T-distribution

Definition

The t-distribution is a continuous probability distribution that is used to make inferences about the mean of a population when the sample size is small and the population standard deviation is unknown. It is closely related to the normal distribution and is commonly used in statistical hypothesis testing and the construction of confidence intervals.

5 Must Know Facts For Your Next Test

1. The t-distribution is used when the population standard deviation is unknown, and the sample size is small (typically less than 30).
2. As the sample size increases, the t-distribution approaches the standard normal distribution.
3. The shape of the t-distribution depends on the degrees of freedom, which are determined by the sample size minus 1.
4. The t-distribution has heavier tails than the standard normal distribution, indicating a higher probability of observing extreme values.
5. The t-distribution is commonly used in hypothesis testing and the construction of confidence intervals for a single population mean.

Review Questions

• Explain how the t-distribution is used in the context of the Central Limit Theorem for Sample Means (Averages).
  • The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases. However, when the population standard deviation is unknown, the t-distribution is used instead of the standard normal distribution to make inferences about the population mean. The t-distribution accounts for the uncertainty in the estimate of the population standard deviation, which is particularly important when the sample size is small.
• Describe how the t-distribution is used to construct a confidence interval for a single population mean.
  • When the population standard deviation is unknown, the t-distribution is used to construct a confidence interval for a single population mean. The formula for the confidence interval is: $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom. The t-distribution accounts for the uncertainty in the estimate of the population standard deviation, which is necessary when the sample size is small.
• Analyze the role of the t-distribution in hypothesis testing for a single population mean and its relationship to rare events, the sample, decision, and conclusion.
  • In hypothesis testing for a single population mean, the t-distribution is used to determine the probability of observing the sample mean or a more extreme value, given that the null hypothesis is true. The t-statistic is calculated as $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size. The t-statistic follows a t-distribution with $n-1$ degrees of freedom. The p-value, which represents the probability of observing the sample result or a more extreme value under the null hypothesis, is then compared to the chosen significance level to determine whether to reject or fail to reject the null hypothesis. The t-distribution's role in this process is crucial for making valid statistical inferences, particularly when the population standard deviation is unknown and the sample size is small.

Related terms

Standard Normal Distribution: The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is used as a reference distribution in many statistical tests.

Degrees of Freedom: Degrees of freedom are the number of values in the final calculation of a statistic that are free to vary. They are an important parameter in the t-distribution and determine the shape of the distribution.

Confidence Interval: A confidence interval is a range of values that is likely to contain an unknown population parameter, such as the mean, with a specified level of confidence.