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Chi-squared test and linear regression

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Please help me to understand these concepts:
1. What is the null hypothesis in a chi-squared test of independence?
2. How many "tails" can a chi-squared test have, one or two?
3. In a regression analysis, what do the residuals from the fitted line add up to?
4. List the assumptions under which the linear regression model is supposed to work.
5. What is the interpretation of the coefficient of determination? Between what values can it vary?

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In this solution we answer a series of questions that explain elements of the chi-squared test and linear regression.

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Chi-squared test and linear regression
1. What is the null hypothesis in a chi-squared test of independence? 
In a chi-squared test of independence the null hypothesis is that the two variables of interest that are independent of each other, which means that the specific value that one of the variables takes does not influence in any way (i.e. has absolutely nothing to do with) the value that the other variable takes (another way to say this is that they are NOT related in any useful way).

2. How many "tails" can a chi-squared test have, one or two?
The value of the chi-squared test statistic is given by the formula: 
(please see the attached file) where (please see the attached file) is the observed frequency in the (please see the attached file) row and the column, and is the expected frequency in the (please see the attached file) row and ...

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