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)

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: chi^2 = sum frac{(O_i - E_i)^2}{E_i}

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

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

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

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

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

5. text{Find the Critical Value:}
- 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.}

6. text{Make a Decision:}
- 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.}

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. text{Calculate } chi^2:
chi^2 = frac{(30-25)^2}{25} + frac{(15-20)^2}{20} + frac{(25-30)^2}{30}

2. text{Degrees of Freedom:}
df = 3 - 1 = 2

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

4. text{Decision:}
- 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.}