Sometimes the null hypothesis is that the mean is at least (or at most) a specified value. (e.g., there are at least two scoops of raisins in Kellogg's Raisin Bran, there are at most 5 ppm lead in drinking water.) In this case one would reject the null hypothesis only if x-bar is too small (or too large, respectively). In this circumstance a one-tailed test is employed. The null hypothesis (H0) for a one tailed test is that the mean is greater (or less) than or equal to µ, and the alternative hypothesis is that the mean is < (or >, respectively) µ. The text will always use = to state the null hypothesis, and use the alternative hypothesis to identify whether it is a one-tailed test, and which tail. Often a subscript 0 will be appended to µ to emphasize that it refers to the mean under the null hypothesis.
Examples:
Competencies: If the standard deviation is known to be equal to 12, and your null hypothesis is that the mean of the population is less than or equal to 15 i.e., the alternative hypothesis is that the mean is greater than 15, at what level (p-value) is x-bar = 13.5 based on a sample of size 200 significant? Would you reject the null hypothesis at the 10% significance level? 5% significance level? 1% significance level?
If the standard deviation is known to be equal to 12, and your null hypothesis is that the mean of the population is greater than or equal to 15 i.e., the alternative hypothesis is that the mean is less than 15, at what level (p-value) is x-bar = 13.5 based on a sample of size 200 significant? Would you reject the null hypothesis at the 10% significance level? 5% significance level? 1% significance level?
Reflection: How are one and two tailed tests related? Where in the calculations is the one- or two-tailedness of the test manifested?