Solve the System of Equations $\left\{ \begin{array} { r } x + 2 y + z + w = 0 \\x + 2 y - z - w = 0 \\x + 4 y + z + 2 w = - 1 \\x + 3 y + z + w = 2\end{array} \right.$

question 11

Essay

Solve the system of equations $\left\{ \begin{array} { r } x + 2 y + z + w = 0 \\x + 2 y - z - w = 0 \\x + 4 y + z + 2 w = - 1 \\x + 3 y + z + w = 2\end{array} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c c c } \frac { 5 } { 2 } & \frac { 1 } { 2 } & 0 & - 2 \\- 1 & 0 & 0 & 1 \\- \frac { 1 } { 2 } & - \frac { 1 } { 2 } & - 1 & 2 \\1 & 0 & 1 & - 2\end{array} \right]$ .

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Definitions:

Type I Error

The incorrect rejection of a true null hypothesis, also known as a "false positive" in hypothesis testing.

P-Value

The probability of observing a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

Significance Level

The probability of rejecting the null hypothesis in a statistical test when it is actually true, a measure of the risk of making a Type I error.

Alternative Hypothesis

The hypothesis that there is a significant difference or effect, contrasting with the null hypothesis and indicating the presence of an observed effect.