T-distribution

Definition

The t-distribution is a probability distribution used when the population standard deviation is unknown and the sample size is small. It is a bell-shaped, symmetric curve that is similar to the normal distribution but has heavier tails, accounting for the additional uncertainty introduced by estimating the population standard deviation from a small sample.

5 Must Know Facts For Your Next Test

1. The t-distribution is used when the population standard deviation is unknown, which is common in real-world situations.
2. As the sample size increases, the t-distribution approaches the standard normal distribution, reflecting greater confidence in the estimate of the population standard deviation.
3. The degrees of freedom for the t-distribution are equal to the sample size minus 1, reflecting the loss of one degree of freedom from estimating the population standard deviation.
4. The t-distribution has heavier tails than the normal distribution, meaning it assigns more probability to values further from the mean, accounting for the additional uncertainty in the estimate of the population standard deviation.
5. The t-distribution is used in a variety of statistical tests, including confidence intervals for a single population mean, comparing two independent population means, and testing the significance of a correlation coefficient.

Review Questions

• Explain why the t-distribution is used when the population standard deviation is unknown and the sample size is small.
  • When the population standard deviation is unknown, the standard normal distribution cannot be used to construct confidence intervals or conduct hypothesis tests, as it requires the population standard deviation to be known. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from a small sample, resulting in a distribution with heavier tails than the normal distribution. This allows for more accurate inferences to be made when the population standard deviation is unknown, particularly for small sample sizes.
• Describe how the t-distribution is used in the context of comparing two independent population means.
  • In the case of comparing two independent population means, the t-distribution is used to construct a confidence interval for the difference between the means or to conduct a hypothesis test. The t-statistic is calculated using the sample means, sample sizes, and an estimate of the pooled standard deviation. The degrees of freedom for the t-distribution are determined by the total sample size minus 2. The t-distribution accounts for the uncertainty in estimating the population standard deviations, allowing for more accurate inferences about the difference between the population means.
• Analyze how the t-distribution is used to test the significance of a correlation coefficient and explain the role of the degrees of freedom in this context.
  • When testing the significance of a correlation coefficient, the t-distribution is used to construct a test statistic that follows a t-distribution under the null hypothesis of no correlation. The degrees of freedom for this test are equal to the sample size minus 2, reflecting the loss of two degrees of freedom from estimating both the correlation coefficient and the population means. The t-statistic is calculated using the correlation coefficient and the sample size, and the resulting p-value is used to determine the statistical significance of the observed correlation. The t-distribution accounts for the uncertainty in estimating the population parameters, allowing for a more accurate assessment of the strength of the relationship between the variables.

Related terms

Confidence Interval: A range of values that is likely to contain an unknown population parameter, calculated from a sample statistic and used to quantify the uncertainty in the estimate.

Degrees of Freedom: The number of values in the final calculation of a statistic that are free to vary, calculated as the sample size minus the number of parameters estimated from the sample.

Standard Error: The standard deviation of the sampling distribution of a statistic, which quantifies the variability in the statistic due to sampling from the population.