T-Statistic Formula

Definition

The t-statistic formula is a statistical measure used to determine the significance of the difference between a sample statistic and a hypothesized population parameter, particularly in the context of hypothesis testing. It is a critical component in evaluating the strength of evidence against the null hypothesis and making inferences about the population.

5 Must Know Facts For Your Next Test

1. The t-statistic formula is $t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
2. The t-statistic is used in hypothesis testing of a single mean when the population standard deviation is unknown, and in hypothesis testing of a single proportion.
3. The t-statistic follows a t-distribution, which is a family of probability distributions that depend on the degrees of freedom.
4. The degrees of freedom for the t-statistic in hypothesis testing of a single mean is $n-1$, and for a single proportion is $n-1$.
5. The p-value associated with the t-statistic is used to determine the statistical significance of the test, and is compared to the chosen significance level to make a decision about the null hypothesis.

Review Questions

• Explain the purpose of the t-statistic formula in the context of hypothesis testing of a single mean.
  • The t-statistic formula is used in hypothesis testing of a single mean when the population standard deviation is unknown. It compares the sample mean to the hypothesized population mean, and takes into account the sample size and sample standard deviation to determine the strength of evidence against the null hypothesis. The t-statistic follows a t-distribution, which is used to calculate the p-value and make a decision about the null hypothesis based on the chosen significance level.
• Describe how the degrees of freedom are calculated for the t-statistic in hypothesis testing of a single proportion.
  • In hypothesis testing of a single proportion, the degrees of freedom for the t-statistic is $n-1$, where $n$ is the sample size. The degrees of freedom represent the number of values in the final calculation of the t-statistic that are free to vary, and are used to determine the appropriate t-distribution for the hypothesis test. The degrees of freedom are an important component of the t-statistic formula, as they affect the shape and spread of the t-distribution and, consequently, the p-value and decision about the null hypothesis.
• Analyze how the t-statistic formula and its associated p-value are used to make a decision about the null hypothesis in the context of hypothesis testing.
  • The t-statistic formula provides a standardized measure of the difference between the sample statistic and the hypothesized population parameter, taking into account the uncertainty in the estimate. The p-value associated with the t-statistic represents the probability of observing a test statistic at least as extreme as the one calculated, assuming the null hypothesis is true. By comparing the p-value to the chosen significance level, researchers can make a decision about whether to reject or fail to reject the null hypothesis. A small p-value (typically less than the significance level) provides strong evidence against the null hypothesis, leading to its rejection, while a large p-value suggests the observed difference is likely due to chance and the null hypothesis cannot be rejected.

Related terms

Hypothesis Testing: The process of using sample data to determine whether to reject or fail to reject a null hypothesis about a population parameter.

Standard Error: The standard deviation of the sampling distribution of a statistic, which is used to quantify the amount of uncertainty in the estimate of a population parameter.

Degrees of Freedom: The number of values in the final calculation of a statistic that are free to vary, which is used to determine the appropriate t-distribution for hypothesis testing.