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Table 2 : Dataset and columns to be completed
(a) The goal of this item is to enhance your ability to use the summation symbol, \( \sum \). Write the extensive forms of the following expressions: \[ \sum_{i=1}^{4} y_{i} x_{i}^{2}, \sum_{i=1}^{4} y_{i}^{2} x_{i}, \sum_{i=1}^{4} x_{i}^{2}, \sum_{j=1}^{4} x_{j}^{2} \text {, and } \frac{\sum_{i=1}^{4}\left(x_{i}-x_{4}\right)}{\sum_{i=1}^{4} x_{i}^{2}} . \] Please, note that you are not asked to evaluate these expressions. For instance, the extensive form of \( \sum_{i=1}^{4} y_{i}^{2} \) is \( 9^{2}+3^{2}+7^{2}+11^{2} \). Hereafter, you can use a calculator. (b) Compute the sample averages \( \bar{x} \) and \( \bar{y} \). (c) Complete the last five columns of the table. (d) Compute \( \sum_{i=1}^{4}\left(x_{i}-\bar{x}\right)^{2}, \sum_{j=1}^{4}\left(x_{j}-\bar{x}\right)^{2} \), and \( \sum_{i=1}^{4}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right) \). (e) Compute the estimates of the sample standard deviations, \( \sqrt{S_{X}^{2}} \) and \( \sqrt{S_{Y}^{2}} \).