Null and alternative hypotheses for a test are given. Give...

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. $H_{0}: \mu_{1}=\mu_{2}$ vs $H_{a}: \mu_{1}>\mu_{2}$

Key Concepts

Null Hypothesis

A foundational concept in hypothesis testing, the null hypothesis is a statement asserting that there is no effect or no difference between specific parameters. It serves as the default claim that any observed differences in the data are due to random chance. In the context of comparing means, it typically posits that the two population means are equal.

Alternative Hypothesis

This is the statement that contradicts the null hypothesis and proposes that there is an effect or a difference between the groups being compared. It represents the outcome the researcher is interested in proving, such as one mean being greater than the other.

Sample Statistic

A numerical value calculated from sample data that serves as an estimate of a population parameter. In hypothesis testing, common sample statistics include the sample mean, median, or difference between means. This statistic is recorded for each simulated sample to help build the distribution of the test statistic under the null hypothesis.

Randomization Distribution

A distribution created by repeatedly simulating samples under the assumption that the null hypothesis is true. This method assesses the variability of the sample statistic and helps determine whether the observed statistic is statistically significant relative to the distribution of values expected by random chance.

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