Distribution of the scores of a difficult exam with most of examinees from same class scoring low marks naturally would have the data points concentrated towards the left side of the centre of the distribution. This gives rise to a shape which deviated from the symmetry towards being a right skewed or positively skewed distribution (Brase & Brase, 2016).
Given a situation where two populations have unequal means, when the inferential test used to test the validity of the conjecture results in the conclusion that the means of the two populations are not different, it means that the test had failed to reject the null hypothesis. However it is given that the means are not in fact equal which makes the conjecture posed as the null to be false thus leading to a case of failure to reject the false null. This kind of an error is termed as type II error (Silvey, 2017).
The p-value of a test is the probability of the statistic of interest in an inferential testing procedure to lie above the observed value of the test statistic. It is an indicator of the results of the test, whereby if the p-value is found to be less than the pre-defined level of significance, the null is rejected at said level of significance; otherwise the test fails to reject the null hypothesis management (Wasserman, 2013).
A test which is found to be statistically significant indicates that the evidence collected and used for the testing suggests that it is more likely that the null hypothesis is untrue than it is likely that it’s true. A test is said to be statistically significant if the observed value of the test statistic is found to be greater than the critical value of the test which is determined keeping in mind that the probability of type I error is restricted to a low value, ?, typically 0.05 or 0.001 (Wasserman, 2013).
Scenario A is a situation where the researcher does not assign any treatment to the subjects but merely observe the subjects and their eating habits as well as the LDL levels that they have. The treatment that a particular subject receives is beyond control of the experimenter. It is thus an observational study design (Brase & Brase, 2016).
Scenario B is a situation where the researcher assigns and hence controls the treatment that a particular subject would receive in the course of the experiment that is whether the subjects would have meals comprising of oatmeal every morning or not and then observes the effects of the treatment on the subjects, which is their LDL levels. It is thus an experimental study design (Brase & Brase, 2016).
Scenario B would serve to be more effective in demonstrating the effects of eating oatmeal on LDL. This is because in scenario B, the experimenter has full control over the kind of treatment applied on the subjects, as well as he or she could note the response before the treatment was applied leaving less room for some other unknown factor to influence the response and the agency to measure the difference made upon applying the treatment on individual subjects.
The response variable (Y) is clearly the LDL levels of the subjects and the predictor variable (X), upon which the LDL levels is believed to be dependent or is being influenced is the therefore, the variable indicating whether the subject regularly eats oatmeal or not.
An independent t-test comparing the reduction in LDL levels between the two treatment groups, namely, the group which had oatmeal daily(group 1) and the group which did not eat oatmeal regularly(group 2) would be appropriate to check whether eating oatmeal regularly could lead to greater reduction in LDL levels.
Let ?i be the average of reduction in LDL in group i (=1,2) . The null and alternate hypothesis are then defined as follows:
H0: ?1 = ?2 (null) versus H1: ?1 > ?2 (alternate)
The observed summary value of reduction in LDL in group 1 was found to be 12units which is greater than the reduction of 8units in group 2. However this is not enough to suggest that the reduction is indeed greater for group 1 since the statistics in this case did not take into account the variation and sampling error in the reduction being considered in the groups.
The null hypothesis of the test ought to be rejected in favour of the alternate at 0.05 level of significance since the p-value was found to be 0.03 which is less than 0.05.
The boxplots and means for each group suggest that the groups have means which differ from one another.
An ANOVA test ought to be conducted to test the tentative equality of the means for the different groups. This is so because merely looking at the means may not truly reflect the difference in the distributions and one must take into account the variation in the data to see whether the groups are truly different (Silvey, 2017).
Let µi be the mean of ith group. Then the hypotheses would be defined as:
H0: µA = µB = µC = µD (null)
H1: not null (alternate)
The test statistic in an ANOVA test is the F statistic which is observed to be equal to 10.63 in this case.
The p-value being < 0.0001 suggests that the null hypothesis specified in part c above ought to be rejected, that is, there is significant difference between the means of the four groups.
The results of part e although is enough to determine that the means of the groups are not equal but fails to determine the relative position of the group means among each other, that is, the efficiency of each group. To determine so, an additional follow up or post hoc test, such as Tukey’s test ought to be carried out (Silvey, 2017).
Silvey, S. D. (2017). Statistical inference management. Routledge.
Wasserman, L. (2013). All of statistics: a concise course in statistical inference. Springer Science & Business Media.
Brase, C. H., & Brase, C. P. (2016). Understanding Basic Statistics, Enhanced. Cengage Learning.