t-Test

One-Sample t-Test

A one-sample t-test occurs when researchers have the mean of the sample population, but do not know the standard deviation. While less common, a one-sample t-test provides information about the sample population significantly differing from the mean. The standard error is used to determine the degree the sample differs from the mean. For example, assume it is your job to assess if the level of contamination in a distribution of wheat exceeds acceptable levels. There are 16 million grams of wheat, the sample mean ($\overline{x}$) contains 3.7 million grams of the contaminant, the acceptable contamination level for mean of a population (μ) of wheat of this size is 2.5 million grams, and the standard error of my population is 1.9 $(s_{\overline{x}})$. Use the following formula to calculate the t value:

$$\begin{array}{l}t=\overline{x}-\mathrm{\mu }=3.7-2.5\\ s_{\overline{x}}=1.9\end{array}$$

If the standard error $(s_{\overline{\mathrm{X}}})$ of the population is unknown, calculate it using the formula: $\frac{s}{\sqrt{N}}$ (s = the

unbiased standard deviation of a sample, N − 1). Computing the example, the t = .88. To determine the significance of this t value, the critical value, or the level t has to be compared in order to reject the null hypothesis. Using the degrees of freedom (N – 1), look up the corresponding t on a critical value table for a one-tailed test (easily found online). In the current example, the degrees of freedom = 15, with t = .88, and looking at a critical value table, it is apparent that t is smaller than the value of 1.75 that would be significant. We would say that t is nonsignificant and would accept the null hypothesis, that the level of contaminant in the wheat sample does not exceed acceptable levels, and is safe to release. The graph in Figure 1 depicts the findings.

...