Part A: Language arts program
Samples: In this question, there are two samples: that of eleventh grade students who took Western Australian state school exams after the introduction of new language arts program, and that of eleventh grade students who took the Western Australian state school exams before the introduction of the new language arts program. In this regard, it is noticeable that the two samples are independent samples, because measurements are supposed to be made on each of the sample. The samples would have been drawn from two different populations: in this case, the first population is the one that undertook the new language art program; the second population is the one that did not have a chance of undertaking the program. Knowing whether the samples are independent or not is usually important when it comes to choosing the best t-test statistic to use. This case requires independent t-test (also called the unpaired t-test).
Independent variables: Time e.g. months or years of learning the new language arts; content of the new language arts covered; student motivation. An independent variable is the variable that the researcher thinks that it can affect the dependent variable. In a good laboratory setting, the researcher usually manipulates the independent variable to realize its effects on the dependent variable. However, in some cases, it may be hard to manipulate the independent variable. In this scenario, however, one may manipulate the time, subject content, and external motivation.
Dependent variables: Scores in language arts program. A dependent variable is the variable that responds to an independent variable, or a group of independent variables. It is the actual variable that the researcher measures in the experiment, because it is the one that is being affected by other variables (independent variables). Thus, it is called dependent variable because it depends on other variables. In a scientific experiment or research, one cannot have a dependent variable without independent variables. In this scenario, it is predicted that scores in the language art program will depend on the time and content covered and student motivation. Thus, the scores in language arts program become a dependent variable.
Hypotheses: Hn There is no significant difference in the scores of students who took new
language arts classes and those that did not.
Ha There is a significant difference in the scores of students who took new
language arts classes and those that did not.
Method of testing hypothesis: Independent T-test for two samples. The independent t-test, also called the student’s t-test or the two sample t-test, is an inferential statistical test that serves to determine whether a significant difference between the means of two unrelated groups exists or not.
Possible confounds: What one needs therefore to test the significant difference in the two means are two arrays of scores for students that did not attend the new language class and scores for those students that attended the new language class. To use the test, one has to ensure that the sample data were drawn from normally distributed populations. This test also assumes that variances of the two samples or populations are equal, which may not be the case. When the variances are not equal, one may commit Type I error by rejecting the null hypothesis when it is actually true. This test is also significant when two samples were randomly selected; the number of items in each sample has to be 30 or less. It is also important to ensure that data of dependent variables are measured on a continuous scale.
Suggestions for improvement: One can run the Mann-Whitney U test to normalize the data to produce significant results. One can also run the Levene’s Test of Equality of Variances to make variances of the two groups homogeneous and avoid committing Type I error.
Part B: The baby’s amount of eye contact with caregiver
RESULTS AND DISCUSSION
The baby’s amount of eye contact with caregiver
Since the same subjects have been tested twice on the same variable (the amount of eye contact) a dependent t-test was the appropriate test for testing changes or variations in eye contacts among babies between the age of 6 and 9 months. The t-test results indicate that the null hypothesis that there are no significant variations or changes in means of the amounts of the babies’ eye contacts within the two periods is rejected, when one pursues one-tailed t-test. However, when one pursues a two-tailed t-test, the null hypothesis that the means and therefore, variances, are the same cannot be rejected (Table 1). In two tailed test, one will have to exclude babies of ages between 1 and 3.5 months and those of ages between 6.5 and 9 months (as indicated by the vertical lines in the graph) to get results that indicate insignificant variations in means of amount of eye conducts due to same variance (Figure 1). However, after the variance correction by the Levene’s test, the null hypothesis cannot be rejected since the calculated p-value is more than the α= 0.05 (Table 1). The observed differences as seen from the graph are just due to random chances, but which cannot be attributed to the time of growth. It can be concluded that the child’s formation of social attachment to the caregiver go with time.
This method assumes that the distribution of differences between the two independent groups will be normally distributed. This may not be achieved given that sample data may not have been taken from a normally or asymptotically distributed population. The researcher also assumed that the data of the two arrays of the variables had no outliers. As already indicated, Mann-Whitney U test are important to normalize the data. The Levene’s test is also quite important in that it checks the homogeneity in variances of two arrays of data to avoid Type I error. Otherwise outliers may be identified by the use of scatter plots for residuals. In this way, in the scatter plots, the observations that indicate outliers may be removed easily.
Section C: Effects of Incentives to Pupils’ Performances
RESULTS AND DISCUSSION
Comparisons among praise, reprove, ignore and control variables
Comparisons between praise and reprove variables
Comparisons among the four groups show that there are significant differences of outcomes in the four variables. This is indicated by the observed p-value of 0.000, which is less than the α value of 0.05, indicating that the differences are significant. This can also be confirmed by the look at their means: they are different to indicate that each group had independent values. The quotient of within group mean squares and between groups mean squares yield an F-value that it is more than the F critical value. This is also another indicator that the null hypothesis that there is homogeneity in variances of the four groups is false; it is apparent from the table that the variances from the four groups are different (Table 2). From the graph, one can realize that there is no clear pattern of a child’s performance with respect to incentives (Figure 2). Same cases have been realized with orthogonal analysis between praise and reprove variables. Smaller F critical value than F-distribution value and p-value < 0.05 indicate significant differences among the groups’ means and variances (Table 3); graphically, there is no clear relationship in trend between them (Figure 3).
There are three major post hoc tests employed to compare significant differences between means of various factor under the ANOVA analyses so that any significant differences among them can be noted. These tests include the Fisher’s Least Significance Difference (LSD) test, Tukey’s Honestly Significant Difference (HSD) and the Scheffe’s test. Although the Scheffe’s test is the most popular among researchers, due to its conservatism, it can become a wasteful statistics technical when it loses its power to lead to type II error. This is likely especially when all pairs are compared. The Tukey’s test is usually preferable in complex comparisons. However, it is just important to use the Fisher’s LSD as the third most important test as. In Tables 2 and 3, it is important to realize that when the difference between means of factors is greater than the Sheffe’s factor, it will be indicated in red. When the value is greater than the HSD, it will be indicated in rose. The yellow color in the table results has indicates that the observed difference between the factors is greater that the LSD value. The greater the observed differences the tests realise the more significant differences in the factors’ means and variances.
Part D: Rumination
RESULTS AND DISCUSSION
Two way factors ANOVA without replication: subject rumination and weeks
The repeated measure design usually involves using the same subjects for all experimental conditions. The benefit of this measure allows the researcher to eliminate likely effects of individual differences that could occur if different people are called upon in the experiment. This research design usually requires a small number of participants for the research itself to be successful. However, the design may not yield better results because of the effect of repetitive experimentation. When participants are involved in repetitive episodes of the experiment, there is no doubt that they would tend to alter their responses. Although this design ensure that those who participate are quickly involved, than recruiting a large number of people, the researcher may also be forced to employ more people and material for efficient and effective administration of the experiment. Although the p-values show differences in means, this ANOVA test also considers differences in variances, so it will be important to look on p-values together with F-values.
The ANOVA p-value for subjects (in rows) is about 0.000; the ANOVA p-value for the weeks (in columns) is 0.0021. In all cases, the null hypotheses are rejected because means of rumination among subjects differ (Table 4). Likewise, means of rumination according to weeks differ. From the graph above, it is evident that the second, third, sixth, eighth, ninety and tenth subjects ruminated most in week 1. The first and the fifth individuals ruminated more in week 2 and week 3 (Figure 4). The same results can be confirmed using F-values. F-critical values in all rows and columns are smaller than those of F-distributed values (Table 4).
The Two-Way ANOVA test tests two factors at a go; in this case, one can note that the test produced results for two factors. Despite the measures of mean differences on subjects, there are also those on weeks. Like other ANOVA methods, the two-way ANOVA assumes that there are equal variances and normal population distributions for all group samples. In the two-way ANOVA analyses, assumptions are usually made that variances and therefore means of various factor espouse homogeneity. The cases of factors must also have been derived from random cases. The samples should be random, which may not be the case under this study. Scores taken from different factors have to have same variances for comparisons of any tow levels to be made. The interaction p-value was needed to show how significant the interaction between the two factors can be. The interaction parameter could indicate how subject rumination and weeks could affect each other. Some software does not usually show post-hoc analyses. Therefore, test statistics for testing the normality of the data has to be devised. There should also be those that can check homogeneity of variances among the factors and normalize them so that they can be compared well. Failure of taking the aforementioned assumptions into consideration will lead one to commit Type I error.
