Comparing Several Means - Practice Lab.docx

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Feb 20, 2024

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Comparing Several Means (ANOVA) Practice Lab Background In a previous study, we found that high school students who spoke Korean as a first language scored higher on standardized English tests than students who spoke English or Spanish as a first language. This was surprising because they had less experience and familiarity with the language than the native English speakers. One possible explanation for this surprising finding is that the Korean students’ parents are highly involved in their education, keeping them accountable for their assignments and pushing them to achieve academically. In this practice lab, we will follow up on the previous study in order to better understand how these students’ experiences at home might explain their academic achievement. For this practice lab, we will use authentic data collected from 1,187 students (52% female, 48% male) at a large high school in the Los Angeles area. Among the students in the sample, 270 speak English as a first language, 859 speak Spanish as a first language, and 58 speak Korean as a first language. Parental involvement was measured using four survey items (e.g., “My parents support my learning at home”) with response options ranging from 1 ( Strongly disagree ) to 4 ( Strongly agree ). Standardized ( M = 0.0, SD = 1.0) composite scores are approximately normally distributed, ranging from -3.48 to 1.47. In this lab, we will determine whether primary language predicts parental academic involvement. To complete this lab, download and open the file called Family_Involvement.sav within SPSS. Analytic Method First, we must determine which statistical test is most appropriate for these data. Be sure to consider the levels of measurement of the independent and dependent variables. 1. Which statistical test is most appropriate for analyzing these data? A one-way ANOVA is the appropriate test for this data. 2. Why is this statistical test appropriate? A one-way ANOVA is appropriate because the independent variable (Primary language) is a nominal-level variable with three groups (English, Spanish< and Korean), and the dependent variable (family involvement) is a scale-level variable. There is only one independent variable. Hypotheses Next, we must state our null and alternative hypotheses both informally and formally. 3. What is the informal null hypothesis?

There is no significant difference in family involvement between the English, Spanish, and Korean primary language groups. 4. What is the formal null hypothesis? H 0 : µ ??𝑔𝑙𝑖?ℎ = µ ??𝑎?𝑖?ℎ = µ 𝐾???𝑎? 5. What is the informal alternate hypothesis? The levels of family involvement of the English, Spanish, and Korean primary language groups are not all equal. 6. What is the formal alternate hypothesis? H a : The means are not all equal Testing Assumptions The next step in our analysis is to generate descriptive statistics and graphs to help us inspect our data to see if they meet the assumptions of this statistical test. 7. What are the assumptions of this statistical test? The assumptions of the one-way ANOVA are normality, absence of outliers, and homogeneity of variance. Normality. We need to determine whether the distribution of family involvement scores within each group is approximately normally distributed. We will first generate a descriptive statistics table broken down by primary language. To do this, click Analyze ± Compare Means ± Means to open up the Means dialogue box. Select “Parent Involvement” from the variables list and move it to the Dependent List box on the upper right. Then select “Primary Language” and move it to the Independent List box on the lower right. Then click Options . Select “Kurtosis” and “Skewness” from the Statistics window and move them into the Cell Statistics box on the right. Click Continue . Then click OK . This should produce two tables in the Output Window. Copy and paste the “Report” table below:

8. Examine the skewness and kurtosis statistics for each of the three groups. Is the assumption of normality supported? All of the skewness and the kurtosis statistics have an absolute value of less than 1.0. They are all within the +/- 1.00 cutoff criterion. Therefore, we can conclude that the data are approximately normally distributed within each group. The assumption of normality is supported. Absence of Outliers. Second, we need to examine a boxplot to see if there are any outliers that could influence our results. To do this, Click Graphs ± Chart Builder . Next, click on Boxplot under the Gallery tab on the bottom left. Drag the first image in the gallery to the Drop zone on the top right. Select the independent variable from the variable list and drag it to the x -axis in the Drop zone. Next, select the dependent variable from the variable list and drag it to the y -axis in the Drop zone. Click OK . This should produce a graph in the Output Window. Copy and paste the bar graph below: 9. Examine the distribution of each language group. Based on visual inspection of the boxplot, are there any potential outliers? There are no apparent outliers (unusual observations) indicated on the boxplot.Thus, the assumption of absence of outliers is also supported.

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Homogeneity of Variance. In order to test the assumption of homogeneity of variance, we will have to run the ANOVA. To do this, click Analyze ± Compare Means ± One-Way ANOVA … to open the One-Way ANOVA dialogue box. Move the independent variable to the Factor box at the bottom. Move the dependent variable to the Dependent List box. Then click Options … to open the Options dialogue box. Check the box next to Homogeneity of variance test and click Continue . Next, click Post Hoc … to open the Post Hoc Multiple Comparison dialogue box. Check the box next to Bonferroni and click Continue . Click OK . This should produce three tables. Paste the “Test of Homogeneity of Variances” table into the document below: 10. Examine the results of Levene’s test for equality of variances (based on the mean). Is the assumption of homogeneity of variance supported? The p-value associated with Levene’s test is .106. Because p is not less than .05, it means that the variance are not significantly different. (i.e., there are the same) Thus, the assumption of the homogeneity of variance is supported. Interpreting the Test Statistic Once the assumptions have been met, we can interpret the results of our statistical test. Paste the “ANOVA” table into the document below: 11. Is the difference between the three groups statistically significant? How do you know? The p-value associated with the F-statistic is .027. Because p is less than .05, we can conclude that the results are statistically significant. The means of the three groups are not all equal.

12. Based on these results, should we reject the null hypothesis? There is only a 2.7% chance that we would observe a test statistic at least this large if the null hypothesis were true. Therefore, we will reject the null hypothesis. Post Hoc Tests Next, we can examine the results of the post hoc tests. Paste the “Multiple Comparisons” table into the document below: 13. According to the post hoc tests, is there a significant difference in parental involvement between the English group and the Spanish group? The p-value associated with the English-Spanish contract is .697. It is not less than .05. Therefore, the group means are not significantly different. The parents of the native English speaking students and the parents of the native Spanish speaking students are equally involved in their children’s academics. English=Spanish 14. According to the post hoc tests, is there a significant difference in parental involvement between the English group and the Korean group? The p-value with the English-Korean contracts is .216. It is not less than .05. Therefore, the group means are not significantly different. The Korean students’ parents and the parents of the native English speaking students are equally involved in their children’s academics. English=Korean 15. According to the post hoc tests, is there a significant difference in parental involvement between the Korean group and the Spanish group?

The p-value associated with the Korean-Spanish contract is .034. It is less than .05. Therefore, the group means are significantly different. The Korean students’ parents are significantly less involved with their children’s education than the parents of the native Spanish speakers. Koren < Spanish Calculating and Interpreting Effect Size Next, we will determine the practical significance of these findings. To do this we will calculate the effect size for the F -statistic using η 2 and the effect size of our key between group contrasts using Cohen’s d . Effect Size for the F-Statistic. First, we will calculate η 2 using the following formula: η2 = ?? 𝐵??𝑤??? 𝐺????? ?? ???𝑎𝑙 Look back at the ANOVA table to find the between groups sum of squares and the total sum of squares. Plug those values into the formula to calculate η 2 : η2 = 7.21 1189.07 .006 η2 = When interpreting effect size in ANOVA, an η 2 of .01 is considered “small,” .06 is considered “medium,” and .14 is considered “large.” 16. Based on the above criteria, how large is this observed effect? Is it practically significant? The observed effect size is close to .01. Therefore, we can interpret this as a small effect. It is not practically significant. Effect Size for Between-Group Contrasts. Next, we will evaluate the practical significance of the difference between the Korean group and the Spanish group using Cohen’s d . To calculate Cohen’s d , we will use the mean and standard deviations reported in the descriptive statistics table at the beginning of our analysis. ? = 𝑀 1 −𝑀 2 ?? ???𝑙?? But we will first have to calculate the pooled standard deviation for the two groups using this formula: ?? ???𝑙?? = (?? 1 2 + ?? 2 2 ) 2 Look back at the descriptive statistics table above to find the standard deviations of the two groups (Korean and Spanish). Plug those values into the formula to calculate the pooled standard deviation.

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?? ???𝑙?? = (.97 2 +1.12 2 ) 2 ?? ???𝑙?? = (.94+1.25) 2 ?? ???𝑙?? = 2.19 ( ) 2 ?? ???𝑙?? = 1. 10 ?? ???𝑙?? = 1. 05 Look back at the descriptive statistics table above to find the means of the two groups. Plug those values, along with the pooled standard deviation, into the formula for Cohen’s d to calculate the effect size. ? = 𝑀 1 −𝑀 2 ?? ???𝑙?? ? = (.03−−.32) 1.05 ? = .35 1.05 ? =−. 33 Cohen suggested that d = .20 be considered a “small” effect size, .50 be considered a “medium” effect size, and .80 be considered a “large” effect size. 17. Based on Cohen’s criteria, how large is the observed effect? The observed effect size is d= .33. Therefore, it should be considered a “small to medium” effect. Reporting the Results Now that our analyses are completed, we can formally report the results of our analysis. 18. Report the results of your analysis. Be sure to provide the key information to your readers using proper APA style. “We conducted a one-way ANOVA to compare the effects of primary language groups on high school students’ family academic involvement. Results showed the effect of the primary language group was significant, F(2, 1184) = 3.61, p = .027. Therefore, we reject the null hypothesis that there is no difference between the three language groups. However, the model only explained a small amount of the variance in parental involvement (n2= .006). Post hoc analyses using Bonferroni correction indicated the parental involvement was significantly lower among native Korean students (M = .32, SD=1.12) than among native Spanish speakers (M = .03, SD = .97). The practical difference between native Korean speakers and native Spanish speakers was small to medium (d= .33)”

Discussing the Results Finally, we will discuss our results in light of our original research question and address the practical implications of our findings. 19. Based on these results, does family involvement differ between primary language groups? Yes, family involvement does differ between primary language groups. The Korean students reported significantly less family involvement than the native-Spanish speaking students. 20. In a previous study, we found that native-Korean speaking students scored significantly higher than the native-English speaking and native-Spanish speaking students on standardized English tests. At the beginning of this study, we theorized that this could be the result of the Korean students’ parents being more involved in their education. Do the results of this study support our theory? No, the results of this study do not support our theory. The Korean students did not report greater family involvement. In fact, they experienced significantly less parental involvement than the native Spanish-speaking students. Family involvement does not appear to explain the achievement gap observed in the previous study.

Comparing Several Means - Practice Lab.docx

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