Question

The null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: \mu=10\) vs \(H_{a}: \mu>10\) Sample: \(\bar{x}=12, s=3.8, n=40\)

Short answer

The center of the randomization distribution is 10, and it is a right-tailed test.

Step-by-step solution

Understand the Concepts

The null hypothesis, denoted by \(H_{0}\), is a statement that the value of a population parameter, such as the population mean \(\mu\), is equal to a claimed value. The alternative hypothesis, denoted by \(H_{a}\), is the statement that the parameter has a value that somehow differs from the null hypothesis. The claim is usually that the parameter is larger, smaller or different from the value given in the null hypothesis.

Calculate Center of Randomization Distribution

The center of the randomization distribution in hypothesis testing will be the same as the mean of the null hypothesis, \(\mu\). So, the center of the randomization distribution is 10.

Determine Type of Hypothesis Test

Looking at the alternative hypothesis, the symbol '>' is an indication that it's a right-tailed test because the interest is in the values to the right (greater) of the center of the null hypothesis.

Key concepts

Null Hypothesis

The null hypothesis, symbolically represented as \(H_{0}\), is a fundamental concept in the realm of hypothesis testing within statistics. It proposes that there is no significant effect or no difference present, asserting that any observed variations in the data are merely due to random chance. In essence, it embodies the assumption of no change or no association.

When formulating a null hypothesis, you typically state that the population parameter, such as the mean \(\mu\), is equal to a specific value. For example, in the given exercise, the null hypothesis is \(H_{0}: \mu=10\), suggesting that the true population mean is presumed to be 10 unless evidence suggests otherwise. A strong grasp of this concept is pivotal as it serves as the starting point for statistical analysis and sets the stage for testing using sample data.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_{a}\) in statistical tests, posits a specific statement contrary to the null hypothesis. It asserts that there is a statistically significant effect, association, or difference that exists in the population. It essentially represents what you aim to support with evidence from your sample data.

In the context of our illustrative exercise, the alternative hypothesis is \(H_{a}: \mu>10\), suggesting the population mean is greater than 10. It's crucial to differentiate between the null and alternative hypotheses, as the entirety of hypothesis testing is designed to evaluate the likelihood of the alternative hypothesis being true given that the null hypothesis is the default position.

Randomization Distribution

Understanding randomization distribution is integral in appreciating how hypothesis testing works. Think of randomization distribution as a map of all possible outcomes that could arise if the null hypothesis were true. In the aesthetics of this map, the 'center' holds a special place, as it corresponds to the value of the parameter that is being hypothesized in the null hypothesis.

In our exercise, since we've established the null hypothesis as \(H_{0}: \mu=10\), the randomization distribution would be centered at this value. It implies that if our samples were drawn randomly under true null conditions, the mean of these randomization samples should hover around 10. Recognizing the center of this distribution is key for determining how extreme our sample statistic is and, consequently, whether there is enough evidence to reject the null hypothesis.

Tail Test

In hypothesis testing, a 'tail test' refers to the direction in which we are looking for evidence against our null hypothesis within the randomization distribution. Depending on the nature of the alternative hypothesis, it can lead to a left-tailed, right-tailed, or two-tailed test.

A left-tailed test is conducted when the alternative hypothesis indicates a parameter is less than the null hypothesis claim, a right-tailed test when it suggests the parameter is more, and a two-tailed test when the parameter is simply not equal to the claim. As outlined in the exercise, with an alternative hypothesis of \(H_{a}: \mu>10\), we are specifically looking for a sample mean to be significantly larger than 10, leading us to a right-tailed test. The 'tail' in a right-tailed test is essentially the area under the curve to the right of the central value where extreme values supporting the alternative hypothesis would lie.