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

Short answer

The notation for a sample statistic we might record for each simulated sample to create the randomization distribution is \( \bar{x_1} - \bar{x_2} \), the difference between the two sample means.

Step-by-step solution

Identify the required statistic

Since we are dealing with two population means here \( \mu_1 \) and \( \mu_2 \), the most appropriate statistic that we can use to create the randomization distribution is the difference between two sample means denoted \( \bar{x_1} - \bar{x_2} \).

Key concepts

Null Hypothesis

Understanding the null hypothesis is critical when conducting statistical tests. It's a statement that there is no effect or no difference, and it's used as a starting point for any significance testing. In other words, it's the assumption that any observed outcomes are due to chance rather than any specific cause. For example, if we're comparing two groups to see if there's a difference in their means, the null hypothesis would state that the two population means are equal, denoted as \(H_0: \mu_1 = \mu_2\).In significance testing, the null hypothesis serves as a benchmark for determining whether the evidence we have is strong enough to reject this initial assumption. Statistically, if we find that the data is highly improbable under the null hypothesis, then we may have reason to consider the alternative hypothesis as more likely.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis \(H_a\) suggests that there is an effect or a difference. It is essentially what we want to prove. When the exercise states the alternative hypothesis as \(H_a: \mu_1 > \mu_2\), it's saying that the researcher believes that the first population mean is greater than the second.This alternative hypothesis is a directional (or one-tailed) hypothesis, since it's testing for the possibility of the relationship in one direction only. As we gather data, if we see a significant pattern that supports \(H_a\), we may reject the null hypothesis in favor of this alternative.

Sample Statistic

Sample statistics are numerical values that summarize data from a sample, and they serve as estimators for population parameters. In the context of our exercise, a key sample statistic we might use is the difference between sample means, denoted as \(\overline{x}_1 - \overline{x}_2\).This specific statistic compares the average outcomes of two different samples. If the populations indeed have no difference in means (null hypothesis), the statistic should typically hover around zero. Otherwise, the statistic might show a consistent deviation from zero, which could suggest that \(\mu_1\) is indeed greater than \(\mu_2\), in line with the alternative hypothesis.

Population Means

Population means \(\mu\) represent the average value within an entire population. They are fixed values, but often unknown to us, which is why we use sample data to estimate them. In studies involving comparison, such as the given exercise, we're considering two different population means, \(\mu_1\) and \(\mu_2\).Our goal with statistical testing is to draw conclusions about these population means based on our sample data. The truth about the population means may be obscured by sample variability, so we use the concept of a randomization distribution derived from sample statistics to make inferences about them.

Difference Between Two Sample Means

When comparing two groups, the difference between two sample means is a powerful tool. It is the calculation of how much one sample mean \(\overline{x}_1\) deviates from another \(\overline{x}_2\). In terms of hypothesis testing, a significant difference provides evidence against the null hypothesis.For the exercise at hand, creating a randomization distribution of the difference in sample means allows us to visualize the variability of this statistic under the null hypothesis. By collecting this data from simulated samples, we're equipping ourselves to better judge how unusual our observed statistic is, and whether it lies within the range of variability that we'd expect if the null hypothesis were true.