## Appendicies

### Appendix 3: Critical Values of t

The table below gives values of $$t\left( \alpha ,\nu \right)$$ where $$\alpha$$ defines the confidence level and $$\nu$$ defines the degrees of freedom. Values for $$\alpha$$ are defined as follows

$$\alpha$$ = 1 - confidence level (as fraction)

For example, for a 95% confidence level, $$\alpha =1-0.95=0.05$$. The degrees of freedom is the number of independent measurements given any constraints that we place on the measurements. For example, if we have $$n$$ measurements and we calculate their mean, $$\overline { x }$$, then we have $$n-1$$ degrees of freedom because the mean, $$\overline { x }$$, and the values for the first four measurements, $${ x }_{ 1 }$$, $${ x }_{ 2 }$$, $${ x }_{ 3 }$$, and $${ x }_{ 4 }$$, removes the independence of the fifth measurement, $${ x }_{ 5 }$$, whose value is defined exactly as

${ x }_{ 5 }=\overline { x } -{ x }_{ 1 }-{ x }_{ 2 }-{ x }_{ 3 }-{ x }_{ 4 }$

The values of $$t$$ in this table are two-tailed in that they define a confidence interval that is symmetrical around the mean. For example, for a 95% confidence interval ($$\alpha=0.05$$), half of the area not included within the confidence interval is at the far right of the distribution and half is at the far left of the distribution. For a one-tailed confidence interval, in which the excluded area is on one side of the distribution, divide the values of $$\alpha$$ in half; thus, for a one-tailed 95% confidence interval, we use values of $$t$$ from the column where $$\alpha=0.1$$.

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