1. A residual for a specific x data point can be calculated from a linear regression by calculating the difference in the y value as predicted by the model called the y predicted and by the y value observed from the data called the y actual.

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2Some residuals are positive numbers and some residuals are negative numbers. When you calculate up all the residuals from a linear regression model, the sum of all these residuals will always be 0.

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3The Coefficient of Determinations (R2) will be +1 for linear regression models that slope upwards and will be -1 for linear regression models that slope downwards.

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4An oil company expects to make $100M in revenue when the economy is good but expects to only make $10M when the economy is bad. If the economy is neither good or bad, in other words, we enter a period of economic stagnation, revenues will only be $20M. Let's say an economist predicts that the likelihood of a good economy is 10% and thelikelihood of a bad economy is 50%. You ask your analyst to come up with what your expected revenues for next year could be. Your analyst determines that your expected revenue is going to be $15M. Do you agree?

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