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Model-based assumptions. These include the following three types: Distributional assumptions. Where a statistical model involves terms relating to random errors, assumptions may be made about the probability distribution of these errors. [5] In some cases, the distributional assumption relates to the observations themselves. Structural assumptions.
The i.i.d. assumption is also used in the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution. [4] The i.i.d. assumption frequently arises in the context of sequences of random variables. Then, "independent and identically ...
Consider an experiment to study the effect of three different levels of a factor on a response (e.g. three levels of a fertilizer on plant growth). If we had 6 observations for each level, we could write the outcome of the experiment in a table like this, where a 1, a 2, and a 3 are the three levels of the factor being studied.
Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-sided dice. We will study two different statistical assumptions about the dice.
Independence of observations – this is an assumption of the model that simplifies the statistical analysis. Normality – the distributions of the residuals are normal . Equality (or "homogeneity") of variances, called homoscedasticity —the variance of data in groups should be the same.
The assumptions underlying a t-test in the simplest form above are that: X follows a normal distribution with mean μ and variance σ 2 /n. s 2 (n − 1)/σ 2 follows a χ 2 distribution with n − 1 degrees of freedom. This assumption is met when the observations used for estimating s 2 come from a normal distribution (and i.i.d. for each group).
The scientific method is an empirical method for acquiring knowledge that has been referred to while doing science since at least the 17th century. The scientific method involves careful observation coupled with rigorous scepticism, because cognitive assumptions can distort the interpretation of the observation.
A person's assumptions or beliefs about the relationship between observations and a hypothesis will affect whether that person takes the observations as evidence. [3] These assumptions or beliefs will also affect how a person utilizes the observations as evidence.