Questions: What sort of statistical analysis would be used to measure how well observed frequencies match the expected frequencies? a) MANOVA b) ANOVA c) Chi-Square d) t-test e) Regression

Solution Steps

The Chi-Square Test Statistic (\(\chi^2\)) is calculated using the formula:

\[ \chi^2 = \sum_i \frac{(O_i - E_i)^2}{E_i} \]

where \(O_i\) represents the observed frequencies and \(E_i\) represents the expected frequencies. For our data, we have:

• Observed frequencies: \(O = [50, 30, 20]\)
• Expected frequencies: \(E = [40, 40, 20]\)

Calculating the statistic gives us:

\[ \chi^2 = 5.0 \]

The degrees of freedom (\(df\)) for the Chi-Square test is calculated as:

\[ df = k - 1 \]

where \(k\) is the number of categories. In this case, we have 3 categories, so:

\[ df = 3 - 1 = 2 \]

The critical value for a Chi-Square test at a significance level of \(\alpha = 0.05\) with \(df = 2\) is obtained from the Chi-Square distribution table:

\[ \chi^2(0.95, 2) = 5.9915 \]

The P-Value is calculated as:

\[ P = P(\chi^2 > 5.0) = 0.0821 \]

Final Answer