# linear regression analysis

i am given data for analysis following data:relationship between height and weight,question is :is relationship between them linear?like $y$=$a$+$b$*$x$+$e$ where e is error,or quadratic?or cubic?data is following(y denoted height, x-weight)

y   x
170 65
167 55
189 85
175 70
166 55
174 55
169 69
170 58
184 74
161 56
170 75
182 68
167 51
187 85
178 62
173 60
172 68
178 55
175 65
176 70


i have calculated a and b and get following result

y=139.6155237+0.5308912572*x


i dont know how to calculate e?or what me result means?please help me how to find e and how to check correctness of my work

EDITED:
i have edited my work,i am sure it is correct,what  now should i do?

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Is this only part of your data? –  Julius Apr 10 '12 at 16:29
not but others are like gender,age and BMI –  dato datuashvili Apr 10 '12 at 16:33
Yes, coefficients are correct now. What is your task? To estimate the model and find residuals? –  Julius Apr 10 '12 at 17:31
yes exactly,i have put x variable but have not got exact y value,so is it not linear yes? –  dato datuashvili Apr 10 '12 at 17:32
That is fine, your model can't be perfect. The difference that you get is residual (-4.12 first data point, -1.81 second etc.). Again, try to plot your data. It is hard to tell whether the relationship is linear, there are too few observations, but linear model, in my opinion, is the best choice here. –  Julius Apr 10 '12 at 17:38

Fast and good way to check the relationship between variables is to plot them like that, here we can see a linear relationship, now this would be a quadratic one. I don't fully understand the situation of your data, if these are the only data you are using for this height ~ weight regression, then your estimates are wrong, $a$ should be almost 170, and $b$ should be almost zero (considering your data). Plotting your data would confirm that. The way to calculate the residuals is $\varepsilon=y-\hat{y}$, here $y$ is your real data and $\hat{y}$ is what the model says. Now your results mean that every additional kilogram reduces 6 centimeters from the weight, which doesn't seem to be right. Then this $\hat{\alpha}=\overline{Y}_t-\hat{\beta}\overline{X}_t$ helps to understand the meaning of $a$. It is hard to say how to check correctness of your work, if you only had to determine the relationship, then you have chosen correct - linear model (but some why coefficients are wrong).

This is the answer to not edited question.. Correcting the answer now

Well, now your data looks fine, but your estimates are even worse: both $a$ and $b$ are negative, so clearly somewhere you have made a mistake. Here you may find two formulas and estimate your model by yourself (hint: $a$ should be between 100 and 150, $b$ between 0 and 1)

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I HAVE CALCULATED FROM HERE USING EXCELL easycalculation.com/statistics/learn-regression.php –  dato datuashvili Apr 10 '12 at 16:59

First you fit the data with your expectation model: $y=a+bx$

Then you compute pertuberations of data points around the model curve (line). Analysis of this will give you variance in data points (the error). The least squares approach expects the errors are distributed normally, so that the errors should be samples drawn from gaussian distribution.

Your (least squares) estimator have two properties: precision and accuracy. I think the precision is what you want to compute. Accuracy is about bias, and this is harder to grasp (e.g. all weights are slighly bigger because of uncalibrated measurement device).

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