Question

In Exercises 4.45 to \(4.49,\) 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: \(H_{0}: p=0.5\) vs \(H_{a}: p<0.5\) Sample: \(\hat{p}=0.4, n=30\) Randomization statistic \(=\hat{p}\)

Short answer

The center of the randomization distribution will be 0.5, and the test is a left-tail test.

Step-by-step solution

Understanding the Hypotheses

The null hypothesis \(H_{0}: p=0.5\) implies that the population proportion \(p\) equals 0.5. The alternative hypothesis \(H_{a}: p<0.5\) suggests that the population proportion \(p\) is less than 0.5. As the alternative hypothesis is less than, it represents a left-tail test.

Identifying the Randomization Distribution Center

The center of the randomization distribution under the null hypothesis is at the assumed value of \(p\) in the null hypothesis, which here is 0.5.

Identifying the type of test

Based on the alternative hypothesis \(H_{a}: p<0.5\), this is a left-tail test, this is because we are looking for values of \(p\) which are less than the hypothesized value in \(H_{0}\).

Key concepts

Null Hypothesis

When diving into statistical hypothesis testing, one must understand the cornerstone concept of the null hypothesis, typically denoted as \(H_0\). The null hypothesis is a statement that there is no effect or no difference, and it sets the stage for statistical comparison. For instance, in the context of a population proportion test, the null hypothesis might assert that the true proportion \(p\) equals a specific value, such as 0.5. In our exercise, the null hypothesis states that \(H_0: p=0.5\), meaning the hypothesis posits that half of the population has a certain characteristic.

In statistical tests, the null hypothesis is what we assume to be true until we have sufficient evidence to suggest otherwise. The thrills of statistical testing often lie in the potential to reject this hypothesis, implying that our data provides convincing evidence of an actual effect or difference.

Alternative Hypothesis

In the exciting world of hypothesis testing, the alternative hypothesis stands in opposition to the null hypothesis. Denoted as \(H_a\) or \(H_1\), it is a statement that suggests there is an effect or a difference, and we are interested in gathering evidence to support it. For problems related to population proportion, it proposes a value of \(p\) that is different from the value stated in the null hypothesis. In the exercise at hand, \(H_a: p<0.5\) theorizes that the population proportion is, in fact, less than 0.5.

This hypothesis is what researchers really hope to find evidence for, since it often represents the possibility of discovering new knowledge or confirming a theory. In our practical scenario, the alternative hypothesis is suggestive of a population proportion that is smaller than the anticipated 50%. If statistical testing supports this hypothesis, it can lead to significant conclusions about the population being studied.

Left-tail Test

Grasping the concept of a left-tail test is crucial for understanding the direction in which we expect our statistical evidence to lead us. A left-tail test, as the name implies, involves looking at the lower end, or 'left tail,' of the distribution of test statistics. It's used when the alternative hypothesis, \(H_a\), indicates that the parameter of interest is less than the null hypothesis value. To visualize it, picture a bell curve: a left-tail test focuses on the tail portion to the left of the center.

In our exercise, we have an alternative hypothesis of \(H_a: p<0.5\), which points to a left-tail test. We are essentially on the lookout for evidence that the sample proportion \(\hat{p}\) is significantly less than the hypothesized proportion of 0.5. If the calculated statistic from our sample falls into this left tail beyond a certain critical value, we may have enough ammunition to reject the null hypothesis.

Population Proportion

The concept of population proportion, symbolized as \(p\), is an estimate that reflects the fraction of a population that exhibits a particular trait. It is a summary statistic that tells us what percentage of the population we are studying falls into a specific category. The population proportion is a key parameter in many types of statistical tests, including tests of proportions.

In situations where we're working with a population proportion, as with the exercise provided, we often deal with a sample from the population to make inferences about this proportion. Our original exercise involves a sample that has been used to estimate the population proportion, which is indicated by the symbol \(\hat{p}\). Here the sample proportion of 0.4, taken from a sample size of 30, is used to make inferences about the population's proportion.

Sample Size

The term 'sample size', denoted as \(n\), plays an influential role in determining the reliability of a statistical test. It refers to the number of observations or measurements that are included in a statistical sample. The larger the sample size, the more confident we can be in our statistical inferences because a larger sample is more likely to represent the true characteristics of the population.

In our given exercise, the sample size is 30. This piece of information is critical because it affects the precision of our sample proportion \(\hat{p}\). When determining how conclusive our evidence is, or when calculating the margin of error or the test statistic, the sample size is an indispensable component to the formulae used. It's essential to remember that while larger samples provide a clearer picture, there are often practical and financial limitations to how large a sample can be.