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I have a simple regression fit for predicting house prices from square feet. The estimated intercept is -44850 and the estimated slope is 280.76

To make predictions for inputs in square meters, what intercept and slope must we use ?

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  • $\begingroup$ So a 159 or less square foot house has a predicted price of being negative? Perhaps there aren't any of those, but it doesn't sound like much of a model $\endgroup$ – Mark L. Stone Dec 3 '15 at 15:02
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    $\begingroup$ @Mark Not all models are intended to be applied to all possible values of the regressors. For instance, if this regression were intended only for mansions, the negative intercept would not be problematic. $\endgroup$ – whuber Dec 3 '15 at 15:29
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    $\begingroup$ An anonymous editor suggests that this is a quiz question on a Coursera machine learning course. Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ – gung May 3 '16 at 15:36
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The intercept has exactly the same units as your response variable, here house price, and is thus unaffected by changing the units of area.

Note incidentally that while the prediction of a negative price for a property with zero area (and with zero values for any other predictors) will be outside the range of the data, an intercept of that magnitude may signal that your regression functional form is a poor choice, as also hinted by @Mark L. Stone. But much depends on what the currency is. Perhaps 44850 is small change in your unstated currency. More generally, I wouldn't expect a simple straight-line model to be automatically a good choice for house price and area. But if you want advice on that front, please show us your data and ask a new question.

The slope associated with area has units (units of currency) / (units of area) and so to convert, given a change to square metres, you must multiply by (feet/metre)$^2$. The conversion factor lies beyond statistics and is easy to Google, but the exact definitions 12 inches $=$ 1 foot and 25.4 mm $=$ 1 inch render it subject to your favourite means for simple calculations.

It is a pity that statistics seems to be not often taught together with thinking about dimensions and units of measurement. For one splendid article with several insights see

Finney, D. J. 1977. Dimensions of statistics. Journal of the Royal Statistical Society Series C (Applied Statistics) 26(3): 285–289. http://doi.org/10.2307/2346969 (Finney 1917$-$2018)

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One way to figure this out (one that works with models more complex than linear regression and conversions more complex than square feet/meters, like Fahrenheit to Celsius) is to write your model and conversion as equations, and do the substitution.

So you have a linear model $y = f(x) = mx + b$, where $x$ is in square feet. What you want is a model $y = g(x')$, where $x'$ is in square meters.

What you now need is the relationship between $x$ and $x'$. For square feet to square meters, this would be $x = 10.7639 \cdot x'$ -- Be careful about the direction of change. What we want to do is take the desired quantity (the area in square meters) and convert it to the quantity we have the model for (the area in square feet).

Now all we have to do is substitute the value of $x$ in the original predictive model with the value of $x$ obtained from our conversion formula. $y = m(10.7639 \cdot x') + b = 10.7639 \cdot m \cdot x' + b$, where $m$ and $b$ are their original (square-foot) values.

This sort of manipulation becomes easier and has a built-in check if you keep in mind dimensional analysis. That is, instead of working with plain numbers, you work with both numbers and units, and consider the units to be part of the values. In the dimensional analysis approach $x$ would would come with the unit '$ft^2$', $m$ would have '$USD/ft^2$' and $b$ would be in '$USD$'. Thus the dimensional analysis of the original equation would be $y = (USD/ft^2) \cdot (ft^2) + USD$. Cancel factors, and permit addition of like units, and you end up with $y = USD$, as you should. In the new formulation, the conversion factor has units of $ft^2/m^2$, so the dimensional analysis of the final equation ($y = 10.7639 \cdot m \cdot x' + b$) would be $y = (ft^2/m^2) \cdot (USD/ft^2) \cdot (m^2) + USD$. Again, cancel like factors through division/multiplication, and allow their combination through addition, and you end up with $y = USD$. If you messed up, then things wouldn't cancel cleanly.

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    $\begingroup$ Helpful (+1). Assuming for the sake of argument that the currency is USD.... $\endgroup$ – Nick Cox Dec 3 '15 at 19:24
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There are 0.092903 square meters in 1 square foot so slope should be (280.76)*(0.092903) and intercept remains same

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  • $\begingroup$ i can't reduce the score on this. but this is a wrong answer: it is 280.76 / 0.092903 $\endgroup$ – endless Dec 2 '16 at 3:42
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I can answer this. In this case, there will not be any change in the intercept.However,only the slope changes with the change in the magnitude. This house costs 280.76 per sqft. Hence it would cost 3022.08 per sq.metre assuming that there is no currency change and keeping in mind the scale of change from sqft to sq.metre (1 sq.ft = 0.092903 sq.metres)

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