is a weighted average of the variances associated with the two means. In this case, No Messages is a control group so: d = X!"#"$ X!"!"##$%" s!"!"##$%" d = 9.33 15.71 2.69 = 2.37 In other words, the number of goats sacrificed in the Satan group was 2.37 standard deviations smaller than in the no message group. This is a very large effect. Completed Summary table: Part C SS df MS F Model 247.38 4 61.85 7.94** Residual 241.36 31 7.79 Total 488.74 35 Levene s test suggests that the assumption of homogeneity of variance is met, because it is not significant, F(4, 31) = 1.80, p =.154. However, the samples in each group are small so Levene s test will lack power to detect differences in the variances. The variance ratio is: VR = 3.44! 1.03! = 11.09 This is pretty huge, which suggests that we can t assume homogeneity of variance. (Note that the SPSS output reports standard deviations so to get the variance I have squared the values. The variance ratio is the ratio of the biggest to smallest variance so I have used the biggest and smallest standard deviations in the SPSS output.) We could write that the effect of subliminal messages (no message, friendly, satanic, goats, and backwards) had a significant effect on the number of goats sacrificed, F(4, 31) = 7.94, p < 0.001. However, at this stage we don t know anything more specific than this, although from the means it looks as though the satanic messages resulted in less goat slaughter. Prof. Andy Field, 2012 www.discoveringstatistics.com Page 1

18 16 14 12 10 8 6 4 2 0 No Message Friendly Satan Goats Backwards Part E You should suggest the following comparisons and codes: 1. To look at hypothesis one they would need to do (weights in brackets) {nice messages: no message(3), Friendly message} vs {satanic message: Satanic (2), Goats (2), Backwards (2)} 2. Having split the groups into 2 chunks, the next contrast needs to decompose one of these two chunks. Let s first decompose the chunk containing the control messages. As such, the second contrast would compare {friendly message (- 1)} vs {no message (1)} with Satanic, Goats and Backwards having weights of zero. 3. We can now look to decompose the second chunk from Contrast 1. There are many ways we could do this, but to test hypothesis two, the appropriate contrast is {backwards message (- 2)} vs {Satanic (1), Goats (1)} with No message and friendly having a weight of zero. 4. To look at hypothesis three they would need to do {goats (- 1)} vs {Satanic (1)} with No message, friendly, and Backwards having a weight of zero. 1 Contrast Coefficients Contrast 1 2 3 4 Type of Message on the Record Friendly No Message message Dark Lord Goats Backward 3 3-2 -2-2 1-1 0 0 0 0 0 1 1-2 0 0 1-1 0 Part F The term mean squares represents the average variability due to either the experimental manipulation (MS M ) or due to unexplained factors or error (MS R ). 1 Even without hypothesis 3 this contrast would have needed to be done to decompose the chunk in Contrast 3 that contained two groups (Satanic and Goats). Remember that to break the variance into its component parts every group has to, at some point, end up singled out in one of the contrasts. Also, you could double check you have enough contrasts by remembering that with 5 groups we should end up with k 1, or 4, contrasts. Prof. Andy Field, 2012 www.discoveringstatistics.com Page 2

Question 2 A three- way 3(Images: positive, neutral and negative) 2 (Group: statistics lecturers vs. students) 2 (time: before vs. after images) mixed ANOVA with repeated measures on the Images and time variables. By testing the same people under different conditions you gain greater control over extraneous variables than in an independent design because things like IQ, gender and other demographic and psychological variables are held constant (because you re testing the same people). In this experiment, for example, by exposing people to different types of imagery we re controlling for things like levels of disgust 9which could affect arousal) across the three types of imagery. The downside is that there could be carry- over effects, for example, arousal after positive imagery might be affected by having just seen some negative imagery. It would be important to counterbalance the order in which people were exposed to the different types of imagery. Part C The assumption of sphericity has been met for all effects 2. Levene s test is not significant for any level of the repeated measures variable, F(1, 18) < 1 for all effects, except for arousal levels before negative imagery for which the variances were significantly different in students and statistics lecturers, F(1, 18) = 10.67, p <.01. Therefore the assumption of homogeneity of variance has been met in most cases. There was a significant main effect of group (F(1, 18) = 18.44, p <.001) indicating that when the type of imagery and time at which arousal is measured is ignored, students and lecturers significantly differed in their levels of positive arousal. Looking at the graph you can see that lecturers showed significantly more positive arousal than students. There was a significant main effect of time (F(1, 18) = 7.58, p <.05) indicating that when the type of imagery and type of people being measured is ignored, positive arousal significantly changed from before the images were shown to after they were shown. Looking at the graph you can see that positive arousal was significantly higher after the images were shown. There was a significant effect of imagery (F(2, 36) = 7.48, p <.01) indicating that when we ignore the group to which participants belong, and the time at which arousal was measured the type of imagery significantly affected the positive arousal levels. Looking at the graph, there was the greatest positive arousal for positive images (as demonstrated by a high mean) and negative and neutral imagery produced similarly small levels of positive arousal. The time group interaction (F(1, 18) = 6.42, p <.05) was significant indicating that the degree to which positive arousal changed over time depended on whether they were a student or a lecturer (ignoring the type of imagery used). The interaction graph shows that when we ignore the type of imagery used, arousal in students didn t really change over time (the line is flat), however, for lecturers there was a large increase in arousal over time. This suggests that if we ignore the type of imagery used, lecturer s positive arousal increased over time, whereas students did not. The imagery group interaction (F(2, 36) = 6.62, p <.01) was significant indicating that when we ignore the time at which arousal was measured the degree to which people displayed positive arousal to different types of stimuli depended on whether they were a student or a lecturer. The interaction graph shows that for positive and neutral imagery students and lecturers are the same: both show more positive arousal to positive imagery compared to neutral imagery. The interaction comes from a difference between statistics lecturers and students in their response 2 For the main effect of Time the assumption doesn t apply because there are only 2 levels, and you need at least three levels of a variable for sphericity to be an issue, for the main effect of Time and the Time Imagery interaction we know sphericity is met because the ps are greater than.05. Prof. Andy Field, 2012 www.discoveringstatistics.com Page 3

to negative imagery: statistics lecturers show positive arousal to negative imagery (possibly because they are sadistic bastards) whereas students show reduced positive arousal to these kinds of stimuli. The imagery time interaction (F(2, 36) = 10.79, p <.001) was significant indicating that when we ignore the group to which people belong, the change in arousal differed across the types of imagery used. The interaction graph shows that for positive imagery there is a large increase in arousal from before to after that imagery is used. For neutral imagery there is no change in arousal (the line is flat). The use of negative imagery seems to cause a slight reduction in positive arousal although only a slight one. Bear in mind these effects lump students and lecturers together though. The group imagery time interaction (F(2, 36) = 8.04, p <.01) was significant indicating that the change in arousal for different types of imagery was different for students and lecturers. (Put another way, the imagery time interaction described above is different for students and lecturers.) If we look at positive imagery first (the circles): both students and lecturers show similar levels of increased positive arousal when this imagery is used. For neutral imagery (squares), again students and lecturers are pretty much the same: for both groups the change in arousal is negligible (the lines on both graphs are more or less flat). Finally, if we look at negative imagery, there is a difference: students show decreased arousal when negative imagery is used, whereas lecturers show increased arousal to this sort of imagery. To sum up, this interaction reflects the fact that although students and statistics lecturers respond in the same way to positive and neutral imagery, they differ with respect to their responses to negative imagery: positive arousal increase in statistics lecturers but decreases in students. Statistics lecturers are, therefore, sadistic bastards. Question 3 A bootstrap confidence interval is one derived empirically from the sample. Numbers are samples from the data (replacing the number back each time) to create a bootstrap sample. The regression parameter is computed within that sample. This process is repeated over many samples (e.g., 1000) and the confidence interval; is derived by looking at the limits between which 95% of bootstrap sample parameters fall. These confidence intervals are robust to the distribution of scores and so should be used when the assumption of normality is doubtful. Based on the final model (which is actually all we re interested in) the following variables predict aggression: Parenting Style, b = 4.35, 95% CI [ 4.51, 4.19], β = 1.40, t = 52.77, p <.001, significantly predicted aggression. The beta value indicates that as parenting style increases by a unit (became more strict), aggression decreased by 4.35 of a unit. Sibling Aggression, b =.30, 95% CI [ 0.33, 0.27], β =.44, t = 18.70, p <.001, significantly predicted aggression. The beta value indicates that as sibling aggression increases by a unit (became more aggressive), aggression decreased by.3 of a unit. Computer games, b = 2.49, 95% CI [2.38, 2.59], β = 1.14, t = 46.82, p <.001, significantly predicted aggression. The beta value indicates that as the time spent playing computer games increases by a unit, aggression increased by 2.49 of a unit E- numbers, b =.15, 95% CI [0.14, 0.16], β =.37, t = 25.35, p <.001, significantly predicted aggression. The beta value indicates that as the e- numbers consumed increases by a unit, aggression increased by.15 of a unit The only factor not to predict aggression was: Television, b =.01, 95% CI [ 0.29, 0.31], β =.00, t = 0.07, p =.941. Based on the standardized betas, the most substantive predictor of aggression was actually parenting style, followed by computer games, sibling aggression and e- numbers. Prof. Andy Field, 2012 www.discoveringstatistics.com Page 4

Part C R 2 is the squared correlation between the observed values of aggression, and the values of aggression predicted by the model. The values in this output tell us that sibling aggression and parenting style in combination explain 44.4% of the variance in aggression. When computer game use is factored in as well 82% of variance in aggression is explained (i.e. an additional 37.6%). When e- numbers are added to the model 94.4% of the variance in aggression is explained (an additional 12.4%). Adding Television into the model does not increase the percentage of variance. The Durbin- Watson statistic tests the assumption of Independence of errors, which means that for any two observations (cases) in the regression, their residuals should be uncorrelated (or independent). In this output the Durbin- Watson statistic falls within Field s (2005) recommended boundaries of 1-3, which suggests that errors are reasonably independent. Part E This is a bit of a naughty question because the scatterplot helps us to assess both Homoscedasticity and Independence of Errors. We ve defined independence of errors above, so we don t need to do that again, but heteroscedasticity is the assumption that at each point along the predictor variable, the spread (or variability) or residuals should be fairly similar. The scatterplot of ZPRED vs. ZRESID does show a random pattern and so indicates no violation of the independence of errors assumption. Also, the errors on the scatterplot do not funnel out indicating homoscedascitity of errors, thus no violations of assumptions. Prof. Andy Field, 2012 www.discoveringstatistics.com Page 5