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In a Chi-squared test for goodness of fit, expected frequency refers to the theoretical count of occurrences for each category in your sample. These frequencies are derived from the known or hypothesized proportions. To clarify, let's consider the candy color exercise. Here, the expected frequencies are calculated using the percentages provided by the manufacturer. For instance, if brown candies are expected to make up 12% of a total of 72 candies, the expected frequency for brown is calculated as: \[ 0.12 \times 72 = 8.64 \] This mathematical step is repeated for each color based on their specified percentages. These computed expected frequencies allow us to later compare against the observed frequencies in our statistical test.

Observed frequency represents the actual number of times a particular outcome occurs in a data set. In our exercise with M&M's, the observed frequency refers to the real count of each candy color seen in the sample bag. For example, the observed frequency of brown candies was 14. Understanding observed frequencies is critical because they are directly compared with expected frequencies. This comparison helps to determine if there is a significant difference between what we see in reality versus what was expected. Observed frequencies are the backbone of the Chi-squared test, as they provide the empirical evidence needed to evaluate hypotheses.

Degrees of freedom are a concept that helps determine the flexibility of a statistical test. In the context of the Chi-squared test for goodness of fit, degrees of freedom indicate the number of values that can vary in your final calculation once constraints are applied. The formula used is: \[ k - 1 \] where \( k \) is the total number of categories. Applied to our problem, we have six color categories: brown, yellow, red, blue, orange, and green. Hence: \[ 6 - 1 = 5 \] This result means our test will have 5 degrees of freedom. It's crucial because it influences the critical value we refer to when determining whether to reject or accept the null hypothesis.

The significance level, often denoted as \( \alpha \), is a threshold for determining the strength of evidence against the null hypothesis in a hypothesis test. In this exercise, we use a significance level of 0.05, which is a common choice in many statistical tests. This means there is a 5% risk of concluding that a difference exists when there is none (Type I error). The chosen significance level directly impacts the critical value obtained from Chi-squared distribution tables. If the calculated Chi-squared statistic exceeds the critical value at the given significance level, we reject the null hypothesis. Here, since our calculated \( \chi^2 \) value of 11.389 is greater than the critical value of 11.07 at 5% significance, it indicates a statistically significant difference from the expected candy distributions.