Hi,

So long as you have one condition which is not satisfied I suggest you to use a non parametric test

If the normality assumption is failed non-parametric test is powerful than parametric.

The term parametric has at least two meanings in this context. One, arguably, more correct is that the model assumes a particular distribution for one or more parameters in the model - this needn't be a normal distribution. The second, arguably a poor use of terminology, is to describe models that assume normality (usually of the residuals of the model).

Non-normal data is often modeled using parametric models such as Poisson regression, logistic regression and so forth. Alternatively, it may be sensible to use a transformation before applying a parametric model. This is often helpful for skewed data (e.g., log tranforms often deal well with positive skew).

Sometimes it is preferable to use non-parametric methods (e.g., the bootstrap) - and this is often attractive if the the problem is with kurtosis (particularly leptokurtosis).

Also - as noted by earlier commenters with sufficiently large samples it may not matter much (though if you have leptokurtotic distributions the samples may need to be very large and the CLT may not be assumed in some common leptokurtotic distributions).

Dear Jessy,

As all contributors point out, particularly nicely formulated by Thom, your question as stated cannot be answered other than in generalities. The answer is: Sure. Therefore, something is missing. What is missing - for yourself and us - is clarity regarding the context and, specifically, the hypothesis that you wish to test. Presumably, you have formulated a Null hypothesis and done so before gathering any data. Does the hypothesis entail a distribution model and does this model have one or more parameters? What is/are the parameter(s)? Next, I would like to point you to Tukey's "Exploratory Data Analysis". Why? Exploration of your data - without prejudice or worries about the magic word "test" - will tell you of the (non-)existence of patterns and how those patterns might intuitively, at first, relate to your Null hypothesis. After all, it is the latter or its alternative that guides your interpretation of the data - potentially even in the framework of a model. Could you kindly comment on your context and your hypothesis?

Best regards, Hans

Normally distributed  data has skewness of 0 and kurtosis of 3. Kurtosis is indicative of the width of the tails. It is possible to have this value for various shapes of the distribution. Normality indicates shape not just moment matching. Exploratory Data Analysis is a good option with possible data transformation.