Chi-Squared Test Study Notes

Chi-Squared Test Study Notes

Null and Alternate Hypothesis

  • Understanding Statistical Significance:

    • A method to determine if there is a statistically significant difference between two treatments: use the Standard Error of the Mean.

    • Error Bars Overlap:

    • If error bars for two sets of data overlap, it indicates no statistical difference.

    • Observed differences are likely due to random chance.

    • Error Bars Do Not Overlap:

    • If they do not overlap, there is a statistically significant difference.

    • Direct Testing of Hypothesis:

    • In some cases, hypotheses can be directly tested using statistical tests.

    • The Chi-Squared Test is emphasized as the only statistical test to be used in this course.

Formulating Hypotheses

  • Begin by crafting a hypothesis, which you will test for statistical significance.

    • Two types of hypotheses for a given phenomenon:

    • Null Hypothesis (H0):

      • States there is no relationship between two variables.

      • Example: "Cheese kept at room temperature has the same amount of mold as cheese in a refrigerator for a week."

    • Alternative Hypothesis:

      • States there is a relationship between two variables and the finding did not occur by chance.

      • Example: "Cheese kept at room temperature has a different amount of mold than cheese kept in a refrigerator for a week."

Chi-Squared Test Overview

  • Definition: The Chi-Squared Test assesses whether the differences between observed and expected results are statistically significant.

  • Function: It helps determine if the observed differences arise from random chance or another factor.

  • Method:

    • Propose a hypothesis.

    • Determine the probability that the hypothesis is wrong, providing a probability that the alternative hypothesis is correct.

Choosing a Hypothesis

  • The hypothesis should generate expected results for comparison against actual observed results.

    • Example: Hypothesis about a die roll states each number is equally likely (1/6 probability).

    • A comparison of actual frequencies against expected frequencies of 1/6 can lend or reduce support for the hypothesis.

    • If you assume unequal frequencies, it becomes untestable without specific expected values.

Chi-Squared Formula

  • Formula: χ^2 = Σ \frac{(o - e)^2}{e}

    • Where:

    • χ^2 = Chi-Squared statistic

    • Σ =