A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.

A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).

All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference.

Could you explain each sentence in simple terms?
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Sure, I'd be happy to explain each sentence in simpler terms:

1. "A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population)."

A statistical model is a way to represent how data is generated. It is a mathematical model that includes some assumptions about how the data is generated.

2. "A statistical model represents, often in considerably idealized form, the data-generating process."

The statistical model is a simplified version of the actual process that generates the data. It may not be exactly how the data is generated, but it is close enough to help us understand the data better.

3. "A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables."

The statistical model includes some variables that are random, meaning their values are not known in advance, and other variables that are non-random, meaning their values are known or fixed.

4. "As such, a statistical model is 'a formal representation of a theory' (Herman Adèr quoting Kenneth Bollen)."

The statistical model is a formal representation of a theory because it is a mathematical representation of how we think the data is generated.

5. "All statistical hypothesis tests and all statistical estimators are derived via statistical models."

All statistical tests and estimators rely on statistical models. These tests and estimators help us to determine if there is a significant difference between groups or if a relationship exists between variables.

6. "More generally, statistical models are part of the foundation of statistical inference."

Statistical models are an important part of statistical inference because they help us to make conclusions about a population based on a sample of data. Without statistical models, it would be difficult to make inferences about a larger population based on a small sample.
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