Other One-Sample Tests for Location
When testing a hypothesis about a population mean based on sample data, more often than not we will not know the variance of the population. Rather, we will use the sample data to estimate that variance. As presented in the previous chapter, for testing hypotheses of the form:
H0: μ = μ0 (12.1) H0: μ = μ0
We presented a formula that transforms our data into a z-statistic:
When we do not know the value of σ, we must substitute an estimate of σ. We learned much earlier in the book that the sample value of s2 was an unbiased estimate of σ2. Thus we may want to use s as an estimate of σ. Recall that:
Thus, our expression for transforming our data would be:
We have placed a “hat” over the zobs as we are not really certain that this statistic will be distributed as z. To demonstrate this point we can look at the sampling distribution of s2 using simulations with R. Let’s assume that we are working with a random variable that is normally distributed with a mean of 50 and a variance of 25 (Y ~ N(50, 25)). We used the R script (Script 12.1) that is contained in the supplementary materials for this chapter (see book website), repeating it with sample sizes of 2, 5, 10, and 25. The descriptive statistics are presented in Table 12.1, and the results are depicted graphically in Figure 12.1.