Testing associations between nominal and ordinal variables I know there have been many questions on this topic. Sorry for another one. 
I have done a survey in which both questions using Likert scale (strongly disagree-strongly agree, so 5 point) have been asked and yes/no (binary) questions. I want to explore whether one of the yes/no questions (if a certain quality team exist) is associated with one of the likert scale questions (an ordinal question: if a leader helps to facilitate quality). What would be the best test to explore this? Is a Pearson chi-squared test sufficient for this? 
 A: As far as I know, a chi-squared test is best for comparing two categorical variables because it only tells you if the distribution of your likert variable is different for the different groups. It doesn't tell you which group has a higher score than the other. If the scale is (sort of) normally distributed, you can do a t-test to test for differences between groups. if it's highly skewed, I would suggest an ordinal regression. You can check the distribution with a histogram. 
A: Yes and No on each of the dichotic questions divide the dataset into two groups. In each of these groups we find independent ordinal values to compare. That is a classical setting for Wilcoxon rank sum test (aka Mann-Whitney U). If you plan to calculate a lot of them, think about how you want to deal with alpha error inflation. 
One single answer on a scale of 5 possible answers is hardly a Likert scale. A Likert scale is the sum of a number of such answers and is then often treated as quasi-metric. With only one answer on a 5-step scale, your answers are not metric, which is why a t-test is not recommended.
I suppose, yuo want to know, whether Yes or No leads to higher or lower scores on the 5-point scale. A $\chi^2$ test is not the optimal solution. Consider yes-Answers to be all "1" and "5" on the 5-point-scale and no-answers to be all "3". The $\chi^2$ will tell you, they are different, even though none of them has generally "higher" or "lower" answers.
A: You need to conduct some univariate analyses to investigate if your data qualifies the assumptions for parametric testing. If it does, I would suggest to compare the means among/between your groups using T-test or ANOVA. If the assumptions are violated, you need to look at non-parametric tests. 
A: If you believe that what is being measured by the Likert scale and the binary questions are a continuous latent variable (e.g., the amount of agreement with the statement, or the extent to which the quality exists) then you might want to consider the use of polychoric correlation. This would not only tell you if there was an association, but you would get a correlation coefficient which provides information on its strength and direction.
You can use polychoric correlation with any combination discrete variables (i.e., with 2 or more levels), as long as you assume that each variable is representing a continuous latent variable.
