Statistical errors are common in scientific literature, and about 50% of the published articles have at least one error. These tests are called parametric tests, because their validity depends on the distribution of the data. A frequent error is to use statistical tests that assume a normal distribution on data that are actually skewed. The outcome of these preliminary test then determines which method should be used for assessing the main hypothesis.

With large enough sample sizes (n > 30) the violation of the normality assumption should not cause major problems. The central limit theorem tells us that no matter what distribution things have, the sampling distribution tends to be normal if the sample is large enough (n > 30).

However, to be consistent, normality can be checked by visual inspection [normal plots (histogram), Q-Q plot (quantile-quantile plot)] or by significance tests. The histogram plot (frequency distribution) provides a visual judgment about whether the distribution is bell shaped. The significance test compares the sample distribution to a normal one in order to ascertain whether data show or not a serious deviation from normality. There are several methods for normality test such as Kolmogorov-Smirnov (K-S) normality test and Shapiro-Wilk?s test.

Shapiro-Wilk?s method is widely recommended for normality test and it provides better power than K-S.

Normality test and the others assumptions made by parametric tests should be pretested before continuing with the main test. The standard two independent samples t test assumes also that the samples have equal variances. If the samples being compared are not normally distributed, a non-parametric test like Wilcoxon test is recommended as an alternative to the t test. If the two samples are normally distributed, but with unequal variances, the Welch t test can be used (figure below). If the two variances are significantly different: the Welch t-test is automatically applied. Note that an online software is available here to calculate Student?s t-test without any installation.

Instructors do need to explain the tilt in the F-test ellipse because the direction of the tilt is the opposite of students tend to expect as it is the opposite of the sign of the correlation coefficient.

The final concept covered in the Statistics Level 1 material relates to the results of the F-test and the two t-tests.

Further, we can also see in Figure 3 through Figure 6 that the areas corresponding to Geary and Leser’s outcome (e), the region in which we conclude that the slope coefficients are jointly significant but individually insignificant increases in size as the correlation coefficient increases in absolute value away from zero. Reworking Equations (8a) and (8b) allows the area of the t-test rectangle to be defined as in Equation (10) below.

Students readily understand that the area of the t-test rectangle increases as the correlation coefficient increases. The homework materials ask the students to record the sizes of both the F-test ellipse and the t-test rectangle. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Student?s t test is one of the common statistical test used for comparing the means of two independent or paired samples. Many of the statistical procedures including correlation, regression, t test, and analysis of variance assume that the data are normally distributed.

There are different types of t-test : one-sample t test, the independent two samples t test and the paired t test.

Unfortunately, these pretests are not performed automatically by the built-in t.test() R function. This implies that we can ignore the distribution of the data and use parametric tests if we are dealing with large sample sizes. For example, in medical research, normally distributed data are the exception rather than the rule. If the two samples, being compared, follow normal distribution, F test can be performed to compare the variances. If the test is significant, null hypothesis are rejected and then we can conclude that the variances are significantly different. Welch t-test is an adaptation of Student?s t test used when the equality of variances of the two samples cannot be assumed. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor. Statistics Level 3 augments those materials with Bonferroni-type corrections of significance levels. The other worksheets within that spreadsheet are those mentioned in the appendices for student use.

The notation for the significance level allows for the possibility that the analyst might use one level of significance for the F-test and lower levels of significance for the t-test. The first homework question asks the students to examine how changing the level of significance changes the size of the rectangle.

To help the students, the teacher can remind the students of two concepts and then bring those concepts together to explain the tilt. The labels (a)-(e) in Figure 2 correspond to the first five outcomes while the sixth arises only with multicollinearity and is discussed below. This result is exactly the case that motivated this note; it is one in which the two variables work together to have a statistically significant joint influence on the dependent variable even if neither variable’s marginal influence is significant.

This outcome can occur with multicollinearity, but unlike outcome (f), multicollinearity is not a prerequisite. Clearly, the teacher needs to emphasize that the context of the analysis will be the driving force in distinguishing between the two possibilities in any specific case.

This possibility occurs at the extreme ends of the F-test ellipse (as the stretching of the F-test ellipse is more obvious at higher correlation coefficient values this possibility is denoted by the arrows only in Figure 6). The spreadsheet has the area calculations discussed below already programmed into the appropriate cells, so teachers don’t need to worry the their students’ programming accuracy.

One ad-hoc version of the Bonferroni Correction to the significance level of the t-tests is developed by beginning with definition of the union of rejecting the two t-test null hypotheses, as noted below. The exercises first ask the students to solve for Equation (12) and the implied level of significance and then incorporate those results into their analysis. Therefore, it?s important to combine visual inspection and significance test in order to take the right decision. In such situations, the use of parametric methods is discouraged, and non-parametric tests (which are also referred to as distribution-free tests) such as the two-samples Wilcoxon test are recommended instead.

First, the students should remember that the regions outside the F-test ellipse represent regions where the two slope coefficients are jointly significant. However, obtaining these types of results is unlikely as, in the context of Figure 6, that outcome would require the two slope coefficients to have different signs when the variables are positively correlated (and the same signs when the variables are negatively correlated as in Figure 5).

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