Use the chi-squared test to test the significance of differences between observed and expected results (the formula for the chi-squared test will be provided, as shown in the Mathematical requirements)

[latex]The chi-squared test is commonly used to determine if there is a significant difference between observed and expected frequencies in categorical data. The formula for the chi-squared statistic (chi^2) is given by:[/latex]

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

[latex]Where:[/latex]
[latex]O_i = text{observed frequency for category } i[/latex]
[latex]E_i = text{expected frequency for category } i[/latex]
[latex]text{The summation is over all categories.}[/latex]

[latex]textbf{Steps to Perform the Chi-Squared Test}[/latex]

[latex]1. text{State the Hypotheses:}[/latex]
[latex]- text{Null Hypothesis } (H_0): text{ There is no significant difference between observed and expected frequencies.}[/latex]
[latex]- text{Alternative Hypothesis } (H_a): text{ There is a significant difference between observed and expected frequencies.}[/latex]

[latex]2. text{Collect Data:}[/latex]
[latex]- text{Obtain the observed frequencies } (O) text{ for each category.}[/latex]
[latex]- text{Determine the expected frequencies } (E) text{ based on the theoretical distribution or proportions.}[/latex]

[latex]3. text{Calculate the Chi-Squared Statistic:}[/latex]
[latex]- text{Use the formula provided to compute } chi^2.[/latex]

[latex]4. text{Determine the Degrees of Freedom:}[/latex]
[latex]- text{Degrees of freedom } (df) text{ are calculated as:}[/latex]
[latex]df = k – 1[/latex]
[latex]text{where } k text{ is the number of categories.}[/latex]

[latex]5. text{Find the Critical Value:}[/latex]
[latex]- text{Using a chi-squared distribution table, find the critical value for } chi^2 text{ at a specific significance level (commonly } alpha = 0.05) text{ and the calculated degrees of freedom.}[/latex]

[latex]6. text{Make a Decision:}[/latex]
[latex]- text{If the calculated } chi^2 text{ is greater than the critical value from the table, reject the null hypothesis } (H_0). text{ Otherwise, fail to reject it.}[/latex]

[latex]textbf{Example}[/latex]

[latex]text{Let’s say we have the following observed and expected frequencies for a survey:}[/latex]

[latex]begin{array}{|c|c|c|}[/latex]
[latex]hline[/latex]
[latex] text{Category} & text{Observed (O)} & text{Expected (E)} \[/latex]
[latex]hline[/latex]
[latex] text{Category 1} & 30 & 25 \[/latex]
[latex] text{Category 2} & 15 & 20 \[/latex]
[latex] text{Category 3} & 25 & 30 \[/latex]
[latex]hline[/latex]
[latex]end{array}[/latex]

[latex]1. text{Calculate } chi^2:[/latex]
[latex]chi^2 = frac{(30-25)^2}{25} + frac{(15-20)^2}{20} + frac{(25-30)^2}{30}[/latex]

[latex]chi^2 = frac{25}{25} + frac{25}{20} + frac{25}{30} = 1 + 1.25 + 0.8333 approx 3.0833[/latex]

[latex]2. text{Degrees of Freedom:}[/latex]
[latex]df = 3 – 1 = 2[/latex]

[latex]3. text{Find the Critical Value:}[/latex]
[latex]- text{For } alpha = 0.05 text{ and } df = 2, text{ the critical value is approximately } 5.991.[/latex]

[latex]4. text{Decision:}[/latex]
[latex]- text{Since } 3.0833 < 5.991, text{ we fail to reject the null hypothesis } (H_0). text{ There is no significant difference between the observed and expected frequencies.}[/latex]