Overview
This lecture explains how to use StatKey software to calculate critical values for various statistical distributions, primarily for constructing confidence intervals.
Using StatKey for Critical Values
- StatKey is a web-based program for calculating critical values and exploring theoretical distributions.
- Traditional critical values are found in tables, but StatKey offers a technological alternative.
- Theoretical distributions of interest include the normal (z-scores), t-distribution (t-scores), chi-squared, and F-distribution.
Normal Distribution (Z-Scores)
- Use the "Normal" setting in StatKey with mean = 0 and standard deviation = 1 for z-score critical values.
- Common confidence levels and their z* critical values:
- 90%: ±1.645
- 95%: ±1.96
- 99%: ±2.576
- Higher confidence levels require larger critical values, increasing the margin of error.
T-Distribution (T-Scores)
- T-scores are used for mean (average) confidence intervals and depend on sample size (degrees of freedom = n-1 for one sample).
- Each sample size has a unique t* critical value.
- Examples for degrees of freedom (df) = 39:
- 90%: ±1.685
- 95%: ±2.023
- 99%: ±2.709
- Smaller sample sizes yield larger t* values, reflecting increased uncertainty.
Chi-Squared Distribution
- Chi-squared distribution is used for variance or standard deviation confidence intervals and categorical association tests.
- Degrees of freedom for chi-squared depend on context (e.g., n-1 for variance, k-1 for goodness-of-fit).
- The chi-squared curve is right-skewed, especially for small degrees of freedom.
Key Terms & Definitions
- Critical value — A threshold value that defines the boundaries of a confidence interval.
- Confidence interval — An interval estimate likely to include the population parameter, calculated using critical values.
- Z-score — Standardized value for normal distributions, used primarily for proportions.
- T-score — Used for means, adjusts for sample size via degrees of freedom.
- Degrees of freedom — Number of independent values in a calculation (often n-1 for samples).
- Chi-squared distribution — Used in categorical data and variance inference; not a normal distribution and is skewed right.
Action Items / Next Steps
- Practice using StatKey to find critical values for different distributions and sample sizes.
- Review the textbook sections on confidence intervals with z, t, and chi-squared distributions.
- Try calculating confidence intervals using both charts and StatKey to compare methods.