12.20 For the following data, assuming a between-groups design, determine: Group 1: 11, 17, 22, 15 Group 2: 21, 15, 16 Group 3: 7, 8, 3, 10, 6, 4 Group 4: 13, 6, 17, 27, 20 a. df between b. df within c. df total d. The critical value, assuming a p value of 0.05 e. The mean for each group and the grand mean f. The total sum of squares g. The within-groups sum of squares h. The between-groups sum of squares i. The rest of the ANOVA source table for these data j. Tukey HSD values 12.24 An incomplete one-way between-groups ANOVA source table is shown below. Compute the missing values. 12.40 Consideration of Future Consequences and two kinds of hypothesis testing: Two samples of students, one comprised of social science majors and one comprised of students with other majors, completed the Consideration of Future Consequences scale (CFC). The accompanying tables include the output from soft-ware for an independent-samples t test and a one-way between-groups ANOVA on these data. a. Demonstrate that the results of the independent-samples t test and the one-way between-groups ANOVA are the same. (Hint: Find the t statistic for the t test and the F statistic for the ANOVA.) b. In statistical software output, \"Sig.\" refers to the actual p level of the statistic. We can compare the actual p level to a cutoff p level such as 0.05 to decide whether to reject the null hypothesis. What are the \"Sig.\" levels for the two tests here the independent-samples t test and the one-way betweengroups ANOVA? Are they the same or different? Explain why this is the case. c. In the CFC ANOVA, the column titled \"Mean Square\" includes the estimates of variance. Show how the F statistic was calculated from two types of variance. (Hint: Look at the far-left column to determine which estimate of variance is which.) d. Looking at the table titled \"Group Statistics,\" how many participants were in each sample? e. Looking at the table titled \"Group Statistics,\" what is the mean CFC score for the social science majors? 15.18 Decide which of the three correlation coefcient values below goes with each of the scatterplots presented in Exercise 15.17 (scatter plots above). a. 0.545 b. 0.018 c. - 0.20 15.20 For each of the pairs of correlation coefficients provided, determine which one indicates a stronger relation between variables: a. - 0.28 and - 0.31 b. 0.79 and 0.61 c. 1.0 and - 1.0 d. - 0.15 and 0.13 15.34 Externalizing behavior, anxiety, and correlation: A study on the relation between rejection and depression in adolescents (Nolan, Flynn, & Garber, 2003) also collected data on externalizing behaviors (e.g., acting out in negative ways, such as causing ghts) and anxiety. They wondered whether externalizing behaviors were related to feelings of anxiety. Some of the data are presented in the accompanying table. a. Create a scatterplot of these data. Be sure to label both axes. b. What does the scatterplot suggest about the relation between these two variables? c. Would it be appropriate to calculate a Pearson correlation coefcient? Explain your answer. d. Construct a second scatterplot, but this time add a participant who scored 1 on externalizing behaviors and 45 on anxiety. Would you expect the correlation coefcient to be positive or negative now? Small in magnitude or large in magnitude? e. The Pearson correlation coefcient for the rst set of data is 0.65; for the second set of data it is 0.12. Explain why the correlation changed so much with the addition of just one participant. 15.36 Direction of a correlation: For each of the following pairs of variables, would you expect a positive correlation or a negative correlation between the two variables? Explain your answer. a. How hard the rain is falling and your commuting time b. How often you say no to dessert and your body fat c. The amount of wine you consume with dinner and your alertness after dinner 15.56 Mental health and partial correlation: A study by Nolan and colleagues (2003) examined the relation be-tween externalizing behaviors (acting out) and anxiety in adolescents. Depression has been shown to relate to both of these variables. What role might depression play in the observed positive relation between these variables? The correlation matrix below displays the Pearson correlation coefcients, as calculated by computer software, for each pair of the variables of interest: depression, externalizing, and anxiety. The Pearson correlation coefcients for each pair of variables are at the intersection in the chart of the two variables. For example, the correlation coefcient for the association between depression (top row) and externalizing (second column of correlations) is 0.635, a very strong positive correlation. a. Given that the authors calculated correlation coefcients, what kind of variables are depression, anxiety, and externalizing? Explain your answer. b. What is the correlation coefcient for the association between depression and anxiety? Explain what this correlation coefcient tells us about the relation between these variables. c. What is the correlation coefcient for the association between anxiety and externalizing? Explain what this correlation coefcient tells us about the relation between these variables. d. The partial correlation of anxiety and externalizing is 0.17, controlling for the variable of depression. How is this different from the original Pearson correlation coefcient between these two variables? e. Why is the partial correlation coefcient different from the original Pearson correlation coefcient between these two variables? What did we learn by calculating a partial correlation? f. Why can we not draw causal conclusions with respect to these ndings? 12.20 For the following data, assuming a between-groups design, determine: Group 1: 11, 17, 22, 15 Group 2: 21, 15, 16 Group 3: 7, 8, 3, 10, 6, 4 Group 4: 13, 6, 17, 27, 20 a. df between D.F=k-1=4-1=3 b. df within d.f=n-k-1=17-3-1=13 c. df total d.f=n-1=17-1=16 d. The critical value, assuming a p value of 0.05 F 0.05, 3, 13=3.4105 d. The mean for each group and the grand mean Group s Group 1 Group 2 Group 3 Group 4 Grand mean Count Sum Average 4 65 16.25 3 52 5 35 17.3333 3 7 5 83 16.6 14.2958 3 e. The total sum of squares TSS=yij2 - (yij)2/n TSS= 3929 2352 =680.4706 17 f. The within-groups sum of squares 2 SS-Within= T i 2 ( yij ) n = 348.6167 =1029.0872 - 680.4706=348.6167 g. The between-groups sum of squares SSG=TSS-SSW 680.4706 - 348.6167=331.8539 h. The rest of the ANOVA source table for these data Source of SS df MS F Variation Between Groups 331.85 3 110.61 4.1249 39 8 71 Within Groups 348.61 13 26.816 67 67 Total 680.47 16 06 Pvalue 0.0292 89 F crit 3.4105 34 i. Tukey HSD values MSE HSD = q n 110.618 HSD=3.4105* =27.56 17 12.24 An incomplete one-way between-groups ANOVA source table is shown below. Compute the missing values. SOURCE SS DF MS F BETWEEN WITHIN TOTAL 191.450 104.72 296.17 4 32 36 47.863 3.2725 14.6258 12.40 Consideration of Future Consequences and two kinds of hypothesis testing: Two samples of students, one comprised of social science majors and one comprised of students with other majors, completed the Consideration of Future Consequences scale (CFC). The accompanying tables include the output from soft-ware for an independent-samples t test and a one-way between-groups ANOVA on these data. a. Demonstrate that the results of the independent-samples t test and the one-way betweengroups ANOVA are the same. (Hint: Find the t statistic for the t test and the F statistic for the ANOVA.) From the t-test the tests statistics=-0.650 is not significant since it has a p-value that is greater than 0.05. The same case with ANOVA, the test statistics 0.422 is not significant since it has p-value that is greater than 0.05. Therefore for both test we will conclude that there is no significant difference between samples b. In statistical software output, \"Sig.\" refers to the actual p level of the statistic. We can compare the actual p level to a cutoff p level such as 0.05 to decide whether to reject the null hypothesis. What are the \"Sig.\" levels for the two tests here the independentsamples t test and the one-way between-groups ANOVA? Are they the same or different? Explain why this is the case. They are same. The p-value for t-test is 0.521 and the p-value for ANOVA is also 0.521, this is because in both test we are comparing the same populations hence the probability of rejecting or accepting the null hypothesis is equal in both tests. c. In the CFC ANOVA, the column titled \"Mean Square\" includes the estimates of variance. Show how the F statistic was calculated from two types of variance. (Hint: Look at the far-left column to determine which estimate of variance is which.) F= Mean square between groups 0.204 =0.422 = Mean square within groups 0.883 d. Looking at the table titled \"Group Statistics,\" how many participants were in each sample? Group named other had 10 participants Grouped named social science had 20 participants e. Looking at the table titled \"Group Statistics,\" what is the mean CFC score for the social science majors? Mean =3.3750 15.18 Decide which of the three correlation coefcient values below goes with each of the scatterplots presented in Exercise 15.17 (scatter plots above). a. 0.545 B b. 0.018 C c. - 0.20 A 15.20 For each of the pairs of correlation coefficients provided, determine which one indicates a stronger relation between variables: a. - 0.28 and - 0.3. b. 0.79 and 0.61 c. 1.0 and - 1.0 d. - 0.15 and 0.13 Answer= - 0.31 Answer= 0.79 Answer= Both Answer= -0.15 15.34 Externalizing behavior, anxiety, and correlation: A study on the relation between rejection and depression in adolescents (Nolan, Flynn, & Garber, 2003) also collected data on externalizing behaviors (e.g., acting out in negative ways, such as causing ghts) and anxiety. They wondered whether externalizing behaviors were related to feelings of anxiety. Some of the data are presented in the accompanying table. a. Create a scatterplot of these data. Be sure to label both axes. Scatterplot of Anxiety vs Externalizing Behaviours 45 Anxiety 40 35 30 25 20 2 3 4 5 6 7 8 Externalizing Behaviours 9 10 11 b. What does the scatterplot suggest about the relation between these two variables? The relationship is fairly strong as points are fairly arranged in a straight line. The direction of relationship is positive since scatter plots are upward sloping c. Would it be appropriate to calculate a Pearson correlation coefcient? Explain your answer. Yes since in b above we have seen there is linear relationship, it will be appropriate so that we can measure the strength of relationship. d. Construct a second scatterplot, but this time add a participant who scored 1 on externalizing behaviors and 45 on anxiety. Would you expect the correlation coefcient to be positive or negative now? Small in magnitude or large in magnitude? Scatterplot of Anxiety vs Externalizing Behaviours 45 Anxiety 40 35 30 25 20 0 2 4 6 8 10 12 Externalizing Behaviours I would expect the correlation coefficient to be positive since majority of scatter points are upward sloping I would expect the correlation coefficient to be smaller in magnitude since we have added an outlier which is weakening the relationship. e. The Pearson correlation coefcient for the rst set of data is 0.65; for the second set of data it is 0.12. Explain why the correlation changed so much with the addition of just one participant. This is because the one person who have been added is an outlier and thus having a great negative effect on the relationship between variable 15.36 Direction of a correlation: For each of the following pairs of variables, would you expect a positive correlation or a negative correlation between the two variables? Explain your answer. a. How hard the rain is falling and your commuting time Negative since as the rain falls for a long time I will have less time to commutate. b. How often you say no to dessert and your body fat Negative since the more I say no to dessert, the less I will be eating them and thus I will have less body fat. c. The amount of wine you consume with dinner and your alertness after dinner Negative since if I drink more wine the more I will get drunk lowering my alertness 15.56 Mental health and partial correlation: A study by Nolan and colleagues (2003) examined the relation be-tween externalizing behaviors (acting out) and anxiety in adolescents. Depression has been shown to relate to both of these variables. What role might depression play in the observed positive relation between these variables? The correlation matrix below displays the Pearson correlation coefcients, as calculated by computer software, for each pair of the variables of interest: depression, externalizing, and anxiety. The Pearson correlation coefcients for each pair of variables are at the intersection in the chart of the two variables. For example, the correlation coefcient for the association between depression (top row) and externalizing (second column of correlations) is 0.635, a very strong positive correlation. a. Given that the authors calculated correlation coefcients, what kind of variables are depression, anxiety, and externalizing? Explain your answer. Dependent variables since for there to be a correlation, the variables must depend on each other. The variables must also be numerical. b. What is the correlation coefcient for the association between depression and anxiety? Explain what this correlation coefcient tells us about the relation between these variables. R=0.368. This means there is weak but positive relationship between depression and anxiety. b. What is the correlation coefcient for the association between anxiety and externalizing? Explain what this correlation coefcient tells us about the relation between these variables. R=0.356. This means there is weak but positive relationship between externalizing and anxiety. c. The partial correlation of anxiety and externalizing is 0.17, controlling for the variable of depression. How is this different from the original Pearson correlation coefcient between these two variables? Partial correlation 0f 0.17 is the measure of association between two variables anxiety and externalizing, while controlling or adjusting the effect one more additional variables depression while Pearson correlation is the correlation between anxiety and externalizing without adjusting the effect depression. d. Why is the partial correlation coefcient different from the original Pearson correlation coefcient between these two variables? It's because in partial correlation, influence of the remaining variable is eliminated The effect of depression is also considered which is not considered in Pearson correlation Hence the difference is due to the correlation of a third variables, depression with both anxiety and externalizing . e. What did we learn by calculating a partial correlation? I have learnt that: A partial correlation between two variables can differ substantially from their simple correlation. Sometimes the correlation between two variables X and Y may be partly due to the correlation of a third variables, Z with both X and Y. f. Why can we not draw causal conclusions with respect to these findings? Since in causal correlation the relation is one sided and in Pearson correlation, the relationship is mutual in other words causation occur when X can cause Y can't cause X but in our case for example anxiety can cause depression and at the same time depression can cause anxiety. 12.20 For the following data, assuming a between-groups design, determine: Group 1: 11, 17, 22, 15 Group 2: 21, 15, 16 Group 3: 7, 8, 3, 10, 6, 4 Group 4: 13, 6, 17, 27, 20 a. df between D.F=k-1=4-1=3 b. df within d.f=n-k-1=17-3-1=13 c. df total d.f=n-1=17-1=16 d. The critical value, assuming a p value of 0.05 F 0.05, 3, 13=3.4105 d. The mean for each group and the grand mean Group s Group 1 Group 2 Group 3 Group 4 Grand mean Count Sum Average 4 65 16.25 3 52 5 35 17.3333 3 7 5 83 16.6 14.2958 3 e. The total sum of squares TSS=yij2 - (yij)2/n TSS= 3929 2352 =680.4706 17 f. The within-groups sum of squares 2 SS-Within= T i 2 ( yij ) n = 348.6167 =1029.0872 - 680.4706=348.6167 g. The between-groups sum of squares SSG=TSS-SSW 680.4706 - 348.6167=331.8539 h. The rest of the ANOVA source table for these data Source of SS df MS F Variation Between Groups 331.85 3 110.61 4.1249 39 8 71 Within Groups 348.61 13 26.816 67 67 Total 680.47 16 06 Pvalue 0.0292 89 F crit 3.4105 34 i. Tukey HSD values MSE HSD = q n 110.618 HSD=3.4105* =27.56 17 12.24 An incomplete one-way between-groups ANOVA source table is shown below. Compute the missing values. SOURCE SS DF MS F BETWEEN WITHIN TOTAL 191.450 104.72 296.17 4 32 36 47.863 3.2725 14.6258 12.40 Consideration of Future Consequences and two kinds of hypothesis testing: Two samples of students, one comprised of social science majors and one comprised of students with other majors, completed the Consideration of Future Consequences scale (CFC). The accompanying tables include the output from soft-ware for an independent-samples t test and a one-way between-groups ANOVA on these data. a. Demonstrate that the results of the independent-samples t test and the one-way betweengroups ANOVA are the same. (Hint: Find the t statistic for the t test and the F statistic for the ANOVA.) From the t-test the tests statistics=-0.650 is not significant since it has a p-value that is greater than 0.05. The same case with ANOVA, the test statistics 0.422 is not significant since it has p-value that is greater than 0.05. Therefore for both test we will conclude that there is no significant difference between samples b. In statistical software output, \"Sig.\" refers to the actual p level of the statistic. We can compare the actual p level to a cutoff p level such as 0.05 to decide whether to reject the null hypothesis. What are the \"Sig.\" levels for the two tests here the independentsamples t test and the one-way between-groups ANOVA? Are they the same or different? Explain why this is the case. They are same. The p-value for t-test is 0.521 and the p-value for ANOVA is also 0.521, this is because in both test we are comparing the same populations hence the probability of rejecting or accepting the null hypothesis is equal in both tests. c. In the CFC ANOVA, the column titled \"Mean Square\" includes the estimates of variance. Show how the F statistic was calculated from two types of variance. (Hint: Look at the far-left column to determine which estimate of variance is which.) F= Mean square between groups 0.204 =0.422 = Mean square within groups 0.883 d. Looking at the table titled \"Group Statistics,\" how many participants were in each sample? Group named other had 10 participants Grouped named social science had 20 participants e. Looking at the table titled \"Group Statistics,\" what is the mean CFC score for the social science majors? Mean =3.3750 15.18 Decide which of the three correlation coefcient values below goes with each of the scatterplots presented in Exercise 15.17 (scatter plots above). a. 0.545 B b. 0.018 C c. - 0.20 A 15.20 For each of the pairs of correlation coefficients provided, determine which one indicates a stronger relation between variables: a. - 0.28 and - 0.3. b. 0.79 and 0.61 c. 1.0 and - 1.0 d. - 0.15 and 0.13 Answer= - 0.31 Answer= 0.79 Answer= Both Answer= -0.15 15.34 Externalizing behavior, anxiety, and correlation: A study on the relation between rejection and depression in adolescents (Nolan, Flynn, & Garber, 2003) also collected data on externalizing behaviors (e.g., acting out in negative ways, such as causing ghts) and anxiety. They wondered whether externalizing behaviors were related to feelings of anxiety. Some of the data are presented in the accompanying table. a. Create a scatterplot of these data. Be sure to label both axes. Scatterplot of Anxiety vs Externalizing Behaviours 45 Anxiety 40 35 30 25 20 2 3 4 5 6 7 8 Externalizing Behaviours 9 10 11 b. What does the scatterplot suggest about the relation between these two variables? The relationship is fairly strong as points are fairly arranged in a straight line. The direction of relationship is positive since scatter plots are upward sloping c. Would it be appropriate to calculate a Pearson correlation coefcient? Explain your answer. Yes since in b above we have seen there is linear relationship, it will be appropriate so that we can measure the strength of relationship. d. Construct a second scatterplot, but this time add a participant who scored 1 on externalizing behaviors and 45 on anxiety. Would you expect the correlation coefcient to be positive or negative now? Small in magnitude or large in magnitude? Scatterplot of Anxiety vs Externalizing Behaviours 45 Anxiety 40 35 30 25 20 0 2 4 6 8 10 12 Externalizing Behaviours I would expect the correlation coefficient to be positive since majority of scatter points are upward sloping I would expect the correlation coefficient to be smaller in magnitude since we have added an outlier which is weakening the relationship. e. The Pearson correlation coefcient for the rst set of data is 0.65; for the second set of data it is 0.12. Explain why the correlation changed so much with the addition of just one participant. This is because the one person who have been added is an outlier and thus having a great negative effect on the relationship between variable 15.36 Direction of a correlation: For each of the following pairs of variables, would you expect a positive correlation or a negative correlation between the two variables? Explain your answer. a. How hard the rain is falling and your commuting time Negative since as the rain falls for a long time I will have less time to commutate. b. How often you say no to dessert and your body fat Negative since the more I say no to dessert, the less I will be eating them and thus I will have less body fat. c. The amount of wine you consume with dinner and your alertness after dinner Negative since if I drink more wine the more I will get drunk lowering my alertness 15.56 Mental health and partial correlation: A study by Nolan and colleagues (2003) examined the relation be-tween externalizing behaviors (acting out) and anxiety in adolescents. Depression has been shown to relate to both of these variables. What role might depression play in the observed positive relation between these variables? The correlation matrix below displays the Pearson correlation coefcients, as calculated by computer software, for each pair of the variables of interest: depression, externalizing, and anxiety. The Pearson correlation coefcients for each pair of variables are at the intersection in the chart of the two variables. For example, the correlation coefcient for the association between depression (top row) and externalizing (second column of correlations) is 0.635, a very strong positive correlation. a. Given that the authors calculated correlation coefcients, what kind of variables are depression, anxiety, and externalizing? Explain your answer. Dependent variables since for there to be a correlation, the variables must depend on each other. The variables must also be numerical. b. What is the correlation coefcient for the association between depression and anxiety? Explain what this correlation coefcient tells us about the relation between these variables. R=0.368. This means there is weak but positive relationship between depression and anxiety. b. What is the correlation coefcient for the association between anxiety and externalizing? Explain what this correlation coefcient tells us about the relation between these variables. R=0.356. This means there is weak but positive relationship between externalizing and anxiety. c. The partial correlation of anxiety and externalizing is 0.17, controlling for the variable of depression. How is this different from the original Pearson correlation coefcient between these two variables? Partial correlation 0f 0.17 is the measure of association between two variables anxiety and externalizing, while controlling or adjusting the effect one more additional variables depression while Pearson correlation is the correlation between anxiety and externalizing without adjusting the effect depression. d. Why is the partial correlation coefcient different from the original Pearson correlation coefcient between these two variables? It's because in partial correlation, influence of the remaining variable is eliminated The effect of depression is also considered which is not considered in Pearson correlation Hence the difference is due to the correlation of a third variables, depression with both anxiety and externalizing . e. What did we learn by calculating a partial correlation? I have learnt that: A partial correlation between two variables can differ substantially from their simple correlation. Sometimes the correlation between two variables X and Y may be partly due to the correlation of a third variables, Z with both X and Y. f. Why can we not draw causal conclusions with respect to these findings? Since in causal correlation the relation is one sided and in Pearson correlation, the relationship is mutual in other words causation occur when X can cause Y can't cause X but in our case for example anxiety can cause depression and at the same time depression can cause anxiety