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Y Hat: Definition

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Y hat (written ŷ ) is the predicted value of y (the dependent variable) in a regression equation. It can also be considered to be the average value of the response variable.

The regression equation is just the equation which models the data set. The equation is calculated during regression analysis. A simple linear regression equation can be written as:

ŷ = b0 + b1x.

Since b0 and b1 are constants defined by your analysis, finding ŷ for any particular point simply involves plugging in the relevant value of x.


Finding Y Hat for a Linear Regression Line

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Suppose we want to predict first grade reading abilities from the number of hours per week a child spends reading in preschool. We have a set of data points: reading ability scores (on some hopefully unbiased scale) and survey data from homes which tell us hours per day of preschool reading. With this information, we use simple linear regression and the least squares method to find the regression equation that best fits the data. In almost all cases, you won’t do this by hand, but rather with software like SPSS or R.

Suppose our line is:

ŷ = 2.45 x -0.16


Let’s say ŷ is the predicted average reading level for a child who has read half an hour a day in preschool. To find this value, all we would need to do is to plug in x = 0.5. So we’d get

ŷ = 2.45 (0.5) -0.16 = 1.065

The result tell us that the regression line predicts that a child read to half an hour a day in preschool would have a 1.065 reading level in kindergarten (on our hypothetical, fictional reading scale).

References

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.
Hughes, Elisha. Notes on Basic Practices of Statistics– Chapter 5. Regression. Retrieved from https://www.math.utah.edu/~hughes/Chapter_05.pdf on May 25, 2018
Page, Thomas. MSC 317 Class Notes: Simple Linear Regression. Retrieved from https://msu.edu/course/msc/317/slr-reg.htm on May 25, 2018.
Saint-Germaine, Michelle. Simple Regression. PPA 696 Research Methods Class Notes. Retrieved from https://web.csulb.edu/~msaintg/ppa696/696regs.htm on May 27, 2018.

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Stephanie Glen. "Y Hat: Definition" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/y-hat-definition/
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