Question

Null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. \(H_{0}: \mu=15\) vs \(H_{a}: \mu<15\)

Short answer

The notation for the sample statistic that we might record for each simulated sample in order to create the randomization distribution is \(\bar{x}\), which represents the sample mean.

Step-by-step solution

Understanding the problem

The first step is to understand the problem. In this case, we have a set of null and alternative hypotheses about the population mean, and we need to figure out the appropriate sample statistic that we can use for creating a randomization distribution. Here, the population mean (\(\mu\)) could be measured in any number of ways, depending on the context. However, regardless of the measure, we are testing whether it is equal to 15 (null hypothesis) or less than 15 (alternative hypothesis).

Choosing the appropriate sample statistic

For this type of problem, we normally use the sample mean (\(\bar{x}\)) as our sample statistic to create a randomization distribution. The sample mean is a good statistic because it provides an estimate of the population mean, which is what our hypotheses are about.

The Notation for the Sample Statistic

In this situation, the notation for the sample statistic that could be recorded for each simulated sample to create the randomization distribution is \(\bar{x}\). This is the symbol that is generally used to represent the sample mean in statistics.