T-distribution

Definition

The t-distribution, also known as the Student's t-distribution, is a probability distribution used to make statistical inferences about the mean of a population when the sample size is small and the population standard deviation is unknown. It is a bell-shaped, symmetric distribution that is similar to the normal distribution but has heavier tails, accounting for the increased uncertainty associated with small sample sizes.

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. The t-distribution has a heavier tail than the normal distribution, which means it assigns more probability to values further from the mean, reflecting the increased uncertainty associated with small sample sizes.
3. The degrees of freedom for the t-distribution are equal to the sample size minus 1, and the t-distribution approaches the standard normal distribution as the degrees of freedom increase.
4. The t-distribution is used in the calculation of confidence intervals for a single population mean and in hypothesis testing for a single population mean.
5. The t-distribution is also used in the analysis of two population means with unknown standard deviations, as well as in the testing of the significance of correlation coefficients.

Review Questions

• Explain how the t-distribution is used in the context of a single population mean with an unknown standard deviation.
  • When the population standard deviation is unknown and the sample size is small (typically less than 30), the t-distribution is used to make inferences about the population mean. This is done by calculating a t-statistic, which takes into account the sample mean, sample size, and sample standard deviation. The t-statistic is then compared to critical values from the t-distribution, which depend on the chosen significance level and the degrees of freedom (sample size minus 1). This allows for the construction of confidence intervals and the testing of hypotheses about the population mean.
• Describe how the t-distribution is used in the context of confidence intervals for home costs and women's heights.
  • The t-distribution is used to construct confidence intervals for the mean of a population when the population standard deviation is unknown and the sample size is small. For example, in the case of home costs, a random sample of home prices is taken, and the sample mean and sample standard deviation are calculated. The t-distribution is then used to determine the appropriate margin of error, which is added and subtracted from the sample mean to create a confidence interval for the true population mean of home costs. Similarly, the t-distribution is used to construct confidence intervals for the mean height of women, where a random sample of women's heights is taken, and the t-distribution is used to account for the unknown population standard deviation.
• Explain how the t-distribution is used in the context of hypothesis testing, including the concepts of null and alternative hypotheses, and the distribution needed for testing.
  • The t-distribution is a key component of hypothesis testing, particularly when the population standard deviation is unknown and the sample size is small. In this context, the t-distribution is used to determine the appropriate test statistic and critical values for evaluating the null and alternative hypotheses. The null hypothesis typically states that the population mean is equal to a specified value, while the alternative hypothesis states that the population mean is different from the specified value. The test statistic, which follows a t-distribution, is calculated using the sample mean, sample size, and sample standard deviation. This test statistic is then compared to critical values from the t-distribution to determine whether the null hypothesis should be rejected in favor of the alternative hypothesis, based on the chosen significance level.

Related terms

Degrees of Freedom:

The number of values in the final calculation of a statistic 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 range of values, calculated from a sample, that is likely to contain an unknown population parameter, such as the mean, with a specified level of confidence.

Hypothesis Testing:

The use of statistical methods to determine whether a claim or hypothesis about a population parameter is supported by the sample data.