T-distribution

Definition

The t-distribution, also known as the Student's t-distribution, is a probability distribution used in statistical inference when the sample size is small, and the population standard deviation is unknown. It is a symmetric, bell-shaped curve that is similar to the normal distribution but has heavier tails, allowing for greater variability in the data.

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 (z-distribution).
3. The shape of the t-distribution depends on the degrees of freedom, which are equal to the sample size minus 1.
4. The t-distribution has heavier tails compared to the standard normal distribution, meaning it assigns more probability to values further from the mean.
5. The t-distribution is used to construct confidence intervals and perform hypothesis tests when the population standard deviation is unknown.

Review Questions

• Explain the purpose and application of the t-distribution in statistical analysis.
  • The t-distribution is used in statistical analysis when the population standard deviation is unknown, and the sample size is small (typically less than 30). In this scenario, the t-distribution is more appropriate than the standard normal distribution (z-distribution) for making inferences about the population mean, constructing confidence intervals, and performing hypothesis tests. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample data, and its heavier tails allow for greater variability in the data compared to the normal distribution.
• Describe the relationship between the t-distribution and the standard normal distribution as the sample size increases.
  • As the sample size increases, the t-distribution approaches the standard normal distribution (z-distribution). This is because with a larger sample size, the sample standard deviation becomes a more reliable estimate of the population standard deviation, and the additional uncertainty introduced by estimating the standard deviation becomes less significant. Specifically, when the sample size is large (typically greater than 30), the t-distribution and the standard normal distribution become virtually indistinguishable, and the z-distribution can be used instead of the t-distribution for statistical inference.
• Analyze the impact of the degrees of freedom on the shape and properties of the t-distribution.
  • The degrees of freedom, which are equal to the sample size minus 1, have a significant impact on the shape and properties of the t-distribution. As the degrees of freedom increase, the t-distribution becomes more symmetric and approaches the standard normal distribution. With fewer degrees of freedom, the t-distribution has heavier tails, meaning it assigns more probability to values further from the mean. This reflects the greater uncertainty in the estimate of the population standard deviation when the sample size is smaller. The degrees of freedom are a crucial factor in determining the appropriate t-statistic and the corresponding p-values or confidence intervals when using the t-distribution for statistical inference.

Related terms

Standard Normal Distribution: The standard normal distribution, also known as the z-distribution, is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is used to standardize data and make inferences about population parameters.

Degrees of Freedom:

Degrees of freedom refer to the number of values in a statistical calculation that are free to vary. In the context of the t-distribution, the degrees of freedom are equal to the sample size minus 1.

Confidence Interval:

A confidence interval is a range of values that is likely to contain an unknown population parameter, such as the mean or proportion, with a specified level of confidence. The t-distribution is used to construct confidence intervals when the population standard deviation is unknown.