Sourav PanNovember 1, 2024
Answered
Thechi−squaredtestiscommonlyusedtodetermineifthereisasignificantdifferencebetweenobservedandexpectedfrequenciesincategoricaldata.Theformulaforthechi−squaredstatistic(χ2)isgivenby:
χ2=∑Ei(Oi−Ei)2
Where:
Oi=observed frequency for category i
Ei=expected frequency for category i
The summation is over all categories.
Steps to Perform the Chi-Squared Test
1.State the Hypotheses:
−Null Hypothesis (H0): There is no significant difference between observed and expected frequencies.
−Alternative Hypothesis (Ha): There is a significant difference between observed and expected frequencies.
2.Collect Data:
−Obtain the observed frequencies (O) for each category.
−Determine the expected frequencies (E) based on the theoretical distribution or proportions.
3.Calculate the Chi-Squared Statistic:
−Use the formula provided to compute χ2.
4.Determine the Degrees of Freedom:
−Degrees of freedom (df) are calculated as:
df=k−1
where k is the number of categories.
5.Find the Critical Value:
−Using a chi-squared distribution table, find the critical value for χ2 at a specific significance level (commonly α=0.05) and the calculated degrees of freedom.
6.Make a Decision:
−If the calculated χ2 is greater than the critical value from the table, reject the null hypothesis (H0). Otherwise, fail to reject it.
Example
Let’s say we have the following observed and expected frequencies for a survey:
\begin{array}{|c|c|c|}
\hline
\text{Category} & \text{Observed (O)} & \text{Expected (E)} \\
\hline
\text{Category 1} & 30 & 25 \\
\text{Category 2} & 15 & 20 \\
\text{Category 3} & 25 & 30 \\
\hline
\end{array}
1.Calculate χ2:
χ2=25(30−25)2+20(15−20)2+30(25−30)2
χ2=2525+2025+3025=1+1.25+0.8333≈3.0833
2.Degrees of Freedom:
df=3−1=2
3.Find the Critical Value:
−For α=0.05 and df=2, the critical value is approximately 5.991.
4.Decision:
−Since 3.0833<5.991, we fail to reject the null hypothesis (H0). There is no significant difference between the observed and expected frequencies.