How linear regression can be used - explanation for grandmothers Educational question. 
Suppose you have to explain linear regression to your granny. She is well educated, she knows even the idea of the hypothesis testing, but before you start to tell her what regression is she asks you: "Why do I need to study a regression? How could it help me?"
The problem is to explain the goals and benefits of a linear regression without mathematical terminology (without definition of linearity, for example). Can it be done?
My variant is "Suppose we have Y and X - vectors of equal length (continuous measurements). We want to study if Y has a relationship with X. Linear regression can help us to establish properties of one particular type of a relationship".
 A: I have had to do this before.  I thought that comparing to a cab ride was easy.  
You have a flat fee to get into the cab even if you go nowhere and then you pay a fee per mile.  
Now, what if you don't know the fees but have a history of trip prices?
That was easy for people to understand since that's something that they deal with often here where I live.
A: Linear regression fits a line to data:
Linear regression as estimating a conditional expectation function:


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*This is immediately useful for simple forecasting. Tell me x, and I can say something about y. 

*For example, forecast plant growth based upon sunlight exposure.


Linear regression as estimating causal effects:


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*In some situations (eg. when variation in x is exogenously introduced by the researcher), linear regression can estimate causal effects.

*If sunlight exposure is randomly assigned by researchers, the regression line would estimate the causal effect of sun exposure on plant growth (i.e. the local average treatment effect).


Linear, ordinary least squares regression is the starting place for all sorts of more advanced estimation techniques. Linear regression is to data analysis as salt is to cooking. You don't have to add it, but it's everywhere.
A: I would start with some games: give the person you want to teach many continuous numbers which represent a group of people's weight, and ask him/her to predict next. I would say, the person is very likely to use average of all numbers for the prediction. Ask the person why. Try to plant a seed to say using mean is essentially to have a predictor: $y=\beta_0$
Now, we will go to next game: in addition to only give the person predicting target, i.e., weight, give him/her additional information: height. Apparently height is very useful, in most cases, the taller suggests heavier. But the question is: what is the predictor / formula.
You can give him/her some formula got from the regression, such as $y=\beta_0+\beta_1x$ and let the him/her to use this for prediction. I think him/her will like the idea.
When him/her ask you how did you get the formula, you can tell the whole "linear regression" story, i.e., what the $\beta_0$,$\beta_1$ represent for and how to get them.
