Chi-Squared Test

Introduction

• The chi-squared test is a method used in hypothesis testing where observed frequencies are compared with expected frequencies for experimental outcomes.

Hypothesis Testing

• Involves using sample data to draw conclusions about a population parameter or probability distribution.
• Null Hypothesis (H0): A tentative assumption about the parameter or distribution.
• Alternative Hypothesis (Ha): Opposite of what is stated in the null hypothesis.
• Procedure determines whether H0 can be rejected; if rejected, Ha is considered true.

Chi-Squared Test

• A hypothesis test where one selects a p-value to measure likelihood of sample results falling in a predicted range assuming H0 is true.
• p-value: Smaller value indicates less likelihood of sample results falling within a predicted range.
• Chi-Squared Formula: [ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} ]
  • (O_i): Observed frequency
  • (E_i): Expected frequency

Types of Chi-Squared Tests

1. Goodness of Fit Test

• Determines if a single variable value is within a given distribution.
• Example: Testing if the volume of soda cans falls within an acceptable range.

2. Test of Independence

• Evaluates if two variables could be related.
• Example: Correlation between book choices and seasonal changes.

Degrees of Freedom

• Number of independent quantities needed to express values of all variable properties of a system.
• In statistics, degrees of freedom equate to ( n - m ), where ( n ) is the number of variables and ( m ) is the number of constraints.
• Example: A simple pendulum has one degree of freedom.

Conclusion

• Chi-squared tests are crucial in hypothesis testing, offering a method to statistically evaluate assumptions about data distributions and relationships between variables.