Test Statistic

Definition

A test statistic is a numerical value calculated from a sample data that is used to determine whether to reject or fail to reject the null hypothesis in a hypothesis test. It is a crucial component in various statistical analyses, as it provides the basis for making inferences about population parameters.

5 Must Know Facts For Your Next Test

1. The test statistic is calculated based on the sample data and is compared to a critical value from the appropriate probability distribution to determine the p-value and make a decision about the null hypothesis.
2. The specific test statistic used depends on the type of hypothesis test being conducted, the underlying probability distribution, and the parameters being estimated.
3. Common test statistics include the z-statistic, t-statistic, chi-square statistic, and F-statistic, each with their own unique properties and applications.
4. The test statistic follows a known probability distribution (e.g., normal, t, chi-square, or F) under the null hypothesis, which allows for the calculation of the p-value and the determination of statistical significance.
5. The magnitude of the test statistic, relative to the critical value, determines the strength of the evidence against the null hypothesis and the likelihood of observing the sample data under the null hypothesis.

Review Questions

• Explain the role of the test statistic in the context of the Central Limit Theorem (Pocket Change).
  • In the context of the Central Limit Theorem, the test statistic is used to make inferences about the population mean when the sample size is large enough. Specifically, the test statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean. This test statistic follows a standard normal distribution (z-distribution) under the null hypothesis, allowing for the calculation of the p-value and the determination of whether to reject or fail to reject the null hypothesis about the population mean.
• Describe how the test statistic is used in the context of Null and Alternative Hypotheses.
  • The test statistic is the key component in the hypothesis testing process. The null hypothesis and alternative hypothesis represent the two competing claims about a population parameter or the relationship between variables. The test statistic is calculated from the sample data and is used to determine the likelihood of observing the sample data under the null hypothesis. If the test statistic falls in the critical region, the null hypothesis is rejected in favor of the alternative hypothesis. The magnitude and direction of the test statistic provide the strength and nature of the evidence against the null hypothesis.
• Analyze the role of the test statistic in the context of Type I and Type II Errors.
  • The test statistic is crucial in the determination of Type I and Type II errors in hypothesis testing. A Type I error occurs when the null hypothesis is true, but it is rejected, while a Type II error occurs when the null hypothesis is false, but it is not rejected. The critical value of the test statistic, which is based on the chosen significance level, determines the threshold for rejecting the null hypothesis. If the test statistic falls in the critical region, the null hypothesis is rejected, potentially leading to a Type I error. Conversely, if the test statistic does not fall in the critical region, the null hypothesis is not rejected, potentially leading to a Type II error. Understanding the properties of the test statistic and its distribution is essential for controlling the risks of these two types of errors.

Related terms

Null Hypothesis: The null hypothesis is a statement that there is no significant difference or relationship between the variables being tested.

Alternative Hypothesis: The alternative hypothesis is a statement that there is a significant difference or relationship between the variables being tested.

Sampling Distribution: The sampling distribution is the probability distribution of a statistic calculated from a random sample, which is used to determine the test statistic's expected behavior under the null hypothesis.