M&M Colors. Observing that the proportion of I3.29 Mam...

M\&M Colors. Observing that the proportion of I3.29 Mam Colors. blue M\&Ms in his bowl of candy appeared to be less than that of the other colors, R. Fricker, Jr., decided to compare the color distribution in randomly chosen bags of M\&Ms to the theoretical distribution reported by M\&M/MARS consumer affairs. Fricker published his findings in the article "The Mysterious Case of the Blue M\&Ms" (Chance, Vol. 9(4), pp. 19-22). The following is the theoretical distribution. $$\begin{array}{l|c} \hline \text { Color } & \text { Percentage } \\ \hline \text { Brown } & 30 \\ \text { Yellow } & 20 \\ \text { Red } & 20 \\ \text { Orange } & 10 \\ \text { Green } & 10 \\ \text { Blue } & 10 \\ \hline \end{array}$$ For his study, Fricker bought three bags of M\&Ms from local stores and counted the number of each color. The average number of each color in the three bags was distributed as follows.

M\&M Colors. Observing that the proportion of I3.29 Mam Colors. blue M\&Ms in his bowl of candy appeared to be less than that of the other colors, R. Fricker, Jr., decided to compare the color distribution in randomly chosen bags of M\&Ms to the theoretical distribution reported by M\&M/MARS consumer affairs. Fricker published his findings in the article "The Mysterious Case of the Blue M\&Ms" (Chance, Vol. 9(4), pp. 19-22). The following is the theoretical distribution.
$$\begin{array}{l|c}
\hline \text { Color } & \text { Percentage } \\
\hline \text { Brown } & 30 \\
\text { Yellow } & 20 \\
\text { Red } & 20 \\
\text { Orange } & 10 \\
\text { Green } & 10 \\
\text { Blue } & 10 \\
\hline
\end{array}$$
For his study, Fricker bought three bags of M\&Ms from local stores and counted the number of each color. The average number of each color in the three bags was distributed as follows.

Key Concepts

Chi-Square Goodness-of-Fit Test

A chi-square goodness-of-fit test is a statistical method used to determine if observed data differ significantly from a theoretical or expected distribution. It involves computing a chi-square statistic by comparing observed frequencies with expected frequencies and then determining whether the differences can be attributed to chance or indicate a significant discrepancy.

Theoretical Distribution

A theoretical distribution represents a model or expected pattern of data based on known or assumed probabilities. In statistical testing, this distribution serves as a benchmark against which observed data are compared. Its parameters are often derived from previous studies, known probabilities, or assumptions about the process generating the data.

Observed Frequencies

Observed frequencies are the actual counts or proportions of events or outcomes collected from experiments or natural observations. These values are used in statistical tests to assess whether the experimental or survey outcomes are consistent with expected results, revealing potential anomalies or trends in the data.

Expected Frequencies

Expected frequencies are the counts or proportions predicted by a theoretical distribution or model, under the assumption that the observed data follow the proposed probabilities. They serve as a standard of comparison in hypothesis testing, helping to identify when the actual data diverge significantly from what was anticipated.

Random Sampling

Random sampling is a method used to select a subset of individuals or items from a larger population, ensuring that every member of the population has an equal chance of being chosen. This technique is critical in statistical analysis as it helps to obtain a representative sample, reduce bias, and allow the results to be generalized to the broader population.

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