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In statistics, the term **y hat** (written as **ŷ**) refers to the estimated value of a response variable in a linear regression model.

We typically write an estimated regression equation as follows:

ŷ = β_{0} + β_{1}x

where:

**ŷ**: The estimated value of the response variable**β**: The average value of the response variable when the predictor variable is zero_{0}**β**: The average change in the response variable associated with a one unit increase in the predictor variable_{1}

For example, suppose we have the following dataset that shows the number of hours studied by six different students along with their final exam scores:

Suppose we use some statistical software (like R, Excel, Python, or even by hand) to fit the following regression model using *hours studied* as the predictor variable and *exam score* as the response variable:

**Score = 66.615 + 5.0769*(Hours)**

The way to interpret the regression coefficients in this model is as follows:

- The average exam score for a student who studies zero hours is
**66.615**. - Exam score increases by an average of
**5.0769**points for each additional hour studied.

We can use this regression equation to *estimate* the score of a given student based on the number of hours they studied.

For example, a student who studies for 3 hours is predicted to get a score of:

Score = 66.615 + 5.0769*(3) = **81.85**

**Why is Y Hat Used?**

The “hat” symbol in statistics is used to denote any term that is “estimated.” For example, **ŷ** is used to denote an estimated response variable.

Typically when we fit linear regression models, we use a sample of data from a population since it’s more convenient and less time-consuming than collecting data for every possible observation in a population.

So, when we find a regression equation we’re only *estimating* the true relationship between a predictor variable and a response variable.

This is why we use the term ŷ in the regression equation instead of y.

**Additional Resources**

Introduction to Simple Linear Regression

Introduction to Multiple Linear Regression

Introduction to Explanatory & Response Variables