Introduction

The Student’s t distribution, also known simply as the t distribution, is a fundamental probability distribution commonly used in statistics for hypothesis testing, confidence interval estimation, and sample size determination. It plays a crucial role when dealing with small sample sizes, where the underlying population distribution might not be normal. Understanding the t distribution is essential for making accurate inferences about population parameters based on limited sample data.

Properties and Characteristics

The t distribution is similar to the normal distribution in shape, but its tails are slightly thicker, especially for small sample sizes. It’s characterized by its degrees of freedom (df), which determine the shape of the distribution. The degrees of freedom depend on the sample size, specifically (n - 1), where n is the sample size. As the degrees of freedom increase, the t distribution approaches the standard normal distribution.

Formula and Probability Density Function (PDF)

The probability density function of the t distribution is given by:

$$f(t)=\frac{\Gamma\left(\frac{d f+1}{2}\right)}{\sqrt{d f \pi} \Gamma\left(\frac{d f}{2}\right)}\left(1+\frac{t^2}{d f}\right)^{-\frac{d f+1}{2}}$$

Here, $\Gamma$ represents the gamma function. The t distribution has heavier tails than the normal distribution, making it more suitable for small sample sizes or situations where the population standard deviation is unknown.

Applications and Examples

The t distribution finds its applications in various scenarios:

• Confidence Intervals: When the population standard deviation is unknown, the t distribution is used to construct confidence intervals for population parameters, such as the mean. For example, when estimating the average income based on a small sample, the t distribution helps provide a range within which the true mean is likely to fall.

• Hypothesis Testing: In hypothesis testing, the t distribution helps assess whether a sample mean is significantly different from a known or hypothesized value. This is particularly valuable when working with small samples, where assumptions of normality might not hold.

• Comparing Means: The t distribution is used to compare means of two independent samples. It helps determine whether the observed differences are statistically significant or just due to chance.

Example: Confidence Interval Calculation

Suppose a sample of 20 students was taken to estimate the average time spent studying per week. The sample mean is 15 hours, and the sample standard deviation is 3 hours. Using the t distribution with 19 degrees of freedom (20 - 1), we can calculate a 95% confidence interval for the population mean.