Let's Suppose That You Work In The Quality Control Departmen

Lets Suppose That You Work In The Quality Control Department For The

Suppose you work in the quality control department for the M&M’s/Mars Company. You have been tasked with reviewing product consistency regarding the packaging of plain M&M’s candy. Your job is to determine if the number of blue, green, brown, orange, and yellow M&M’s in each bag is relatively equal. Not every bag will contain the exact same number of M&M’s, divided into equal quantities. However, as part of your quality control responsibilities, you want to determine if there is a significant difference in the number of colors in each bag.

Using a one-variable chi-square test, you aim to analyze whether the number of each color (blue, green, brown, orange, and yellow) differs significantly from what would be expected if the colors were evenly distributed. Your null hypothesis states that there is no significant difference between the number of each color in the bags, implying an even distribution. The alternative hypothesis asserts that a significant difference exists, indicating some colors are over- or under-represented.

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To conduct this analysis, the chi-square goodness-of-fit test is an appropriate statistical method. This test assesses whether observed frequencies of categorical data differ significantly from expected frequencies under a specified hypothesis—in this case, equal distribution among the five colors in each bag.

The first step involves collecting data. This requires sampling a sufficient number of M&M’s bags and recording the number of each color in each bag. For accurate analysis, the sample size should be large enough to meet the assumptions of the chi-square test, typically more than 20 observations. Data collection must be systematic and randomized to ensure representativeness of the sample to the entire production line.

Next, the observed frequencies for each color are tabulated. For example, if analyzing 100 bags with a total of 5,000 M&M’s, the observed counts per color are summed across all bags. The observed counts of blue, green, brown, orange, and yellow M&M’s are then compared to the expected counts—assuming equal distribution. If the total M&M count is 5,000, then each color's expected frequency, under the null hypothesis, would be 1,000 (i.e., 5,000 total divided evenly among 5 colors).

Once the data is tabulated, the chi-square statistic is calculated using the formula:

χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ]

Where Oᵢ is the observed frequency for each color, and Eᵢ is the expected frequency. This sum runs over all categories (colors). The calculated χ² value is then compared against the critical value from the chi-square distribution table, based on the degrees of freedom (number of categories minus one, which is 4 in this case) and the chosen significance level (commonly 0.05).

If the calculated χ² exceeds the critical value, the null hypothesis is rejected, indicating that the distribution of colors is significantly different, and some colors are over- or under-represented. If the χ² value is less than the critical value, we fail to reject the null hypothesis, suggesting that the observed counts are consistent with an even distribution among colors.

In conclusion, data collection, defining expected frequencies, calculating the chi-square statistic, and comparing it to a critical value provides a structured approach to assessing whether the color distribution of M&M’s in the bags is uniform or significantly skewed. This analysis helps ensure product consistency and quality control in packaging, subsequently maintaining customer satisfaction and brand integrity.

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