Posted inBlog

Testing the Goodness of Fit

Testing the goodness of fit is a statistical procedure used to determine how well an observed set of data fits a theoretical or expected distribution.

Buy MCO-022 Pdf

Get the full solved assignment PDF of MCO-022 of 2023-24 session now.

This type of test assesses whether there are significant differences between the observed data and what would be expected under a certain theoretical model or hypothesis.

Chi-Square Test for Goodness of Fit:
One commonly used method for testing goodness of fit is the chi-square test. The chi-square ((\chi^2)) statistic is calculated by comparing the observed frequencies in each category or interval with the expected frequencies under the theoretical distribution.

The formula for the chi-square statistic is:

[ \chi^2 = \sum \frac{{(O_i – E_i)^2}}{{E_i}} ]

Where:

  • (O_i) is the observed frequency in category (i).
  • (E_i) is the expected frequency in category (i).
  • The sum is taken over all categories.

The chi-square statistic follows a chi-square distribution with degrees of freedom equal to the number of categories minus 1.

Steps for Chi-Square Goodness of Fit Test:

  1. Formulate Hypotheses:
  • Null Hypothesis ((H_0)): The observed data follows the expected distribution.
  • Alternative Hypothesis ((H_a)): The observed data differs significantly from the expected distribution.
  1. Determine Expected Frequencies:
  • Calculate the expected frequencies under the assumed distribution.
  1. Calculate Chi-Square Statistic:
  • Use the formula to compute the chi-square statistic.
  1. Compare with Critical Value or P-Value:
  • Determine the critical value from the chi-square distribution table or calculate the p-value associated with the chi-square statistic.
  1. Make a Decision:
  • If the p-value is less than the significance level ((\alpha)), reject the null hypothesis. If the chi-square statistic is greater than the critical value, reject the null hypothesis.

Example:
Suppose you have observed frequencies of different eye colors in a sample and want to test whether these frequencies match the expected distribution in the population.

  1. Formulate Hypotheses:
  • (H_0): The observed distribution matches the expected distribution.
  • (H_a): The observed distribution differs from the expected distribution.
  1. Determine Expected Frequencies:
  • Use population percentages for each eye color to calculate expected frequencies.
  1. Calculate Chi-Square Statistic:
  • Use the formula to calculate the chi-square statistic.
  1. Compare with Critical Value or P-Value:
  • Compare the chi-square statistic to the critical value or calculate the p-value.
  1. Make a Decision:
  • If the p-value is small (typically less than 0.05), or if the chi-square statistic is greater than the critical value, reject the null hypothesis.

The chi-square test for goodness of fit is widely used in various fields, including biology, psychology, and quality control, to assess whether observed data conforms to expected distributions.