module 3 stat part 2

Overview of the Student t Distribution

• The Student t distribution is foundational in statistics, particularly concerning the t test.

• Focus on understanding its shape, properties, and parameters.

Degrees of Freedom

• Definition: Degrees of freedom (df) indicate how many of the data points are flexible.

  • For continuous data: Example with 5 data points having a mean of 20.

    • 4 values can vary (indicated by vases), while the 5th is fixed to maintain the mean (e.g., 18).

    • Degrees of freedom in this case = 4.

  • For categorical data: Example of a 3x2 contingency table.

    • Knowing the column totals and row totals allows calculation of the free numbers.

    • Degrees of freedom here = 2.

    • Formula: (Number of Rows - 1) × (Number of Columns - 1).

T Test as a Parametric Test

• The t test is assumed to be valid when data follows a normal distribution, particularly with large samples (n > 30).

• Smaller sample sizes strictly follow a t distribution.

The Shape of the T Distribution

• Compared to the normal distribution, the t distribution displays fatter tails:

  • Likelihood of extreme values (e.g., -3, -4) is higher in a t distribution.

• Degrees of Freedom Effect:

  • The heaviness of the tails is influenced by degrees of freedom (df).

  • For the one sample t test, df = sample size - 1.

  • Larger sample sizes = larger df = t distribution resembles the normal distribution

  • After df=30, t distribution approximates the normal distribution closely.

Visual Representation of Distributions

• T Distribution with df = 3: Noticeable differences in tails compared to normal distribution.

• T Distribution with df = 10: Similar shape to normal distribution.

• T Distribution with df = 30: Almost identical to the normal distribution, affirming the assumption that normal distribution can be used for large sample sizes.

Summary of Key Points

• Defined degrees of freedom in both continuous and categorical contexts.

• Analyzed properties of the t distribution, its relationship to degrees of freedom, and implications for statistical analysis.

• Next topic entails practical analysis comparing means of two groups or datasets.