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. \(H_{0}: \rho=0 \quad\) vs \(H_{a}: \rho \neq 0\)

Short answer

The sample statistic we might record for each simulated sample to create the randomization distribution in this scenario is the sample correlation coefficient, \( r \).

Step-by-step solution

Understanding the Hypotheses

Here, we have two hypotheses about the population correlation coefficient \( \rho \). The null hypothesis \( H_0 \) is that \( \rho = 0 \), which means there is no correlation between the two variables. The alternative hypothesis \( H_a \) is that \( \rho \neq 0 \), indicating there is a correlation, either positive or negative.

Decide on the Sample Statistic

For the sample statistic, we need something that can help us estimate the correlation in our sample data and compare it to the hypothesized value. The appropriate statistic in this case is the sample correlation coefficient, \( r \). This statistic can provide a measure of the strength and direction of the relationship between the two variables in the sample data.

Creating the Randomization Distribution

We would use this \( r \) value as our test statistic and record its value for each simulated sample. By doing this, we can create a randomization distribution of \( r \) values under the assumption that the null hypothesis is true (i.e., that there is no correlation). This distribution can then be used to determine the probability of observing a result as extreme as our sample data if the null hypothesis is true, which forms the basis for deciding whether to reject the null hypothesis.